# Existence, uniqueness and characterisation of local minimisers in higher
order Calculus
of Variations in $\mathrm{L}^{\infty}$

Nikos Katzourakis  N.K., Department of Mathematics and Statistics, University
of Reading, Whiteknights Campus, Pepper Lane, Reading RG6 6AX, UNITED KINGDOM
<EMAIL_ADDRESS>and  Roger Moser  R.M., Department of Mathematical
Sciences, University of Bath, Bath BA2 7AY, UNITED KINGDOM<EMAIL_ADDRESS>
###### Abstract.

We study variational problems for second order supremal functionals
$\mathrm{F}_{\infty}(u)=\|F(\cdot,u,\mathrm{D}u,\mathrm{A}\\!:\\!\mathrm{D}^{2}u)\|_{\mathrm{L}^{\infty}(\Omega)}$,
where $F$ satisfies certain natural assumptions, $\mathrm{A}$ is a positive
matrix, and $\Omega\Subset\mathbb{R}^{n}$. Higher order problems are very
novel in the Calculus of Variations in $\mathrm{L}^{\infty}$, and exhibit a
strikingly different behaviour compared to first order problems, for which
there exists an established theory, pioneered by Aronsson in 1960s. The aim of
this paper is to develop a complete theory for $\mathrm{F}_{\infty}$. We prove
that, under appropriate conditions, “localised” minimisers can be
characterised as solutions to a nonlinear system of PDEs, which is different
from the corresponding Aronsson equation for $\mathrm{F}_{\infty}$; the latter
is only a necessary, but not a sufficient condition for minimality. We also
establish the existence and uniqueness of localised minimisers subject to
Dirichlet conditions on $\partial\Omega$, and also their partial regularity
outside a singular set of codimension one, which may be non-empty even if
$n=1$.

###### Key words and phrases:

Calculus of Variations in $\mathrm{L}^{\infty}$; higher order problems; local
minimisers; absolute minimisers; Euler-Lagrange equations; Aronsson equations.

###### 2020 Mathematics Subject Classification:

49K20; 35A15; 35B38; 35D99; 35J94; 49J27.

N.K. has been partially financially supported through the EPSRC grant
EP/X017206/1. R.M. has been partially financially supported through the EPSRC
grant EP/X017109/1.

###### Contents

  1. 1 Introduction and main results
  2. 2 Preliminaries and auxiliary results
    1. 2.1 Preliminaries and notational conventions
    2. 2.2 Discussion of assumptions and examples
      1. 2.2.1 The restricted convexity assumptions ()-(1.13)
      2. 2.2.2 The Whitney differentiability assumption (1.17)
      3. 2.2.3 Explicit classes of supremands
    3. 2.3 The functional space setup and duality
      1. 2.3.1 Completeness and a priori estimates
      2. 2.3.2 Identification of the predual of $\mathcal{W}^{2,\infty}_{0}(\Omega)$ and weak* approximation
    4. 2.4 Compact sets in $\mathcal{W}^{2,\infty}_{0}(\Omega)$ and examples
    5. 2.5 $\mathrm{L}^{p}$-limits of $\mathrm{L}^{\infty}$ constrained minimisers
    6. 2.6 A lemma of strong compactness in $\mathrm{L}^{1}$
    7. 2.7 Nodal sets and non-trivial solutions
  3. 3 Proofs of the main results

##  1\. Introduction and main results

Let $\Omega\Subset\mathbb{R}^{n}$ be a bounded open set with Lipschitz
boundary $\partial\Omega$, $n\in\mathbb{N}$. Suppose also that
$F:\Omega\times(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R})\longrightarrow\mathbb{R}$
is a given Carathéodory function, whose arguments will be denoted by
$(x,\eta,\mathrm{p},\xi)$. In this paper we are interested in studying
$\mathrm{L}^{\infty}$ variational problems for supremal functionals involving
higher order derivatives of admissible functions. Specifically, we will
consider the second order functional

(1.1) $\ \ \ \
\mathrm{F}_{\infty}(u):=\big{\|}F(\cdot,u,\mathrm{D}u,\mathrm{A}\\!:\\!\mathrm{D}^{2}u)\big{\|}_{\mathrm{L}^{\infty}(\Omega)},\
\ \ \ u\in\mathcal{W}^{2,\infty}(\Omega),$

where the class $\mathcal{W}^{2,\infty}(\Omega)$ of weakly twice
differentiable functions is defined as

(1.2)
$\mathcal{W}^{2,\infty}(\Omega)\,:=\,\bigcap_{1<p<\infty}\Big{\\{}u\in\mathrm{W}^{2,p}(\Omega)\
\big{|}\
\mathrm{A}\\!:\\!\mathrm{D}^{2}u\in\mathrm{L}^{\infty}(\Omega)\Big{\\}}.$

In (1.1)-(1.2), $\mathrm{A}\in\mathbb{R}^{n\times n}$ denotes a positive
symmetric matrix, and “$:$” denotes the Euclidean inner product in
$\mathbb{R}^{n\times n}$ (for details on all the symbolisations we employ in
this paper, we refer to Subsection 2.1). Variational problems for second (or
higher) order supremal functionals are very novel in the Calculus of
Variations in $\mathrm{L}^{\infty}$, and no general theory is yet available.
The purpose of the present paper is to develop a complete theory for the
general second order functional (1.1).

Since $\mathrm{F}_{\infty}$ is not Gateaux differentiable, general critical
points cannot be defined in the classical sense. The lack of smoothness is a
standard intricacy of all supremal functionals, and persists even if e.g. $F$
is $\mathrm{C}^{\infty}$ and we consider only the restriction
$\smash{\mathrm{F}_{\infty}|_{\mathrm{C}^{2}(\overline{\Omega})}}$: by
Danskin’s theorem (see [20]), the Gateaux derivative exists only when the
argmax set is a singleton. We therefore turn our attention to the existence,
uniqueness and regularity of appropriately defined minimisers, as well in
their variational characterisation through PDEs.

Before introducing our assumptions and results, and for the sake of
comparison, let us discuss briefly the more classical case of first order
$\mathrm{L}^{\infty}$ functionals, which presents some striking differences.
In this case, at least when restricted to scalar-valued admissible functions,
there exists a rather complete theory. Let
$\mathrm{H}:\Omega\times\mathbb{R}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}$
be given, and consider

(1.3) $\ \ \
\mathrm{E}_{\infty}(u,\mathcal{O}):=\big{\|}\mathrm{H}(\cdot,u,\mathrm{D}u)\big{\|}_{\mathrm{L}^{\infty}(\Omega)},\
\ \ \ u\in\mathrm{W}^{1,\infty}(\Omega),\ \mathcal{O}\Subset\Omega.$

Variational problems for (1.3) were pioneered by Aronsson in the 1960s, who is
considered the founder of the theory (see e.g. [3]-[5]). The trademark feature
of (1.3) is that minimisers of $\mathrm{E}_{\infty}(\cdot,\Omega)$ are in
general highly non-unique and may not satisfy any PDE, in the sense of some
analogue of the Euler-Lagrange equations of integral functionals. If however
one builds locality in the minimality notion, namely require _by definition_
that $\mathrm{E}_{\infty}(\cdot,\mathcal{O})$ is minimised for all
$\mathcal{O}\Subset\Omega$ with respect to the function’s own boundary values
on $\partial\mathcal{O}$, then any such function $u$, satisfies the _Aronsson
equation_ in the viscosity sense and is called an _absolute minimiser_ of
(1.3) (for a pedagogical introduction, we refer e.g. to [28]). The Aronsson
equation arising from (1.3) is the following quasilinear second order
degenerate elliptic PDE

(1.4)
$\mathrm{H}_{\mathrm{p}}(\cdot,u,\mathrm{D}u)\mathrm{D}\big{(}\mathrm{H}(\cdot,u,\mathrm{D}u)\big{)}=0,\
\ \text{ in }\Omega.$

It turns out that the opposite is true as well, and the PDE (1.4) is
sufficient for absolute minimality, whilst the corresponding Dirichlet problem
has a unique solution. All these deep results require appropriate assumptions
to be imposed, typically involving convexity of $\mathrm{H}$ in $\mathrm{P}$,
and no explicit dependence in $u$. Without any attempt to be exhaustive, for
some important contributions in the area, we refer e.g. to [2, 8, 12, 13, 14,
27, 38, 40, 45, 46, 47].

The phenomena arising in the study of higher order supremal functionals are
radically different from those governing the first order case of (1.3)-(1.4).
Firstly, at least for the case of (1.1) where the dependence on second
derivatives is through the elliptic operator
$u\mapsto\mathrm{A}\\!:\\!\mathrm{D}^{2}u$, absolute minimisers are not a
fundamental notion, as _usual (global) minimisers of ( 1.1) satisfy astounding
uniqueness properties_, even with $u$-dependence being allowed and under very
weak convexity requirements. In fact, one can take this one step further, and
show that even _localised minimisers_ , namely those which minimise only in an
appropriate neighbourhood of $\mathcal{W}^{2,\infty}(\Omega)$, are also
locally unique. On the other hand, the corresponding higher order Aronsson
equation for (1.1), although it can be easily derived, it plays only an
ancillary role in this setting, as it _is a necessary but not a sufficient
condition for minimality_ , even if we restrict our attention to classical
solutions. Additionally, it is fully nonlinear and of third order, failing to
be elliptic in any reasonable sense. Furthermore, the very definition of
generalised solutions for the Aronsson equation is a non-trivial matter, as
neither weak nor viscosity solutions apply.

In this paper, following an entirely different approach, _we identify a
different set of equations which completely characterises minimisers of (
1.1), even localised ones, and it allows to ascertain their uniqueness for
given prescribed Dirichlet boundary conditions_, under natural assumptions. We
may also include explicit $(u,\mathrm{D}u)$-dependence, a fact which is
interesting in itself, as $u$-dependence typically precludes uniqueness
assertions in the first order case (see e.g. [52]). Our investigations are in
part inspired by curvature problems in Riemannian geometry, specifically the
Yamabe and Nirenberg problems, see [42, 49] and subsequently. However, our
results herein have a different focus and as they stand they do not directly
apply to differential geometry.

We now present the main contributions of this work, beginning with the minima-
lity notions. To state them concisely, let us define the following subspace of
(1.2):

(1.5)
$\mathcal{W}^{2,\infty}_{0}(\Omega)\,:=\,\mathcal{W}^{2,\infty}(\Omega)\cap\mathrm{W}^{2,2}_{0}(\Omega).$

Then, $\mathcal{W}^{2,\infty}_{0}(\Omega)$ becomes a dual Banach space with
separable predual when endowed with the norm (we postpone a thorough
discussion until Subsection 2.4):

$\|u\|_{\mathcal{W}^{2,\infty}_{0}(\Omega)}:=\,\|\mathrm{A}\\!:\\!\mathrm{D}^{2}u\|_{\mathrm{L}^{\infty}(\Omega)}.$

######  Definition 1 (Global, narrow, local minimisers).

Consider a function $u\in\mathcal{W}^{2,\infty}(\Omega)$.

(i) We say that $u$ is a _(global) minimiser_ of (1.1) when

$\ \mathrm{F}_{\infty}(u)\leq\mathrm{F}_{\infty}(u+\psi),\ \ \ \ \text{for all
}\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega).$

(ii) We say that $u$ is a _narrow minimiser_ of (1.1) when there exists
$\rho>0$ such that

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\mathrm{F}_{\infty}(u)\leq\mathrm{F}_{\infty}(u+\psi),\ \ \ \ \text{for all
}\psi\in\mathbb{B}^{\mathrm{W}^{1,\infty}(\Omega)}_{\rho}\cap\mathcal{W}^{2,\infty}_{0}(\Omega).$

(iii) We say that $u$ is a _local minimiser_ of (1.1) when there exists
$\rho>0$ such that

$\ \mathrm{F}_{\infty}(u)\leq\mathrm{F}_{\infty}(u+\psi),\ \ \ \ \text{for all
}\psi\in\mathbb{B}^{\mathcal{W}^{2,\infty}_{0}(\Omega)}_{\rho}.$

(iv) If additionally the inequalities in either of (i)-(ii)-(iii) are strict
when $\psi\neq 0$, then $u$ will be called a _strict global/narrow/local
minimiser_ respectively.

One can define global/narrow/local minimisers of integral functionals in an
analogous fashion. Evidently, every global minimiser is narrow, and every
narrow minimiser is local, but in general neither of the converse implications
are true. The significance of the concept of narrow minimisers is justified by
the fact that they satisfy stronger existence, uniqueness and stability
properties than local minimisers. Narrow minimisers relate to the concept of
strong local minimisers (see e.g. [39]), but they are different. The space
$\mathrm{W}^{1,\infty}(\Omega)$ appearing in their definition is not
essential, and can be replaced by any Banach space
$(\mathfrak{X},|\|\cdot\||)$ into which $\mathcal{W}^{2,\infty}_{0}(\Omega)$
has a compact embedding (e.g.
$\mathrm{L}^{\infty}(\Omega),\mathrm{C}^{0,\gamma}(\overline{\Omega}),\mathrm{C}^{1,\gamma}(\overline{\Omega}),\mathrm{W}^{2-\varepsilon,p}(\Omega)$,
etc.), and analogous results are obtainable in this case too. Let us now state
our assumptions:

(1.6) $\left\\{\
\begin{split}&\text{$F:\Omega\times(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R})\longrightarrow\mathbb{R}$
is a Carath\'{e}odory function, satisfying }\\\ &\text{that
$F(x,\cdot)\in\mathrm{C}^{2}(\mathbb{R}\times\mathbb{R}^{n}\\!\times\mathbb{R})$,
a.e.\,\,$x\in\Omega$, and
$F(\cdot,\eta,\mathrm{p},\xi)\in\mathrm{L}^{\infty}(\Omega)$}\\\
&\text{uniformly when $(\eta,\mathrm{p},\xi)\in\mathcal{M}$, for any bounded
$\mathcal{M}\subseteq\mathbb{R}\times\mathbb{R}^{n}\\!\times\mathbb{R}$.}\end{split}\right.$
(1.7) $\left\\{\ \begin{split}\text{There exist
$C\in\mathrm{C}^{\nnearrow}[0,\infty)$ and $c>0$}&\text{ such that}\\\
\partial_{\xi}F(x,\eta,\mathrm{p},\xi)\geq\ &c,\phantom{{}^{\big{|}}}\\\
F(x,\eta,\mathrm{p},\cdot)^{-1}(\\{0\\})\subseteq\
&\big{[}\\!-\\!C\big{(}|\eta|+|\mathrm{p}|\big{)},C\big{(}|\eta|+|\mathrm{p}|\big{)}\big{]},\\\
\big{|}\partial_{*}F(x,\eta,\mathrm{p},\xi)\big{|}\,\leq\
&C\big{(}|\eta|+|\mathrm{p}|+|\xi|\big{)},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\
\big{|}\partial^{2}_{**}F(x,\eta,\mathrm{p},\xi)\big{|}\,\leq\
&C\big{(}|\eta|+|\mathrm{p}|+|\xi|\big{)},\phantom{{}_{\big{|}}}\\\ \text{for
a.e.\ $x\in\Omega$ and all $(\eta,\mathrm{p},\xi)\in$
}\mathbb{R}\times\,&\mathbb{R}^{n}\times\mathbb{R}.\end{split}\right.$

In (1.7) we used the notation $\mathrm{C}^{\nnearrow}[0,\infty)$ for the set
of continuous increasing functions on $[0,\infty)$, and also the following
compact symbolisations for the _partial_ gradient $\partial_{*}F$ and the
_partial_ Hessian $\partial^{2}_{**}F$ with respect to the variables
$(\eta,\mathrm{p},\xi)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}$:

$\partial_{*}F:=\big{(}0,\partial_{\eta}F,\partial_{\mathrm{p}}F,\partial_{\xi}F\big{)},\
\ \ \partial^{2}_{**}F:=\left[\begin{array}[]{c|ccc}0&0&0&0\\\ \hline\cr
0&\partial^{2}_{\eta\eta}F&\partial^{2}_{\eta\mathrm{p}}F&\partial^{2}_{\eta\xi}F\phantom{\Big{]}}\\\
0&\partial^{2}_{\mathrm{p}\eta}F&\partial^{2}_{\mathrm{p}\mathrm{p}}F&\partial^{2}_{\mathrm{p}\xi}F\phantom{\Big{]}}\\\
0&\partial^{2}_{\xi\eta}F&\partial^{2}_{\xi\mathrm{p}}F&\partial^{2}_{\xi\xi}F\phantom{\Big{]}}\end{array}\right].$

For notational convenience, let us also introduce the following
symbolisations:

$\left\\{\ \
\begin{split}\mathrm{J}^{2}u&:=\big{(}\cdot,u,\mathrm{D}u,\mathrm{A}\\!:\\!\mathrm{D}^{2}u\big{)}:\
\Omega\longrightarrow\Omega\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R},\\\
\mathrm{J}^{2}_{*}u&:=\big{(}u,\mathrm{D}u,\mathrm{A}\\!:\\!\mathrm{D}^{2}u\big{)}\
\
:\,\,\Omega\longrightarrow\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}.\end{split}\right.$

We will call $\mathrm{J}^{2}u$ the (second order) _Jet_ of $u$ and
$\mathrm{J}^{2}_{*}u$ the (second order) _reduced_ _Jet_ of $u$. In this
notation, the functional (1.1) can be compactly rewritten as

$\mathrm{F}_{\infty}(u)=\|F(\mathrm{J}^{2}u)\|_{\mathrm{L}^{\infty}(\Omega)}.$

Finally, let $\operatorname{sgn}:\mathbb{R}\longrightarrow[-1,1]$, be the sign
function, defined by $\operatorname{sgn}(t):={t}/{|t|}$ for $t\neq 0$ and
$\operatorname{sgn}(0):=0$. We may now state our first main result, which
identifies a system of equations as a sufficient condition for minimality.
Given $u_{\infty}\in\mathcal{W}^{2,\infty}(\Omega)$ and
$f_{\infty}\in\mathrm{L}^{1}(\Omega)$, we consider the differential equations

(1.8) $\displaystyle
F(\mathrm{J}^{2}u_{\infty})=\mathrm{F}_{\infty}(u_{\infty})\operatorname{sgn}(f_{\infty}),$
$\displaystyle\text{ a.e.\ on }\Omega,$ (1.9)
$\displaystyle\mathrm{div}\bigg{(}\\!\mathrm{A}\mathrm{D}f_{\infty}-\frac{\partial_{\mathrm{p}}F(\mathrm{J}^{2}u_{\infty})}{\partial_{\xi}F(\mathrm{J}^{2}u_{\infty})}f_{\infty}\\!\bigg{)}+\frac{\partial_{\eta}F(\mathrm{J}^{2}u_{\infty})}{\partial_{\xi}F(\mathrm{J}^{2}u_{\infty})}f_{\infty}=\,0,$
$\displaystyle\ \ \text{ in
}\big{(}\mathrm{C}^{\infty}_{c}(\Omega)\big{)}^{\\!*}.\\!\\!$

In the above strongly coupled system, (1.8) is a second order fully nonlinear
PDE satisfied in the strong a.e. sense by $u_{\infty}$, and (1.9) is a second
order linear divergence elliptic PDE for $f_{\infty}$, satisfied in the sense
of distributions. The function $f_{\infty}$ should be regarded as a
“parameter” corresponding to $u_{\infty}$ (this will be the minimiser).

######  Theorem 2 (Sufficiency of the equations).

Consider the functional (1.1), where the supremand $F$ is assumed to satisfy
(1.6)-(1.7). Let $u_{\infty}\in\mathcal{W}^{2,\infty}(\Omega)$ and
$f_{\infty}\in\mathrm{L}^{1}(\Omega)\setminus\\{0\\}$ be given functions which
satisfy the system of equations (1.8)-(1.9). Then, for any compact set
$\mathcal{K}\subseteq\mathbb{S}^{\mathcal{W}^{2,\infty}_{0}(\Omega)}$, there
exists $\rho>0$ such that

(1.10)
$\mathrm{F}_{\infty}(u_{\infty})<\mathrm{F}_{\infty}(u_{\infty}+t\psi),\ \ \ \
\ \psi\in\mathcal{K},\ 0<t<\rho.$

We note that the conclusion (1.10) is weaker than local minimisers (cf.
Definition 1(iii)), as it requires a compact set of directions of variations.
However, since $-\mathcal{K}$ is also compact if $\mathcal{K}$ is compact,
inequality (1.10) is in fact true for all $t\in(-\rho,\rho)\setminus\\{0\\}$
(perhaps for a smaller $\rho>0$). In order to obtain that $u_{\infty}$ is a
global/narrow/local minimiser, we need some stronger hypotheses on $F$, see
Theorem 3 that follows. Nevertheless, in Subsection 2.5, we provide a
necessary and sufficient condition for strong compactness in
$\mathcal{W}^{2,\infty}_{0}(\Omega)$, along the lines of the classical Ascoli-
Arzela and Frechét-Kolmogorov theorems. In Corollary 15 (Subsection 2.4) we
demonstrate particular interesting choices of compact sets of variations.

We now state our second main result which gives sufficient conditions for
$u_{\infty}$ to be a local/narrow/global minimiser for $\mathrm{F}_{\infty}$,
under additional assumptions on $F$.

######  Theorem 3 (Sufficiency of the equations and strictness of miminisers).

In the setting of Theorem 2 and under the same assumptions, let
$L,R\subseteq\mathbb{R}$ denote arbitrary open intervals and let
$M\subseteq\mathbb{R}^{n}$ denote an open bounded convex set, and consider the
following hypotheses:

(1.11) $\left\\{\ \ \ \ \ \begin{split}&\text{$\exists\ L,M,R$ such that
$\mathrm{ess}\big{(}\mathrm{J}^{2}_{*}u_{\infty}(\Omega)\big{)}\\!\subseteq
L\\!\times\\!M\\!\times\\!R$, and }\exists\\\ &\text{$\bar{p}\geq 2$:
$|F(x,\cdot)|^{\bar{p}}$ is convex on $L\\!\times\\!M\\!\times\\!R$, for a.e.\
$x\in\Omega$.}\end{split}\right.$ (1.12) $\left\\{\ \ \ \ \
\begin{split}&\text{$\exists\ L,M$ with
$\mathrm{ess}((u_{\infty},\mathrm{D}u_{\infty})(\Omega))\subseteq
L\\!\times\\!M$, and $\forall\,R\subseteq\mathbb{R}$,}\\\ &\text{$\exists\
\bar{p}\geq 2$: $|F(x,\cdot)|^{\bar{p}}$ is convex on
$L\\!\times\\!M\\!\times\\!R$, for a.e.\,$x\in\Omega$.}\end{split}\right.$
(1.13) $\forall\ L,M,R$, $\exists\ \bar{p}\geq 2$: $|F(x,\cdot)|^{\bar{p}}$ is
convex on $L\\!\times\\!M\\!\times\\!R$, for a.e. $x\in\Omega$.

Then, the following are true:

_(i)_ If (LABEL:1.13) is also satisfied, then $u_{\infty}$ is a strict local
minimiser of $\mathrm{F}_{\infty}$, and hence it is unique within the
neighbourhood of $\mathcal{W}^{2,\infty}_{u_{\infty}}(\Omega)$ that minimises.

_(ii)_ If (LABEL:1.14) is also satisfied, then $u_{\infty}$ is a strict narrow
minimiser of $\mathrm{F}_{\infty}$, and hence it is unique within the
neighbourhood of $\mathcal{W}^{2,\infty}_{u_{\infty}}(\Omega)$ that minimises.

_(iii)_ If (1.13) is also satisfied, then $u_{\infty}$ is a strict global
minimiser of $\mathrm{F}_{\infty}$, and hence it is globally unique in the
space $\mathcal{W}^{2,\infty}_{u_{\infty}}(\Omega)$.

The meaning of the inclusions in (LABEL:1.13) and (LABEL:1.14) are understood
in the sense of the essential range of measurable functions. In Subsection
2.2.1 we provide equivalent and sufficient conditions for the convexity of
$|F(x,\cdot)|^{\bar{p}}$ for any $\bar{p}\geq 2$ (see (2.1) and (2.3)). In
particular, this requirement does _not_ impose convexity on $|F(x,\cdot)|$,
but rather a semi-convexity type condition. In Subsection 2.2.3 we discuss
explicit examples of functions which satisfy our assumptions. For convenience,
we define the affine space counterpart of
$\mathcal{W}_{0}^{2,\infty}(\Omega)$, namely for a given
$u_{0}\in\mathcal{W}^{2,\infty}(\Omega)$, we set

(1.14)
$\mathcal{W}^{2,\infty}_{u_{0}}(\Omega)\,:=\,u_{0}+\mathcal{W}^{2,\infty}_{0}(\Omega).$

Now we consider the opposite problem regarding the necessity of equations
(1.8)-(1.9) for minimality, as well as the related problem of existence of
minimisers, given appropriate Dirichlet boundary conditions on
$\partial\Omega$. To this aim, we need to strengthen our assumptions on $F$.
In addition to (1.6)-(1.7), we will suppose that:

(1.15) $\left\\{\ \ \begin{split}&\text{Exist $c>0$ and $\alpha\in(0,1)$ such
that,}\\\ &\hskip
100.0pt\partial^{2}_{\xi\xi}F(x,\eta,\mathrm{p},\xi)\geq-\frac{1}{c},\\\
&\hskip
22.0ptF(x,\eta,\mathrm{p},\cdot)^{-1}(\\{0\\})\subseteq\bigg{[}\\!-\frac{(|\eta|+|\mathrm{p}|)^{\alpha}+1}{c},+\frac{(|\eta|+|\mathrm{p}|)^{\alpha}+1}{c}\bigg{]},\\\
&\text{for a.e.\ $x\in\Omega$ and all
$(\eta,\mathrm{p},\xi)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}$.}\end{split}\right.$

Also, that:

(1.16) $\left\\{\ \ \begin{split}&\text{Exists
$C\in\mathrm{C}^{\nnearrow}[0,\infty)$ such that,}\\\ &\hskip
50.0pt\big{|}\partial_{\xi}F(x,\eta,\mathrm{p},\xi)\big{|}\,\leq\,C\big{(}|\eta|+|\mathrm{p}|\big{)},\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\ &\hskip
50.0pt\big{|}\partial_{\mathrm{p}}F(x,\eta,\mathrm{p},\xi)\big{|}\,\leq\,C\big{(}|\eta|+|\mathrm{p}|\big{)}(1+|\xi|),\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\ &\hskip
50.0pt\big{|}\partial_{\eta}F(x,\eta,\mathrm{p},\xi)\big{|}\,\leq\,C\big{(}|\eta|+|\mathrm{p}|\big{)}(1+|\xi|),\\\
&\text{for a.e.\ $x\in\Omega$ and all
$(\eta,\mathrm{p},\xi)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}$.}\end{split}\right.$

Finally, we we will need to assume that the restriction
$u_{0}|_{\partial\Omega}$ of the boundary data determined by
$u_{0}\in\mathcal{W}^{2,\infty}_{0}(\Omega)$, as well as the boundary
$\partial\Omega$ itself, are more regular:

(1.17) The boundary $\partial\Omega$ is $\mathrm{C}^{2}$, and the restriction
$u_{0}|_{\partial\Omega}$ is $\mathrm{C}^{2}$ on $\partial\Omega$.

We will discuss the exact meaning of the regularity assumption (1.17) in
Subsection 2.2.2. Below is our third main result, which concerns the necessity
of equations (1.8)-(1.9) for minimality.

######  Theorem 4 (Necessity of the equations).

Suppose that $\\{F,\partial\Omega,u_{0}\\}$ satisfy (1.6)-(1.7), and also
(1.15)-(1.17). Let $u_{\infty}\in\mathcal{W}^{2,\infty}_{u_{0}}(\Omega)$ be a
narrow minimiser of (1.1). Then, there exists
$f_{\infty}\in\mathrm{L}^{1}(\Omega)\setminus\\{0\\}$ such that the pair of
functions $(u_{\infty},f_{\infty})$ satisfies equations (1.8)-(1.9), and
$\\{f_{\infty}=0\\}$ is countably $(n-1)$-rectifiable (and hence a Lebesgue
nullset). In addition, the equation (1.9) is satisfied weakly in
$(\mathcal{W}_{0}^{2,\infty}(\Omega))^{*}$, whilst
$f_{\infty}\in\mathrm{W}^{1,p}_{\mathrm{loc}}(\Omega)$ for all
$p\in(1,\infty)$, and the set $\\{f_{\infty}=0\\}$ is closed.

By combining Theorems 3 and 4, we have the following important uniqueness
assertion for narrow and global minimisers (Definition 1).

######  Theorem 5 (Uniqueness of narrow and global minimisers).

 Suppose that $\\{F,\partial\Omega,u_{0}\\}$ satisfy (1.6)-(1.7), and also
(1.15)-(1.17).

_(I)_ If (LABEL:1.14) holds true, then any narrow minimiser
$u_{\infty}\in\mathcal{W}^{2,\infty}_{u_{0}}(\Omega)$ of (1.1) is the unique
narrow minimiser within
$\smash{\mathbb{B}_{\rho}^{\mathrm{W}^{1,\infty}(\Omega)}(u_{\infty})}\cap\mathcal{W}_{u_{0}}^{2,\infty}(\Omega)$
(perhaps for a smaller $\rho>0$).

_(II)_ If (1.13) holds true, then global minimisers of (1.1) in the space
$\mathcal{W}_{u_{0}}^{2,\infty}(\Omega)$ are unique.

Now we consider the question of regularity of local/narrow/global minimisers,
and of solutions to the system of equations (1.8)-(1.9). In [33, Section 8],
it was proved that even when $n=1$ and no lower-order terms are present, the
minimisers for $u\mapsto\|u^{\prime\prime}\|_{L^{\infty}(a,b)}$ in
$\mathrm{W}^{2,\infty}_{u_{0}}(a,b)$ are absolute, unique, but
non-$\mathrm{C}^{2}$. More precisely, minimisers are always piecewise
quadratic with exactly one singular point $\bar{x}\in(a,b)$, and they are
smooth if and only if the boundary data
$\\{u_{0}(a),u_{0}^{\prime}(a),u_{0}(b),u_{0}^{\prime}(b)\\}$ can be
interpolated by a quadratic polynomial, in which case this is the minimiser.
In the generality of this work, we have the following (partial) regularity
result.

######  Corollary 6 (Partial regularity).

Suppose that $F$ satisfies (1.6)-(1.7), and also that
$F\in\mathrm{C}^{k}\big{(}\Omega\times\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}\big{)}$,
for some $k\in\mathbb{N}$.

_(a)_ Let $(u_{\infty},f_{\infty})$
$\in\mathcal{W}^{2,\infty}(\Omega)\times(\mathrm{L}^{1}(\Omega)\setminus\\{0\\})$
satisfy the equations (1.8)-(1.9). Then,

$\ \ \
u_{\infty}\in\mathrm{C}^{k+2,\gamma}\big{(}\Omega\setminus\\{f_{\infty}=0\\}\big{)},\
\ \ \ \text{for all }\gamma\in(0,1),$

and the set $\\{f_{\infty}=0\\}$ is closed and countably $(n-1)$-rectifiable,
hence in particular a Lebesgue nullset in $\Omega$.

_(b)_ Under the additional assumptions of Theorem 5, any narrow/global
minimiser $u_{\infty}$ satisfies the above $C^{k+2,\gamma}$ partial regularity
on $\Omega\setminus\\{f_{\infty}=0\\}$.

We now connect the system of equations (1.8)-(1.9) to the Aronsson equation
which is associated with (1.1). By a standard formal computational argument
based on $\mathrm{L}^{p}$ approximations and an appropriate rescaling of the
Euler-Lagrange equations (as e.g. in [34, 6]), we can derive the Aronsson
equation for (1.1), which is a fully nonlinear third order PDE:

(1.18) $\partial_{\xi}F(\mathrm{J}^{2}u_{\infty})\hskip
1.0pt\mathrm{A}\\!:\\!\mathrm{D}\big{(}|F(\mathrm{J}^{2}u_{\infty})|\big{)}\otimes\mathrm{D}\big{(}|F(\mathrm{J}^{2}u_{\infty})|\big{)}=0,\
\ \text{ in }\Omega.$

Since by assumption (1.7) we have $\partial_{\xi}F>0$, and also
$\mathrm{A}\in\mathbb{R}^{n\times n}$ is a (strictly) positive symmetric
matrix, (1.18) is (at least formally) equivalent to

$\mathrm{D}\big{(}|F(\mathrm{J}^{2}u_{\infty})|\big{)}=0,\ \ \text{ in
}\Omega.$

Therefore, _the Aronsson equation ( 1.18) corresponding to (1.1) is equivalent
to the statement that $|F(\mathrm{J}^{2}u_{\infty})|$ is constant on connected
components of $\Omega$. This is a genuinely weaker statement that the
statement encoded in the system of equations (1.8)-(1.9)_. Indeed, on the one
hand, (1.8) guarantees that $|F(\mathrm{J}^{2}u_{\infty})|$ is not only
constant throughout $\Omega$, but in fact it is everywhere equal to the exact
value $\mathrm{F}_{\infty}(u_{\infty})$ (even if $\Omega$ has more than one
connected components). On the other hand, (1.9), provides sharper information
about the sign of $F(\mathrm{J}^{2}u_{\infty})$, which is determined by the
nodal set of the weak solution $f_{\infty}$ to this divergence PDE. By the
results in [26], the nodal set is actually a lower dimensional set and in
particular has zero Lebesgue measure. _Therefore, the Aronsson equation is
only a necessary but not a sufficient condition for minimality._ More
precisely, directly from Theorem 4, we have that any narrow minimiser of (1.1)
is a strong solution to (1.18) in _contracted_ form (since
$|F(\mathrm{J}^{2}u_{\infty})|$ is constant throughout $\Omega$ and hence it
is differentiable). Further, by using [29, Th. 30, p. 665] and by arguing as
in [34, Sec. 7, p. 18-20], one can additionally show also that $u_{\infty}$ is
in fact a (generalised) $\mathcal{D}$-solution to the _expanded_ counterpart
of the third order fully nonlinear Aronsson equation (1.18).

Now we supplement Theorem 4 with certain approximability properties,
originating from the method of construction through penalised
$\mathrm{L}^{p}$-approximating functionals. Our method is based on a novel
adaptation of the trademark method of $\mathrm{L}^{p}$ approximations, which
consists of the introduction of an extra penalisation term. To the best of our
knowledge, this adaptation first appeared in the literature in [41]. _The
advantage of this modification is that it forces the approximating
$\mathrm{L}^{p}$ minimisers to converge to a pre-selected
$\mathrm{L}^{\infty}$ minimiser as $p\to\infty$_. For any
$u_{0}\in\mathcal{W}^{2,\infty}(\Omega)$,
$\bar{u}\in\mathcal{W}^{2,\infty}_{u_{0}}(\Omega)$, $p>n$, and $\epsilon>0$,
we set

(1.19)
$\mathrm{F}_{p,\epsilon,\bar{u}}(u)\,:=\,\left(\,-\int_{\Omega}|F(\mathrm{J}^{2}u)|^{p}\,\mathrm{d}\mathcal{L}^{n}\\!\right)^{\\!\\!{1}/{p}}\,+\,\frac{\epsilon}{2}\,{-\int_{\Omega}}|u-\bar{u}|^{2}\,\mathrm{d}\mathcal{L}^{n},$

for all $u\in\mathrm{W}^{2,p}_{u_{0}}(\Omega)\
(=\mathrm{W}^{2,p}_{\bar{u}}(\Omega))$.

######  Theorem 7 (Necessity of the equations and $\mathrm{L}^{p}$
approximation with penalisation).

In the setting of Theorem 4 and under the same hypotheses, we have that the
pair $(u_{\infty},f_{\infty})$ satisfies the following approximation
properties:

_(i)_ There exists a family of functions
$\\{u_{p}\\}_{n<p<\infty}\subseteq\mathrm{W}^{2,n}_{u_{0}}(\Omega)$, where for
each $p$, $u_{p}$ is a narrow minimiser of the penalised $\mathrm{L}^{p}$
functional $\mathrm{F}_{p,1,u_{\infty}}$ (given by (1.19)) in the space
$\mathrm{W}^{2,p}_{u_{0}}(\Omega)$, such that,

(1.20) $\left\\{\ \
\begin{split}&\mathrm{F}_{p,1,u_{\infty}}(u_{p})\longrightarrow\mathrm{F}_{\infty}(u_{\infty}),\\\
&u_{p}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{\infty},\ \ \text{ in
}\mathrm{W}^{2,q}(\Omega),\ \forall\ q\in(1,\infty),\\\ &u_{p}\longrightarrow
u_{\infty},\ \ \,\text{ in }\mathrm{C}^{1,\gamma}(\overline{\Omega}),\
\forall\ \gamma\in(0,1),\end{split}\right.$

along a sequence of indices $(p_{j})_{1}^{\infty}$ satisfying $p_{j}\to\infty$
as $j\to\infty$.

_(ii)_ For any $p\in(n,\infty)$, the narrow minimiser $u_{p}$ is a weak
solution to the Euler-Lagrange equations of $\mathrm{F}_{p,1,u_{\infty}}$,
namely

(1.21)
$e_{p}^{1-p}\int_{\Omega}\Big{[}\big{(}|F|^{p-2}F\partial_{*}F\big{)}(\mathrm{J}^{2}u_{p})\cdot\mathrm{J}^{2}_{*}\phi\Big{]}\mathrm{d}\mathcal{L}^{n}+\int_{\Omega}(u_{p}-u_{\infty})\phi\,\mathrm{d}\mathcal{L}^{n}=0,$

for all $\phi\in\mathrm{W}^{2,p}_{0}(\Omega)$, where

(1.22)
$e_{p}\,:=\left(\,-\int_{\Omega}|F(\mathrm{J}^{2}u_{p})|^{p}\,\mathrm{d}\mathcal{L}^{n}\\!\right)^{\\!\\!\frac{1}{p}}.$

Further, by setting

(1.23)
$f_{p}\,:=\,e_{p}^{1-p}\big{(}|F|^{p-2}F\partial_{\xi}F\big{)}(\mathrm{J}^{2}u_{p}),$

we have

(1.24) $\left\\{\ \
\begin{split}&f_{p}\mathcal{L}^{n}\text{\LARGE$\llcorner$}_{\Omega}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}f_{\infty}\mathcal{L}^{n}\text{\LARGE$\llcorner$}_{\Omega},\
\text{ in }\mathcal{M}(\overline{\Omega}),\\\ &f_{p}\longrightarrow
f_{\infty},\hskip 53.0pt\text{ in }\mathrm{L}^{q}_{\mathrm{loc}}(\Omega),\
\forall\ q\in(1,n^{\prime}),\end{split}\right.$

and also

(1.25) $\ \ \
\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{p}\longrightarrow\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{\infty}\
\ \text{ in }\mathrm{L}^{q}\big{(}\Omega\setminus\\{f_{\infty}=0\\}\big{)},\
\forall\ q\in(1,\infty),$

along a sequence of indices $(p_{j})_{1}^{\infty}$ which satisfies
$p_{j}\to\infty$ as $j\to\infty$.

Our final result concerns an immediate generalisation of all our previous
results to more general functionals, which may involve supremands which are
perhaps non-differentiable and with arbitrary growth at infinity.

######  Corollary 8 (Extension to non-differentiable supremands with arbitrary
growth).

Let us consider the supremal functional

$\ \ \ \ \ \ \
\bar{\mathrm{F}}_{\infty}(u):=\|\mathcal{F}(\mathrm{J}^{2}u)\|_{\mathrm{L}^{\infty}(\Omega)},\
\ \ \ u\in\mathcal{W}^{2,\infty}(\Omega),$

and suppose the supremand
$\mathcal{F}:\Omega\times(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R})\longrightarrow\mathbb{R}$
can be factorised as follows:

$\left\\{\ \begin{split}&\text{There exist a strictly increasing odd function
$\Phi:\mathbb{R}\longrightarrow\mathbb{R}$ and a Cara-}\\\ &\text{th\'{e}odory
function
$F:\Omega\times(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R})\longrightarrow\mathbb{R}$,
such that $\mathcal{F}=\Phi\circ F$},\end{split}\right.$

where $F$ satisfies (1.6)-(1.7). Then, all results hold true for
$\bar{\mathrm{F}}_{\infty}$ as well, if $F$ satisfies the additional
corresponding assumptions (LABEL:1.13), (LABEL:1.14), (1.13), (1.15) and
(1.16).

Let us now discuss the state of the art for higher order (scalar-valued)
$\mathrm{L}^{\infty}$ variational problems. The relevant higher order Aronsson
equation was first derived formally by Aronsson himself and Barron in [6], but
noting its peculiarities, it was not explored further. In the paper [34] by
the first author and Pryer, higher order $\mathrm{L}^{\infty}$ problems were
considered systematically for the first time, and several initial results
where established for the functional
$u\mapsto\|H(\mathrm{D}^{2}u)\|_{\mathrm{L}^{\infty}(\Omega)}$, including
existence of generalised solutions to the Dirichlet problem for the
corresponding fully nonlinear Aronsson equation. For this task, the general
framework of $\mathcal{D}$-solutions was utilised, a systematic approach for
fully nonlinear PDE systems, introduced by the first author in [29]. In [44],
Papamikos and Pryer studied special solutions to the same higher order
Aronsson equation, classifying them by using Lie symmetries. In [33], the
first author with Parini studied a higher order $\infty$-eigenvalue problem,
generalised subsequently to the fully nonlinear setting by the first author
with Clark in [17]. In [16], Clark and Muha, jointly with the first author,
inspired by questions arising in data assimilation, studied a second order PDE
constrained $\mathrm{L}^{\infty}$ problem involving the Navier-Stokes
equations. Higher order $\mathrm{L}^{\infty}$ variational problems arising in
Riemannian geometry were previously studied by the second author with
Schwetlick and also by Sakellaris in [42, 49], and more recently again by the
second author in [41]. In [31], the authors considered the special case of
(1.1) given by $u\mapsto\|\mathcal{F}(\cdot,\Delta
u)\|_{\mathrm{L}^{\infty}(\Omega)}$ which, unsurprisingly, is immensely
simpler than the case of $\mathrm{F}_{\infty}$ which involves explicit
$(u,\mathrm{D}u)$-dependence.

This paper is organised as follows. This rather long introduction is followed
by Section 2, in which we discuss in some detail the preliminaries, our
functional space setup, our assumptions, and also provide explicit classes of
integrands (Subsection 2.2). Therein we also establish some auxiliary results
of independent interest. In particular, we identify
$\mathcal{W}^{2,\infty}_{0}(\Omega)$ as a dual Banach space with separable
predual, and we also prove a general lemma regarding the existence of predual
spaces for weakly* closed subspaces in dual Banach spaces (Subsection 2.3). We
also establish certain auxiliary results of independent interest, including
characterisations and examples of compact sets in
$\mathcal{W}^{2,\infty}_{0}(\Omega)$ (Subsection 2.4), approximability of
constrained minimisers in $\mathrm{L}^{\infty}$ via $\mathrm{L}^{p}$
constrained minimisers as $p\to\infty$ (Subsection 2.5), and a lemma about
strong compactness in $\mathrm{L}^{1}_{\mathrm{loc}}(\Omega)$ of solutions to
divergence PDE with $\mathrm{L}^{1}$ bounded coefficients (Subsection 2.6),
and some results regarding the size of nodal sets and other properties of
solutions to equations of the form (1.9) (Subsection 2.7). Finally, in Section
3 we establish all our main results, whose statements are given earlier in
this introduction.

We close this introduction by noting that, in our recent paper [32], we
consider the case of the functional
$u\mapsto\|\mathrm{A}(\cdot)\\!:\\!\mathrm{D}^{2}u+b(\cdot,u,\mathrm{D}u)\|_{\mathrm{L}^{\infty}(\Omega)}$,
which involves a semilinear differential operator. Even though there is some
overlap of [32] with the results obtained herein, the phenomena are
significantly different. In particular, in [32] the natural minimality notion
involves _almost minimisers_ with a first order error term, and not
global/narrow/local minimisers as herein. Additionally, unlike the present
work, uniqueness in [32] is an open question for almost minimisers. Therefore,
for endogenous mathematical reasons, as well as for practicality purposes of
space and size, we treat these related problems separately, as an attempt for
a unified approach would considerably complicate and impair the exposition,
rather than enhance it.

##  2\. Preliminaries and auxiliary results

We begin by discussing in some detail our notation and our functional space
setup, and also establish some auxiliary results of independent interest.

###  2.1. Preliminaries and notational conventions

Let “$:$” denotes the Euclidean (Frobenius) inner product in
$\mathbb{R}^{n\times n}$, namely $A\\!:\\!B=\mathrm{tr}(A^{\top}B)$. For
$a,b\in\mathbb{R}^{n}\equiv\mathbb{R}^{n\times 1}$, we symbolise $a\otimes
b:=ab^{\top}$. Further, for any open $\Omega\subseteq\mathbb{R}^{n}$, for a
twice (weakly) differentiable function $u:\Omega\longrightarrow\mathbb{R}$,
the notation
$\mathrm{D}u=(\mathrm{D}_{1}u,...,\mathrm{D}_{n}u):\Omega\longrightarrow\mathbb{R}^{n}$
will symbolise its gradient, and
$\mathrm{D}^{2}u=(\mathrm{D}^{2}_{ij}u)_{i,j=1}^{n}:\Omega\longrightarrow\mathbb{R}^{n\times
n}$ will denote its Hessian, where obviously
$\mathrm{D}_{i}\equiv\partial/\partial x_{i}$. For any function
$u:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$ and $z\in\mathbb{R}^{n}$, we
define the translation $\tau_{z}u$ by setting $\tau_{z}u(x):=u(x+z)$. The
closed ball, open ball and sphere of radius $r$ centred at a point
$x\in\mathfrak{X}$ of a normed space $(\mathfrak{X},\|\cdot\|)$ will be
denoted by
$\bar{\mathbb{B}}^{\mathfrak{X}}_{\rho}(x),\mathbb{B}^{\mathfrak{X}}_{\rho}(x),\mathbb{S}^{\mathfrak{X}}_{\rho}(x)$
respectively. If $\rho=1$ and/or $x=0$, we will omit explicitly displaying
them, and if $X$ is clear from the context, it may be omitted as well. The
functional space notation we employ is either self-explanatory, or otherwise a
convex combination of standard symbolisations (as e.g. in [1, 19, 24, 35, 48,
50]). For example, $[\,\cdot\,]_{\mathrm{C}^{\alpha}(\overline{\Omega})}$
symbolises the Hölder seminorm over $\Omega$,
$\smash{\mathrm{C}^{\nnearrow}[0,\infty)}$ symbolises the set of continuous
increasing functions on $[0,\infty)$, and
$\smash{\mathrm{C}_{0^{+}}^{\nnearrow}[0,\infty)}$ symbolises the subset of
those that vanish at the origin (i.e. moduli of continuity). The
$n$-dimensional Lebesgue measure will be denoted by $\mathcal{L}^{n}$, and the
$k$-dimensional Hausdorff measure by $\mathcal{H}^{k}$. Further, all
$\mathrm{L}^{p}$ norms appearing in this paper with respect to the Lebesgue
measure $\mathcal{L}^{n}$ over $\Omega$ will always be the normalised ones,
namely with the average over $\Omega$ instead of the integral, as e.g. in
(1.19). Let now $(\mathfrak{X},\|\cdot\|)$ be a normed space, with dual Banach
space $(\mathfrak{X}^{*},\|\cdot\|_{*})$. Given a subset
$\mathcal{A}\subseteq\mathfrak{X}$ and a subset
$\mathcal{B}\subseteq\mathfrak{X}^{*}$, the annihilator
$\mathcal{A}^{\bot}\subseteq\mathfrak{X}^{*}$ and the pre-annihilator
${}^{\bot}\mathcal{B}\subseteq\mathfrak{X}$ are defined as

$\begin{split}\mathcal{A}^{\bot}&:=\big{\\{}f\in\mathfrak{X}^{*}:f(x)=0,\
\forall\,x\in\mathcal{A}\big{\\}}=\big{\\{}f\in\mathfrak{X}^{*}:f^{-1}(\\{0\\})\supseteq\mathcal{A}\big{\\}},\\\
{}^{\bot}\mathcal{B}&:=\big{\\{}x\in\mathfrak{X}:f(x)=0,\
\forall\,f\in\mathcal{B}\big{\\}}=\bigcap\big{\\{}f^{-1}(\\{0\\}):{f\in\mathcal{B}}\big{\\}}.\end{split}$

(Details can be found e.g. in [48, Ch. 1 and Ch. 4].) Then,
$\mathcal{A}^{\bot}$ is a weakly*-closed subspace in $\mathfrak{X}^{*}$, and
${}^{\bot}\mathcal{B}$ is a norm-closed subspace in $\mathfrak{X}$. If
$\mathcal{N}\subseteq\mathfrak{X}$ is any norm-closed subspace, the
equivalence relation $x^{\prime}\sim x^{\prime\prime}\Leftrightarrow
x^{\prime\prime}-x^{\prime}\in\mathcal{N}$ partitions $\mathfrak{X}$ to
cosets, the collection of which comprises the quotient vector space
$\mathfrak{X}/_{\mathcal{N}}:=\\{x+{\mathcal{N}}:x\in\mathfrak{X}\\}$. Let
$\pi:\mathfrak{X}\longrightarrow\mathfrak{X}/_{\mathcal{N}}$ denote the
quotient map, given by $\pi(x):=x+{\mathcal{N}}$. Then,
$\mathfrak{X}/_{\mathcal{N}}$ becomes a normed space, when endowed with the
quotient norm

$\ \ \
\|\pi(x)\|_{\mathfrak{X}/_{\mathcal{N}}}:=\inf\big{\\{}\|x-n\|\,:\,n\in{\mathcal{N}}\big{\\}}=\mathrm{dist}(x,{\mathcal{N}}),\
\ \ x\in\mathfrak{X}.$

###  2.2. Discussion of assumptions and examples

We now focus on assumptions (LABEL:1.13)-(1.13), which involve convexity of
$|F(x,\cdot)|^{\bar{p}}$ on convex subsets of
$\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}$ for large $\bar{p}\geq 2$,
and also on assumption (1.17), which requires additional regularity of the
restriction of the boundary condition $u_{0}|_{\partial\Omega}$ to
$\partial\Omega$.

####  2.2.1. The restricted convexity assumptions (LABEL:1.13)-(1.13)

A necessary and sufficient condition for the convexity of
$|F(x,\cdot)|^{\bar{p}}$ on an open convex subset $L\times M\times
R\subseteq\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}$, is the following
(semi-convexity type) matrix inequality

(2.1) $F\partial^{2}_{**}F+(\bar{p}-1)\partial_{*}F\otimes\partial_{*}F\geq
0,$

which must hold a.e. on $\Omega\times L\times M\times R$. The equivalence
follows from the identity

(2.2)
$\partial^{2}_{**}\big{(}|F|^{p}\big{)}=\,p|F|^{p-2}\Big{(}F\partial^{2}_{**}F+(p-1)\partial_{*}F\otimes\partial_{*}F\Big{)}.$

As a consequence, if $|F(x,\cdot)|^{p}$ is convex on $L\times M\times R$, so
is $|F(x,\cdot)|^{q}$ for any $q\geq p$. In the case that there is no explicit
gradient dependence (and hence no $\mathrm{p}$-variable), a sufficient
condition for (2.1) is the following: let $\star\in\mathbb{R}^{2\times 2}$ be
the Hodge matrix, satisfying $\star^{2}=-\mathrm{I}$ (rotation by $\pi/2$ in
$\mathbb{R}^{2}$). Then, (2.1) follows from:

(2.3) $\left\\{\begin{split}&\text{$(\partial_{*}F,\star\,\partial_{*}F)$ is
an orthogonal eigenframe of $\partial^{2}_{**}(|F|^{2})$ with eigenvalues}\\\
&\text{$(\sigma_{1},\sigma_{2})$, satisfying $\sigma_{2}\geq 0$,
$\sigma_{1}\geq-m$ for $m\in\mathbb{N}$, a.e.\ on $\Omega\times L\times
R$.}\end{split}\right.$

Condition (2.3) is a stronger requirement than (2.1), but might be easier to
verify. The implication “(2.3)$\Rightarrow$(2.1)” follows by the spectral
theorem applied to $\partial^{2}_{**}(|F|^{2})$:

$\begin{split}\partial^{2}_{**}(|F|^{2})&=\frac{\sigma_{1}}{|\partial_{*}F|^{2}}\partial_{*}F\otimes\partial_{*}F\,+\,\frac{\sigma_{2}}{|\partial_{*}F|^{2}}(\star\,\partial_{*}F)\otimes(\star\,\partial_{*}F)\\\
&\geq\frac{\sigma_{1}}{|\partial_{*}F|^{2}}\partial_{*}F\otimes\partial_{*}F\\\
&\geq-\frac{m}{c}\partial_{*}F\otimes\partial_{*}F,\end{split}$

which holds a.e. on $\Omega\times L\times R$. This implies (2.1) when
$\bar{p}\geq(m/c)+1$. In the above inequalities we have used that
$|\partial_{*}F|\geq c>0$, which follows from (1.7).

####  2.2.2. The Whitney differentiability assumption (1.17)

By the (Lipschitz) regularity of $\partial\Omega$, we have the Sobolev
imbedding
$\mathcal{W}^{2,\infty}(\Omega)\subseteq\mathrm{C}^{1,\gamma}(\overline{\Omega})$,
for all $\gamma\in(0,1)$ in the usual sense, namely the precise representative
from each a.e.-equivalence class has an extension to a function in
$\mathrm{C}^{1,\gamma}(\mathbb{R}^{n})$. In particular,
$u_{0}|_{\partial\Omega}$ is at the outset $\mathrm{C}^{1,\gamma}$ on
$\partial\Omega$, for all $\alpha\in(0,1)$. The assumption that
$u_{0}|_{\partial\Omega}$ is $\mathrm{C}^{2}$ on the compact set
$\partial\Omega$ is understood in the standard sense of Whitney [51, Sec. 3,
p. 64], meaning that it possesses (uniform) second order Taylor expansions.
More precisely, we assume that there exists a symmetric matrix-valued map
$X\in\mathrm{C}(\partial\Omega;\mathbb{R}^{n\times n})$ and a modulus of
continuity $\omega\in\mathrm{C}^{\nnearrow}_{0}[0,\infty)$, such that

$\left\\{\ \ \
\begin{split}\left|u_{0}(x+z)-u_{0}(x)-\mathrm{D}u_{0}(x)z-\frac{1}{2}X(x):z\otimes
z\right|&\leq\omega(|z|)|z|^{2},\ \ \ \ \\\
\Big{|}\mathrm{D}u_{0}(x+z)-\mathrm{D}u_{0}(x)-X(x)z\Big{|}&\leq\omega(|z|)|z|,\end{split}\right.$

for all $x,z\in\mathbb{R}^{n}$ such that $x,x+z\in\partial\Omega$. It can be
easily seen that, due to the smoothness of $\partial\Omega$, $\mathrm{C}^{1}$
differentiability for a scalar-valued function in the above sense is
equivalent to tangential differentiability on $\partial\Omega$ (regarded as a
$\mathrm{C}^{2}$ compact manifold with the induced Riemannian metric).
However, requiring a function to be $\mathrm{C}^{2}$ on $\partial\Omega$, or
requiring its Euclidean gradient vector field be tangentially $\mathrm{C}^{1}$
on $\partial\Omega$, are in general slightly stronger requirements that being
tangentially $\mathrm{C}^{2}$ on $\partial\Omega$.

####  2.2.3. Explicit classes of supremands

We demonstrate some classes of functions $F$ which satisfy our assumptions,
noting that the results proved herein apply to supremands of the form
$\mathcal{F}=\Phi\circ F$, for _any_ increasing odd function
$\Phi:\mathbb{R}\longrightarrow\mathbb{R}$ (Corollary 8). For the sake of
clarity, we assume no explicit gradient dependence in these examples (and
hence no $\mathrm{p}$-variable). A similar (but more complicated) analysis can
be performed by assuming explicit dependence on the gradient as well.

Additive ansatz. Let $A:\mathbb{R}\longrightarrow\mathbb{R}$ and
$a:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ be given. We set

$F(x,\eta,\xi):=a(x,\eta)+A(\xi).$

If we have $A\in\mathrm{C}^{2}(\mathbb{R})$ with $A_{\xi}\geq c>0$, and for
all $x\in\Omega$, we have $a(x,\cdot)\in\mathrm{C}^{2}(\mathbb{R})$ with
$a,a_{\eta},a_{\eta\eta}\in C(\overline{\Omega}\times\mathbb{R})$, then
(1.6)-(1.7) are readily satisfied. In particular, since $A$ is strictly
increasing, $F(x,\eta,\xi)=0$ iff $\xi=A^{-1}(-a(x,\eta))$, and therefore

$F(x,\eta,\cdot)^{-1}(\\{0\\})\subseteq\Big{[}A^{-1}\Big{(}-\max_{x\in\overline{\Omega}}a(x,\eta)\Big{)},A^{-1}\Big{(}-\min_{x\in\overline{\Omega}}a(x,\eta)\Big{)}\Big{]}.$

Regarding the satisfaction of (LABEL:1.13)-(LABEL:1.14)-(1.13), a direct
computation gives that, if we set
$\mathfrak{F}:=F\partial_{**}^{2}F+M\partial_{*}F\otimes\partial_{*}F$ where
$M\geq 1$, then we have $\mathfrak{F}\geq 0$ if and only if (suppressing the
arguments $(x,\eta,\xi)$ for brevity)

(2.4)
$\displaystyle\mathrm{tr}(\mathfrak{F})=(a+A)(A_{\xi\xi}+a_{\eta\eta})+M\big{[}(A_{\xi})^{2}+(a_{\eta})^{2}\big{]}$
$\displaystyle\geq 0,$ (2.5)
$\displaystyle\det(\mathfrak{F})=(a+A)^{2}A_{\xi\xi}a_{\eta\eta}+M(a+A)\big{[}a_{\eta\eta}(A_{\xi})^{2}+A_{\xi\xi}(a_{\eta})^{2}\big{]}$
$\displaystyle\geq 0.$

Let us also consider the _hypergraph_ of the function $A^{-1}\circ(-a)$,
namely

$\mathcal{H}:=\Big{\\{}(x,\eta,\xi)\in\Omega\times\mathbb{R}\times\mathbb{R}:\
\xi\geq A^{-1}\big{(}\\!-a(x,\eta)\big{)}\Big{\\}},$

which has the property that $F(x,\eta,\xi)\geq 0$ iff
$(x,\eta,\xi)\in\mathcal{H}$. Then:

(a) Let $L,R\subseteq\mathbb{R}$ be given open intervals such that
$\Omega\times L\times R\subseteq\mathcal{H}$ (the hypergraph). If for all
$x\in\Omega$, $a(x,\cdot)$ is convex on $L$, and $A$ is convex on $R$, then
(2.4)-(2.5) are satisfied on $\Omega\times L\times R$, for all $M>0$. Hence,
(LABEL:1.13) is satisfied.

(b) If $a$ is bounded below by $-m$ for some $m\in\mathbb{R}$, then
$\mathcal{M}:=\Omega\times\mathbb{R}\times(A^{-1}(m),\infty)$
$\subseteq\mathcal{H}$. If for all $x\in\Omega$, $a(x,\cdot)$ is convex on
$\mathbb{R}$, and $A$ is convex on $(A^{-1}(m),\infty)$, then (2.4)-(2.5) are
satisfied on $\mathcal{M}$, for all $M>0$. Hence, (LABEL:1.13) is satisfied.

(c) If for all $x\in\Omega$, $a(x,\cdot)$ is convex on $\mathbb{R}$, and $A$
is also convex on $\mathbb{R}$, then (2.4)-(2.5) are satisfied on the entire
hypergraph $\mathcal{H}$, for all $M>0$. Hence, (LABEL:1.13) is satisfied.

(d) If for all $x\in\Omega$, $a(x,\cdot)$ is affine on $\mathbb{R}$, and $A$
is also affine on $\mathbb{R}$, then (2.4)-(2.5) are satisfied on
$\Omega\times\mathbb{R}\times\mathbb{R}$ for all $M>0$, hence (1.13) is
satisfied.

Regarding the satisfaction of (1.15)-(1.16), they are satisfied when $A$ is
semiconvex on $\mathbb{R}$ ($A_{\xi\xi}$ bounded below) and $A_{\xi}$ is
bounded above, with the growth of $a(x,\cdot)$ smaller than the growth of $A$
at infinity, namely when we have $|A(\xi)|\leq C(1+|\xi|^{p})$ and
$|a(x,\eta)|\leq C(1+|\eta|^{q})$, for some $C>0$ and $0<q<p$.

Multiplicative ansatz. Let $A:\mathbb{R}\longrightarrow\mathbb{R}$ and
$a:\Omega\times\mathbb{R}\longrightarrow\mathbb{R}$ be given. We set

$F(x,\eta,\xi):=a(x,\eta)A(\xi).$

If $A\in\mathrm{C}^{2}(\mathbb{R})$ satisfies $A_{\xi}\geq c>0$, and also for
all $x\in\Omega$, we have $a(x,\cdot)\in C^{2}(\mathbb{R})$, with
$a,a_{\eta},a_{\eta\eta}\in C(\overline{\Omega}\times\mathbb{R})$ and $a\geq
c>0$, then (1.6)-(1.7) are satisfied with
$F(x,\eta,\cdot)^{-1}(\\{0\\})=\\{0\\}$. Regarding the satisfaction of
(LABEL:1.13)-(LABEL:1.14)-(1.13), let again $M\geq 1$. Suppressing the
arguments $(x,\eta,\xi)$ for brevity, a computation gives that, if
$\mathfrak{F}:=F\partial_{**}^{2}F+M\partial_{*}F\otimes\partial_{*}F$, then
$\mathfrak{F}\geq 0$ if and only if

(2.6) $\displaystyle\mathrm{tr}(\mathfrak{F})$
$\displaystyle=A^{2}\big{[}aa_{\eta\eta}+M(a_{\eta})^{2}\big{]}+a^{2}\big{[}AA_{\xi\xi}+M(A_{\xi})^{2}\big{]}\geq
0,$ (2.7)
$\displaystyle\begin{split}\det(\mathfrak{F})&=(aA)^{2}\bigg{\\{}(AA_{\xi\xi})(aa_{\eta\eta})-(A_{\xi})^{2}(a_{\eta})^{2}\\\
&\ \ \
\,+M(A_{\xi})^{2}\big{[}aa_{\eta\eta}-(a_{\eta})^{2}\big{]}+M(a_{\eta})^{2}\big{[}AA_{\xi\xi}-(A_{\xi})^{2}\big{]}\bigg{\\}}\geq
0.\end{split}$

(a) Let $L,R\subseteq\mathbb{R}$ be open intervals, and let us note the
following elementary identities, which are valid for any
$h\in\mathrm{C}^{2}(\mathbb{R})$:
$(|h|^{p})^{\prime\prime}=p|h|^{p-2}[hh^{\prime\prime}+(p-1)(h^{\prime})^{2}]$
on $\mathbb{R}$, and
$(\ln|h|)^{\prime\prime}=h^{-2}[hh^{\prime\prime}-(h^{\prime})^{2}]$ on
$\\{h\neq 0\\}$. It follows that, (2.6)-(2.7) are satisfied on $\Omega\times
L\times R$, if $\ln a(x,\cdot)$ is convex on $L$ for all $x\in\Omega$, and
$\ln|A|$ is convex on $R$. Therefore, we may select $\mathrm{C}^{2}$ functions
$A:=e^{B}$, $a:=e^{b}$, where $B:R\longrightarrow\mathbb{R}$ is convex and
strictly increasing, and $b:\overline{\Omega}\times
L\longrightarrow\mathbb{R}$ is convex in the second variable. We extend $A$ as
a strictly increasing function in $\mathrm{C}^{2}(\mathbb{R})$ which is linear
outside an interval $[-m,m]$ for some $m\in\mathbb{N}$ large, and extend also
$a$ as an arbitrary $\mathrm{C}^{2}$ function on $\Omega\times\mathbb{R}$
bounded below. It follows that (LABEL:1.13) is satisfied.

(b) Let $A$ be affine. It can be easily confirmed that (2.6)-(2.7) can be
deduced from

$(\ln a)_{\eta\eta}\geq\frac{M+1}{M}\big{(}(\ln a)_{\eta}\big{)}^{2}.$

Then, for any interval $L\subseteq\mathbb{R}$, and for any
$b\in\mathrm{C}^{2}(\overline{\Omega}\times\mathbb{R})$ which is bounded below
and is strictly convex in $\eta$, we can find $\varepsilon>0$ small such that
$a:=e^{\varepsilon b}$ satisfies the above inequality on $\Omega\times L$.
Hence, (2.6)-(2.7) hold true on $\Omega\times L\times\mathbb{R}$, and
therefore (LABEL:1.14) is satisfied. Finally, assumptions (1.15)-(1.16) are
also trivially satisfied.

###  2.3. The functional space setup and duality

Now we expound on the functional spaces we utilise herein, and study some of
their properties. We set

(2.8)
$\displaystyle\mathbb{W}_{\mathrm{loc}}^{2,\infty}(\Omega)\,:=\,\bigcap_{1<p<\infty}\mathrm{W}_{\mathrm{loc}}^{2,p}(\Omega),$
(2.9)
$\displaystyle\mathbb{W}^{2,\infty}(\Omega)\,:=\,\bigcap_{1<p<\infty}\mathrm{W}^{2,p}(\Omega),$
(2.10)
$\displaystyle\mathbb{W}_{0}^{2,\infty}(\Omega)\,:=\,\bigcap_{1<p<\infty}\mathrm{W}_{0}^{2,p}(\Omega).$

Then, we have
$\mathbb{W}_{\mathrm{loc}}^{2,\infty}(\Omega)\supset\mathbb{W}^{2,\infty}(\Omega)\supset\mathbb{W}_{0}^{2,\infty}(\Omega)$,
and all three spaces are Fréchet spaces, topologised by the family of
seminorms
$\big{\\{}\|\cdot\|_{\mathrm{W}^{2,p}(\Omega^{\prime})}:\Omega^{\prime}\\!\Subset\Omega,\,p\in(1,\infty)\big{\\}}$.
If now
$\mathbb{A}:\mathrm{W}^{2,1}(\Omega)\longrightarrow\mathrm{L}^{1}(\Omega)$ is
the (strong) linear differential operator defined by
$\mathbb{A}u:=\mathrm{A}\\!:\\!\mathrm{D}^{2}u$, the spaces defined in (1.2)
and (1.5) can be written as

(2.11) $\displaystyle\mathcal{W}^{2,\infty}(\Omega)\,$
$\displaystyle=\,\mathbb{W}^{2,\infty}(\Omega)\cap\mathbb{A}^{-1}\big{(}\mathrm{L}^{\infty}(\Omega)\big{)},$
(2.12) $\displaystyle\mathcal{W}^{2,\infty}_{0}(\Omega)\,$
$\displaystyle=\,\mathbb{W}^{2,\infty}_{0}(\Omega)\cap\mathbb{A}^{-1}\big{(}\mathrm{L}^{\infty}(\Omega)\big{)}.$

As we will show in Subsections 2.3.1-2.3.2,
$\mathcal{W}^{2,\infty}_{0}(\Omega)$ becomes a dual Banach space with
(separable) predual $\mathbb{A}(\mathrm{L}^{1}(\Omega))$ and convenient weak*
density and compactness properties, when endowed with either of the equivalent
norms

(2.13) $\left\\{\ \ \
\begin{split}\|u\|_{\mathcal{W}^{2,\infty}_{0}(\Omega)}&:=\,\|\mathrm{A}\\!:\\!\mathrm{D}^{2}u\|_{\mathrm{L}^{\infty}(\Omega)},\\\
\|u\|^{\prime}_{\mathcal{W}^{2,\infty}_{0}(\Omega)}&:=\,\|\mathrm{J}_{*}^{2}u\|_{\mathrm{L}^{\infty}(\Omega)}.\end{split}\right.$

####  2.3.1. Completeness and a priori estimates

We now study some important properties of the space
$\mathcal{W}_{0}^{2,\infty}(\Omega)$ which have independent interest, but are
also utilised in the proofs of our main results in Section 3.

######  Lemma 9.

The functional space (2.12) is a Banach space when endowed with either of the
Lipschitz-equivalent norms defined in (2.13). Moreover, for any
$p\in(1,\infty)$, there exists a constant $K=K(p)>0$ (depending also on
$\Omega,\mathrm{A}$), such that

$\|u\|_{\mathrm{W}^{2,p}(\Omega)}\leq
K\|\mathrm{A}\\!:\\!\mathrm{D}^{2}u\|_{\mathrm{L}^{p}(\Omega)}\leq
K\|\mathrm{A}\\!:\\!\mathrm{D}^{2}u\|_{\mathrm{L}^{\infty}(\Omega)},$

for all $u\in\mathrm{W}^{2,p}_{0}(\Omega)$.

Proof of Lemma 9. We begin by noting that the Lipschitz equivalence between
the two norms in (2.13) is an immediate consequence of Morrey’s imbedding
theorem, since $\|u\|_{\mathrm{L}^{\infty}(\Omega)}\leq
K\|u\|_{\mathrm{W}^{2,n}(\Omega)}$ for some $K=K(\Omega)>0$. Next, by the
$\mathrm{L}^{p}$ estimates for the Laplacian on a bounded open set
$U\Subset\mathbb{R}^{n}$ (see e.g. [25, Cor. 9.10]), for any $p\in(1,\infty)$,
there exists a $K=K(p,U)>0$ such that

$\|v\|_{\mathrm{W}^{2,p}(U)}\,\leq\,K\|\Delta v\|_{\mathrm{L}^{p}(U)},$

for all $v\in\mathrm{W}^{2,p}_{0}(U)$. Let
$\mathrm{A}=O^{\top}\\!\Lambda^{2}O$ be the spectral representation of
$\mathrm{A}$, namely $O\in\mathrm{O}(n,\mathbb{R})$ and
$\Lambda=\mathrm{diag}(\sigma(\sqrt{\mathrm{A}}))$ is a diagonal matrix with
entries the eigenvalues of $\sqrt{\mathrm{A}}$. Then, by setting for brevity
$\Gamma:=\Lambda O$ and selecting $v(x):=u(\Gamma^{\top}x)$ and also
$U:=\Gamma^{-\\!\top}\Omega$, we have $\mathrm{A}=\Gamma^{\top}\Gamma$ and the
identities $\Delta v(x)=\mathrm{A}\\!:\\!\mathrm{D}^{2}u(\Gamma^{\top}x)$ and
$\mathrm{D}v(x)=\Gamma\hskip 1.0pt\mathrm{D}u(\Gamma^{\top}x)$, for all
$x\in\Omega$. By the change of variables’ formula and Hölder’s inequality,
there exists a new constant $K=K(p,\Omega,\mathrm{A})>0$, such that

$\|u\|_{\mathrm{W}^{2,p}(\Omega)}\,\leq\,K\|\mathrm{A}\\!:\\!\mathrm{D}^{2}u\|_{\mathrm{L}^{p}(\Omega)}\,\leq\,K\|u\|_{\mathcal{W}^{2,\infty}_{0}(\Omega)},$

for all $u\in\mathrm{W}^{2,p}_{0}(\Omega)$. If
$(u_{m})_{1}^{\infty}\subseteq\mathcal{W}^{2,\infty}_{0}(\Omega)$ is a Cauchy
sequence, the above implies

$\|u_{m}-u_{n}\|_{\mathrm{W}^{2,p}(\Omega)}\leq\,K\big{\|}\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{m}-\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{n}\big{\|}_{\mathrm{L}^{\infty}(\Omega)}\longrightarrow
0,$

as $m,n\to\infty$. By a diagonal argument and by the completeness of
$\mathrm{L}^{\infty}(\Omega)$ and of $\mathbb{W}^{2,\infty}_{0}(\Omega)$,
there exist functions $u\in\mathbb{W}^{2,\infty}_{0}(\Omega)$ and
$f\in\mathrm{L}^{\infty}(\Omega)$ such that $u_{m}\longrightarrow u$ in
$\mathrm{W}^{2,p}(\Omega)$ for any fixed $p\in(1,\infty)$, and also
$\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{m}\longrightarrow f$ in
$\mathrm{L}^{\infty}(\Omega)$, both as $m\to\infty$. Further, for any fixed
$\phi\in\mathrm{C}^{\infty}_{c}(\Omega)$, we have

$\begin{split}\int_{\Omega}\phi(\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{m})\,\mathrm{d}\mathcal{L}^{n}&=\int_{\Omega}u_{m}(\mathrm{A}\\!:\\!\mathrm{D}^{2}\phi)\,\mathrm{d}\mathcal{L}^{n}\\\
&\\!\\!\\!\longrightarrow\int_{\Omega}u(\mathrm{A}\\!:\\!\mathrm{D}^{2}\phi)\,\mathrm{d}\mathcal{L}^{n}\\\
&=\int_{\Omega}\phi(\mathrm{A}\\!:\\!\mathrm{D}^{2}u)\,\mathrm{d}\mathcal{L}^{n},\end{split}$

as $m\to\infty$, which establishes that
$\mathrm{A}\\!:\\!\mathrm{D}^{2}u=f\in\mathrm{L}^{\infty}(\Omega)$. This in
turn yields that $u\in\mathcal{W}^{2,\infty}_{0}(\Omega)$, as claimed. ∎

######  Remark 10.

The space $\mathcal{W}^{2,\infty}_{0}(\Omega)$ is not closed in the relative
Fréchet topology of $\mathbb{W}^{2,\infty}_{0}(\Omega)$. Indeed, fix any
function
$u\in\mathbb{W}^{2,\infty}_{c}(\Omega)\setminus\mathcal{W}^{2,\infty}(\Omega)$,
and let
$u^{\varepsilon}:=u*\eta^{\varepsilon}\in\mathrm{C}^{\infty}_{c}(\Omega)\subseteq\mathcal{W}_{0}^{2,\infty}(\Omega)$
be the standard mollifier, for $\varepsilon>0$ small (see e.g. [35, Ch. 9.6]).
Then, $u^{\varepsilon}\longrightarrow u$ in $\mathrm{W}^{2,p}_{0}(\Omega)$ as
$\varepsilon\to 0$ for any fixed $p\in(1,\infty)$, but
$u\not\in\mathcal{W}^{2,\infty}_{0}(\Omega)$.

####  2.3.2. Identification of the predual of
$\mathcal{W}^{2,\infty}_{0}(\Omega)$ and weak* approximation

The next important functional properties have their own independent interest,
but will also be utilised in Section 3.

######  Proposition 11.

_(i)_ The space $\mathcal{W}^{2,\infty}_{0}(\Omega)$ is a dual Banach space,
and in particular

$\mathcal{W}^{2,\infty}_{0}(\Omega)\cong\big{(}\mathbb{A}(\mathrm{L}^{1}(\Omega))\big{)}^{*},$

with the duality pairing given by

$\ \ \ \ \ \ \ \
\langle\psi,\mathbb{A}u\rangle:=\int_{\Omega}u(\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi)\,\mathrm{d}\mathcal{L}^{n},\
\ \ \psi\in\mathcal{W}^{2,\infty}_{0}(\Omega),\,u\in\mathrm{L}^{1}(\Omega).$

In the above,
$\mathbb{A}:\mathrm{L}^{1}(\Omega)\longrightarrow(\mathcal{W}^{2,\infty}_{0}(\Omega))^{*}$
denotes the bounded linear operator given by
$u\mapsto\langle\cdot,\mathbb{A}u\rangle$, and we endow
$\mathbb{A}(\mathrm{L}^{1}(\Omega))$ with the natural norm induced by
inclusion into $(\mathcal{W}^{2,\infty}_{0}(\Omega))^{*}$, for which it
becomes a separable Banach space, namely

$\|\mathbb{A}u\|:=\sup\bigg{\\{}\int_{\Omega}u(\mathrm{A}\\!:\\!\mathrm{D}^{2}\phi)\,\mathrm{d}\mathcal{L}^{n}\
\Big{|}\ \phi\in\mathcal{W}^{2,\infty}_{0}(\Omega),\
\|\phi\|_{\mathcal{W}^{2,\infty}_{0}(\Omega)}\leq 1\bigg{\\}}.$

_(ii)_ The space $\mathrm{C}^{\infty}_{c}(\Omega)$ is sequentially
weakly*-dense in $\mathcal{W}^{2,\infty}_{0}(\Omega)$. Namely, for any
$\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)$, there exists
$(\psi_{m})_{m=1}^{\infty}\subseteq\mathrm{C}^{\infty}_{c}(\Omega)$ such that,
as $m\to\infty$ we have

(2.14)
$\displaystyle\phantom{aaaaaa}\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi_{m}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi,$
$\displaystyle\text{ in
$\mathrm{L}^{\infty}(\Omega)$},\phantom{aaaaaaaaaaaaaa}$ (2.15)
$\displaystyle\phantom{aaaaaa}\psi_{m}\\!\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup\psi,$
$\displaystyle\text{ in $\mathrm{W}_{0}^{2,p}(\Omega)$, $\forall\
p\in(1,\infty)$}.\phantom{aaaaaaaaaaaaaa}$

######  Remark 12.

Using similar techniques to those used to establish Proposition 11, one can
also show that

$\big{(}\mathcal{W}^{2,\infty}_{0}(\Omega)\big{)}^{*}\cong\mathbb{A}(\mathrm{ba}(\Omega)),$

where
$\mathrm{ba}(\Omega)\cong(\mathrm{L}^{\infty}(\Omega))^{*}\cong(\mathrm{L}^{1}(\Omega))^{**}$
is the space of real-valued bounded finitely additive measures, which are
absolutely continuous with respect to the Lebesgue measure on $\Omega$ (see
e.g. [24, 50]). However, the identification of the dual space of
$\mathcal{W}^{2,\infty}_{0}(\Omega)$ is not required for the main results,
therefore we refrain from this task.

The proof of Proposition 11 is based on the following general result of
independent interest about subspaces of dual Banach spaces (Lemma 13). This is
perhaps well-known to the experts of duality theory, but since we could not
locate a specific reference of this fact, we do provide a proof for the sake
of completeness.

######  Lemma 13.

Let $(\mathfrak{X},\|\cdot\|)$ be a Banach space, and let
$(\mathfrak{X}^{*},\|\cdot\|_{*})$ be its dual Banach space. If
$\mathfrak{M}\subseteq\mathfrak{X}^{*}$ is any weakly*-closed subspace, then
$\mathfrak{M}$ is a dual Banach space itself, and in particular

$(\mathfrak{X}/_{{}^{\bot}\mathfrak{M}})^{*}\cong\,\mathfrak{M}.$

The isometric isomorphism above is given by $\phi\mapsto\phi\circ\pi$, where
$\pi:\mathfrak{X}\longrightarrow\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}$ is the
quotient map, and ${}^{\bot}\mathfrak{M}$ denotes the pre-annihilator of
$\mathfrak{M}$ in $\mathfrak{X}$ (Subsection 2.1).

Proof of Lemma 13. We endow $\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}$ with the
quotient norm, and the spaces $\mathfrak{X}^{*}$ and
$(\mathfrak{X}/_{{}^{\bot}\mathfrak{M}})^{*}$ with their natural dual norms,
and define the mapping

$\ \ \ \pi^{*}\ :\ \
(\mathfrak{X}/_{{}^{\bot}\mathfrak{M}})^{*}\longrightarrow\mathfrak{X}^{*},\ \
\ \pi^{*}(\phi):=\phi\circ\pi.$

Then, $\pi^{*}$ is linear and well-defined, as for any $x\in\mathfrak{X}$, we
have

$\big{|}\big{(}\pi^{*}(\phi)\big{)}(x)\big{|}=|\phi(\pi(x))|\leq\|\phi\|_{(\mathfrak{X}/_{{}^{\bot}\mathfrak{M}})^{*}}\|\pi(x)\|_{\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}}\leq\|\phi\|_{(\mathfrak{X}/_{{}^{\bot}\mathfrak{M}})^{*}}\|x\|.$

Hence, $\pi^{*}(\phi)\in\mathfrak{X}^{*}$. We will now show that

(2.16)
$\pi\big{(}\mathbb{B}^{\mathfrak{X}}\big{)}=\mathbb{B}^{\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}}.$

(This fact is claimed without proof in [48, p. 97].) On the one hand, the
inclusion
$\pi(\mathbb{B}^{\mathfrak{X}})\subseteq\mathbb{B}^{\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}}$
is an immediate consequence of the inequality
$\|\pi(x)\|_{\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}}\leq\|x\|$, which is valid
for all $x\in\mathfrak{X}$. Conversely, if we have any
$\pi(x)\in\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}$ which satisfies
$\|\pi(x)\|_{\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}}<1$, for any
$\varepsilon>0$ small enough there exists
${m_{\varepsilon}\in}\,^{\bot}\mathfrak{M}$ such that
$\|x-m_{\varepsilon}\|<\|\pi(x)\|_{\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}}+\varepsilon<1$.
Since $\pi(x-m_{\varepsilon})=\pi(x)$, it follows that there exists a
representative $x^{\prime}:=x-m_{\varepsilon}$ such that
$\pi(x)=\pi(x^{\prime})\in\pi\big{(}\mathbb{B}^{\mathfrak{X}}\big{)}$ and
$\|x^{\prime}\|<1$. This proves the reverse inclusion
$\smash{\mathbb{B}^{\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}}\subseteq\pi(\mathbb{B}^{\mathfrak{X}})}$,
therefore establishing (2.16). We will now show that $\pi^{*}$ is an isometry.
In view of (2.16), we have

$\begin{split}\|\pi^{*}(\phi)\|_{*}&=\sup_{x\in\mathfrak{X},\|x\|<1}|\phi(\pi(x))|=\sup_{\pi(x)\in\mathfrak{X}/_{{}^{\bot}\\!M},\|\pi(x)\|_{\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}}<1}|\phi(\pi(x))|=\|\phi\|_{(\mathfrak{X}/_{{}^{\bot}\mathfrak{M}})^{*}}.\end{split}$

We will now show that the image of $\pi^{*}$ lies in $\mathfrak{M}$, namely

$\pi^{*}\big{(}\big{(}\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}\big{)}^{*}\big{)}\subseteq\mathfrak{M}.$

For any fixed $\phi\in(\mathfrak{X}/_{{}^{\bot}\mathfrak{M}})^{*}$, we have
that $(\pi^{*}(\phi))(x)=\phi(\pi(x))=0$, for all ${x\in}^{\bot}\mathfrak{M}$,
because $\pi(x)=0$ when ${x\in}^{\bot}\mathfrak{M}$. This implies that
$\pi^{*}(\phi)\in(^{\bot}\mathfrak{M})^{\bot}$. Since $\mathfrak{M}$ is by
assumption a weakly* closed subspace of $\mathfrak{X}^{*}$, by [48, Th. 4.7,
p. 96], we have that $(^{\bot}\mathfrak{M})^{\bot}=\mathfrak{M}$. This implies
the claimed inclusion. We finally show that $\pi^{*}$ is onto $\mathfrak{M}$.
To this aim, note that any functional $f\in\mathfrak{M}$ induces a functional
$\bar{f}\in(\mathfrak{X}/_{{}^{\bot}\mathfrak{M}})^{*}$ in a natural way, by
setting $\bar{f}(\pi(x)):=f(x)$, $x\in\mathfrak{X}$. This is well-defined
regardless of the choice of representative, because if we have
$\pi(x^{\prime\prime})=\pi(x^{\prime})$, then
$x^{\prime\prime}-x^{\prime}\in{{}^{\bot}\mathfrak{M}}$ and hence
$f(x^{\prime\prime})=f(x^{\prime})$, because $f|_{{}^{\bot}\mathfrak{M}}\equiv
0$ when $f\in\mathfrak{M}$. Further, by the identity

$|\bar{f}(\pi(x))|=|f(x)|=|f(x-z)|,\ \ \ \forall\
{z\in}{{}^{\bot}\mathfrak{M}},$

for any $x\in\mathfrak{X}$, we estimate

$|\bar{f}(\pi(x))|\leq\|f\|_{*}\inf_{{z\in}{{}^{\bot}\mathfrak{M}}}\|x-z\|=\|f\|_{*}\|\pi(x)\|_{\mathfrak{X}/_{{}^{\bot}\mathfrak{M}}}.$

This shows that $\bar{f}\in(\mathfrak{X}/_{{}^{\bot}\mathfrak{M}})^{*}$ and
$\pi^{*}(\bar{f})=f$, completing the proof. ∎

######  Remark 14.

The assumption that the subspace $\mathfrak{M}\subseteq\mathfrak{X}^{*}$ be
weakly*-closed is essential, and in general it cannot be replaced by norm-
closedness. For example, the space of sequences $c_{0}(\mathbb{N})$ vanishing
at infinity is norm-closed (and weakly*-dense) in $\ell^{\infty}(\mathbb{N})$,
but as it is well known, it possesses no predual space.

Having procured the necessary tools, we main now establish Proposition 11.

Proof of Proposition 11. (i) We begin by noting that $\mathrm{L}^{1}(\Omega)$
is a separable Banach space and $\mathbb{A}$ is a bounded linear operator,
because for any $u\in\mathrm{L}^{1}(\Omega)$, we have

$\begin{split}\|\mathbb{A}u\|=\sup_{\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega),\|\psi\|_{\mathcal{W}^{2,\infty}_{0}(\Omega)}\leq
1}\int_{\Omega}u(\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi)\,\mathrm{d}\mathcal{L}^{n}\leq\|u\|_{\mathrm{L}^{1}(\Omega)}.\end{split}$

Hence, the image $\mathbb{A}(\mathrm{L}^{1}(\Omega))$ is a separable subset of
$(\mathcal{W}^{2,\infty}_{0}(\Omega))^{*}$. Thus,
$\mathbb{A}(\mathrm{L}^{1}(\Omega))$ becomes a separable normed space when
endowed with the dual norm induced by
$(\mathcal{W}^{2,\infty}_{0}(\Omega))^{*}$. Now we show that it is complete,
and hence a Banach space, by showing that it is isometrically isomorphic to a
quotient of $\mathrm{L}^{1}(\Omega)$ by a closed subspace. We define

$\mathcal{W}:=\Big{\\{}f\in\mathrm{L}^{\infty}(\Omega)\ \Big{|}\ \exists\
\phi\in\mathcal{W}^{2,\infty}_{0}(\Omega)\,:\,f=\mathrm{A}\\!:\\!\mathrm{D}^{2}\phi\text{
a.e.\ on
}\Omega\Big{\\}}=\mathbb{A}\big{(}\mathcal{W}^{2,\infty}_{0}(\Omega)\big{)}.$

Then, by arguing as in the proof of Lemma 9, we have that $\mathcal{W}$ is
sequentially weakly* closed in $\mathrm{L}^{\infty}(\Omega)$, and therefore it
is also weakly* closed, because its predual $\mathrm{L}^{1}(\Omega)$ is
separable and complete, as a consequence of the Krein-Smulian theorem (we
refer e.g. [18, Sec. 12, p. 159] or [43]). By applying Lemma 13, it follows
that
$\pi^{*}:\mathcal{W}\longrightarrow\left(\mathrm{L}^{1}(\Omega)/_{\\!{}^{\bot}{\mathcal{W}}}\right)^{*}$
is an isometric isomorphism, and we also have that

$\\!{}^{\bot}{\mathcal{W}}=\bigg{\\{}u\in\mathrm{L}^{1}(\Omega):\int_{\Omega}u(\mathrm{A}\\!:\\!\mathrm{D}^{2}\phi)\,\mathrm{d}\mathcal{L}^{n}=0,\
\forall\
\phi\in\mathcal{W}^{2,\infty}_{0}(\Omega)\bigg{\\}}=\mathrm{N}(\mathbb{A}).$

Hence,
$\mathcal{W}\cong(\mathrm{L}^{1}(\Omega)/_{\mathrm{N}(\mathbb{A})})^{*}$.
Next, note that the induced map on the quotient space

$\mathbb{A}\ :\ \
\mathrm{L}^{1}(\Omega)/_{\mathrm{N}(\mathbb{A})}\longrightarrow\mathbb{A}(\mathrm{L}^{1}(\Omega)),\
\ \ \ \mathbb{A}(u+\mathrm{N}(\mathbb{A})):=\mathbb{A}u,$

is a well-defined linear bijection, independent of the representative used in
the coset. Further, for any fixed $u\in\mathrm{L}^{1}(\Omega)$, by standard
duality results (see e.g. [35, Corollary 11.46]) and by the expression of the
quotient norm on $\mathrm{L}^{1}(\Omega)/_{\mathrm{N}(\mathbb{A})}$, it
follows that

$\begin{split}\|\mathbb{A}u\|&=\sup_{\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega),\,\|\mathrm{A}:\mathrm{D}^{2}\psi\|_{\mathrm{L}^{\infty}(\Omega)}\leq
1}\int_{\Omega}u(\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi)\,\mathrm{d}\mathcal{L}^{n}\\\
&=\sup_{f\in\mathcal{W},\,\|f\|_{\mathrm{L}^{\infty}(\Omega)}\leq
1}\int_{\Omega}uf\,\mathrm{d}\mathcal{L}^{n}.\end{split}$

Hence,

$\begin{split}\|\mathbb{A}u\|&=\sup_{f\in(\mathrm{L}^{1}(\Omega)/_{\mathrm{N}(\mathbb{A})})^{*},\,\|f\|_{(\mathrm{L}^{1}(\Omega)/_{\mathrm{N}(\mathbb{A})})^{*}}\leq
1}\int_{\Omega}uf\,\mathrm{d}\mathcal{L}^{n}\\\
&=\big{\|}u+\mathrm{N}(\mathbb{A})\big{\|}_{\mathrm{L}^{1}(\Omega)/_{\mathrm{N}(\mathbb{A})}}.\end{split}$

Thus, the mapping
$\mathbb{A}:\mathrm{L}^{1}(\Omega)/_{\mathrm{N}(\mathbb{A})}\longrightarrow\mathbb{A}(\mathrm{L}^{1}(\Omega))$
is an isometric isomorphism, and therefore
$\mathbb{A}(\mathrm{L}^{1}(\Omega))$ is a Banach space. It follows that the
induced dual map

$\mathbb{A}^{*}:\ \
\big{(}\mathbb{A}(\mathrm{L}^{1}(\Omega))\big{)}^{*}\longrightarrow\big{(}\mathrm{L}^{1}(\Omega)/_{\mathrm{N}(\mathbb{A})}\big{)}^{*},$

given by

$\ \ \ \
\langle\mathbb{A}^{*}(\psi),u\rangle:=\langle\psi,\mathbb{A}u\rangle,\ \ \ \
u\in\mathrm{L}^{1}(\Omega),$

is also an isometric isomorphism between the corresponding dual spaces.
Further, the linear map

$\mathbb{A}\ :\ \
\mathcal{W}^{2,\infty}_{0}(\Omega)\longrightarrow\mathcal{W},\ \ \
\phi\mapsto\mathbb{A}\phi,$

is an isometric isomorphism between Banach spaces, since
$\|\mathbb{A}\phi\|_{\mathrm{L}^{\infty}(\Omega)}=\|\phi\|_{\mathcal{W}^{2,\infty}_{0}(\Omega)}$.
In conclusion, we have the following isometric isomorphism

$(\mathbb{A}^{*})^{-1}\circ\pi^{*}\circ\mathbb{A}\ :\ \
\mathcal{W}^{2,\infty}_{0}(\Omega)\longrightarrow\big{(}\mathbb{A}(\mathrm{L}^{1}(\Omega))\big{)}^{*},$

which, in view of the expression that the induced duality pairing takes,
establishes the desired conclusion.

(ii) We will use the regularisation scheme introduced and studied in [30, Sec.
5], whose definition we recall for the convenience of the reader. Since
$\partial\Omega$ is assumed to be Lipschitz, there exists a smooth vector
field $\xi\in\mathrm{C}^{\infty}_{c}(\mathbb{R}^{n};\mathbb{R}^{n})$ which is
transversal to the boundary $\partial\Omega$, namely
$\xi\cdot\mathrm{n}\geq\delta_{0}$ which holds $\mathcal{H}^{n-1}$-a.e. on
$\partial\Omega$, for some $\delta_{0}>0$, where
$\mathrm{n}:\partial\Omega\longrightarrow\mathbb{S}$ is the unit outer normal
vector field, and additionally we have $|\xi|\equiv 1$ in an open collar
around the boundary $\partial\Omega$. Further, there exist
$\ell,\varepsilon_{0}>0$ such that, for all
$\varepsilon\in(0,\varepsilon_{0})$, we have
$\mathrm{dist}(x+\varepsilon\ell\xi(x),\partial\Omega)\geq 2\varepsilon$, for
$x\in\partial\Omega$. Then, for any
$\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)$, extended by zero on
$\mathbb{R}^{n}\setminus\Omega$, we set

$\mathrm{K}_{\varepsilon}\psi(x):=\int_{\mathbb{R}^{n}}\psi\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\eta(y)\,\mathrm{d}y,$

where $\varepsilon\in(0,\varepsilon_{0})$ and
$\eta\in\mathrm{C}^{\infty}_{c}(\mathbb{B})$ is a fixed mollifying kernel
which satisfies $\eta\geq 0$ and
$\|\eta\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}=1$. This is a variant of the
standard mollifier, the difference being that it “compresses” $\psi$ to a
compactly supported function in $\Omega$ before mollifying. This method gives
rise to compactly supported smooth approximations, which converge globally on
$\Omega$. In [30, Prop. 12] it is shown that
$\mathrm{K}_{\varepsilon}v\in\mathrm{C}^{\infty}_{c}(\Omega)$ and that
$\mathrm{K}_{\varepsilon}\psi\longrightarrow\psi$ in
$\mathrm{W}^{1,q}_{0}(\Omega)$ as $\varepsilon\to 0$, for any
$q\in(1,\infty)$. By iterating this result, we also have that
$\mathrm{K}_{\varepsilon}\psi\longrightarrow\psi$ in
$\mathrm{W}^{2,q}_{0}(\Omega)$ as $\varepsilon\to 0$, for any
$q\in(1,\infty)$. Hence, it remains to show that
${\\{\mathrm{A}\\!:\\!\mathrm{D}^{2}(\mathrm{K}_{\varepsilon}\psi)\\}_{\varepsilon\in(0,\varepsilon_{0})}}$
is bounded in $\mathrm{L}^{\infty}(\Omega)$. To this aim, we compute

$\begin{split}\mathrm{A}\\!:\\!\mathrm{D}^{2}(\mathrm{K}_{\varepsilon}\psi)(x)=&\
\big{(}\mathrm{I}+\varepsilon\ell\mathrm{D}\xi(x)\big{)}^{\\!\\!\top}\\!\mathrm{A}\big{(}\mathrm{I}+\varepsilon\ell\mathrm{D}\xi(x)\big{)}\\\
&\
:\int_{\mathbb{R}^{n}}\mathrm{D}^{2}\psi\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\eta(y)\,\mathrm{d}y,\\\ &\
+\varepsilon\ell\big{(}\mathrm{A}\\!:\\!\mathrm{D}^{2}\xi(x)\big{)}\cdot\int_{\mathbb{R}^{n}}\mathrm{D}\psi\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\eta(y)\,\mathrm{d}y,\end{split}$

for any $x\in\Omega$. Hence,

$\begin{split}\mathrm{A}\\!:\\!\mathrm{D}^{2}(\mathrm{K}_{\varepsilon}\psi)(x)=\,&\int_{\mathbb{R}^{n}}\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\eta(y)\,\mathrm{d}y\\\
&+\,\varepsilon\ell\Big{(}\mathrm{D}\xi(x)^{\top}\\!\mathrm{A}+\mathrm{A}\mathrm{D}\xi(x)+\varepsilon\ell\mathrm{D}\xi(x)^{\top}\\!\mathrm{A}\mathrm{D}\xi\Big{)}\\\
&\
:\int_{\mathbb{R}^{n}}\mathrm{D}^{2}\psi\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\eta(y)\,\mathrm{d}y\\\
&+\varepsilon\ell\big{(}\mathrm{A}\\!:\\!\mathrm{D}^{2}\xi(x)\big{)}\cdot\int_{\mathbb{R}^{n}}\mathrm{D}\psi\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\eta(y)\,\mathrm{d}y,\end{split}$

for any fixed $x\in\Omega$. Since
$\eta\in\mathrm{C}^{\infty}_{c}(\mathbb{B})$, an integration by parts yields

$\begin{split}\mathrm{A}\\!:\\!\mathrm{D}^{2}(\mathrm{K}_{\varepsilon}\psi)(x)=&\,\int_{\mathbb{R}^{n}}\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\eta(y)\,\mathrm{d}y\\\
&-\,\ell\Big{(}\mathrm{D}\xi(x)^{\top}\\!\mathrm{A}+\mathrm{A}\mathrm{D}\xi(x)+\varepsilon\ell\mathrm{D}\xi(x)^{\top}\\!A\mathrm{D}\xi\Big{)}\\\
&\ :\int_{\mathbb{B}}\mathrm{D}\psi\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\cdot\mathrm{D}\eta(y)\,\mathrm{d}y\\\
&+\varepsilon\ell\big{(}\mathrm{A}\\!:\\!\mathrm{D}^{2}\xi(x)\big{)}\cdot\int_{\mathbb{R}^{n}}\mathrm{D}\psi\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\eta(y)\,\mathrm{d}y,\end{split}$

for any $x\in\Omega$. Since $\eta,|\mathrm{D}\eta|$ are in
$\mathrm{L}^{1}(\mathbb{B})$, by Young’s inequality, we get the estimate

$\begin{split}\big{|}\mathrm{A}\\!:\\!\mathrm{D}^{2}(\mathrm{K}_{\varepsilon}\psi)(x)\big{|}\leq&\
\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi\|_{\mathrm{L}^{\infty}(\Omega)}\,+\,\varepsilon_{0}\ell|\mathrm{A}|\|\mathrm{D}^{2}\xi\|_{\mathrm{L}^{\infty}(\mathbb{R}^{n})}\|\mathrm{D}\psi\|_{\mathrm{L}^{\infty}(\Omega)}\\\
&+\,\ell|\mathrm{A}|\|\mathrm{D}\xi\|_{\mathrm{L}^{\infty}(\mathbb{R}^{n})}\big{(}2+\varepsilon_{0}\ell\|\mathrm{D}\xi\|_{\mathrm{L}^{\infty}(\mathbb{R}^{n})}\big{)}\|\mathrm{D}\psi\|_{\mathrm{L}^{\infty}(\Omega)}\|\mathrm{D}\eta\|_{\mathrm{L}^{1}(\mathbb{R}^{n})},\end{split}$

for any $x\in\Omega$. This implies that
$\mathrm{A}\\!:\\!\mathrm{D}^{2}(\mathrm{K}_{\varepsilon}\psi)$ is indeed
bounded in $\mathrm{L}^{\infty}(\Omega)$ uniformly in
$\varepsilon\in(0,\varepsilon_{0})$. The conclusion ensues. ∎

###  2.4. Compact sets in $\mathcal{W}^{2,\infty}_{0}(\Omega)$ and examples

By using the analogue of the Ascoli-Arzela theorem in
$\mathrm{L}^{\infty}(\Omega)$ (see [15]), and the results established in
Subsection 2.3, we can explicitly characterise compact sets
$\mathcal{K}\subseteq\mathcal{W}^{2,\infty}_{0}(\Omega)$ (arising in Theorem
2). To state this concisely, we need some additional notation. For any
$f\in\mathrm{L}^{\infty}(\Omega)$, the _essential oscillation_ of $f$ over a
Lebesgue measurable set $E\subseteq\Omega$ is defined as

$\underset{E}{\mathrm{ess}\,\mathrm{osc}}\,[f]\,:=\,\underset{x,y\in
E}{\mathrm{ess}\sup}\big{|}f(x)-f(y)\big{|}.$

Then, a closed subset $\mathcal{K}\subseteq\mathcal{W}^{2,\infty}_{0}(\Omega)$
is (strongly) compact if and only if

(2.17) $\inf_{\big{\\{}\\{\Omega_{i}\\}_{i\in I}\text{ finite Borel partition
of }\Omega\big{\\}}}\sup_{f\in\mathcal{K}}\max_{i\in
I}\underset{\Omega_{i}}{\mathrm{ess}\,\mathrm{osc}}\,[\mathrm{A}\\!:\\!\mathrm{D}^{2}f]=0.\
\ \ \ \ $

The seemingly complicated condition (2.17) is equivalent to the following
statement: for any $\varepsilon>0$, there exists a finite partition of
$\Omega$ to Borel sets $\\{\Omega_{i}\\}_{i\in I}$, such that

$\big{|}\mathrm{A}\\!:\\!\mathrm{D}^{2}f(x)-\mathrm{A}\\!:\\!\mathrm{D}^{2}f(y)\big{|}\,\leq\,\varepsilon,\
\ \ \text{a.e.\ }x,y\in\Omega_{i},$

for all functions $f\in\mathcal{K}$ and all indices $i\in I$. By utilising the
above, we have the following immediate consequence of Theorem 2 regarding
interesting compact sets of variations.

######  Corollary 15 (Examples of compact sets of directions of test functions
in $\mathcal{W}^{2,\infty}_{0}(\Omega)$).

In the setting of Theorem 2 and under the same assumptions, the following are
true:

_(i)_ For any $\theta>0$ and $p>n$, there exists $\rho>0$ such that

$\mathrm{F}_{\infty}(u_{\infty})<\mathrm{F}_{\infty}(u_{\infty}+\psi),$

among all
$\psi\in\big{(}\mathrm{W}^{3,p}\cap\mathcal{W}^{2,\infty}_{0}\big{)}(\Omega)\setminus\\{0\\}$
satisfying $\|\psi\|_{\mathcal{W}^{2,\infty}_{0}(\Omega)}<\rho$,
$\|\mathrm{D}^{3}\psi\|_{\mathrm{L}^{p}(\Omega)}<\theta\rho$.

_(ii)_ Let $\\{\Omega_{i}:i\in I\\}$ be a finite family of disjoint open
subsets of $\Omega$ with Lipschitz boundaries $\partial\Omega_{i}$, whose
closures cover $\Omega$. Then, for any $\theta>0$, there exists $\rho>0$ such
that

$\mathrm{F}_{\infty}(u_{\infty})<\mathrm{F}_{\infty}(u_{\infty}+\psi),$

among all $\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)\setminus\\{0\\}$ having
piecewise Hölder continuous Hessians (namely
$\psi|_{\Omega_{i}}\in\mathrm{C}^{2,\alpha}(\overline{\Omega}_{i})$ for all
$i\in I$), satisfying $\|\psi\|_{\mathcal{W}^{2,\infty}_{0}(\Omega)}<\rho$,
$\underset{i\in
I}{\max}\,[\mathrm{D}^{2}\psi]_{\mathrm{C}^{\alpha}(\overline{\Omega}_{i})}<\theta\rho$.

Proof of Corollary 15. (i) Given $\theta>0$ and $p>n$, we define

$\begin{split}\mathcal{K}^{\prime}&\,:=\,\Big{\\{}\psi\in\mathbb{S}^{\mathcal{W}^{2,\infty}_{0}(\Omega)}\cap\mathrm{W}^{3,p}(\Omega)\,:\
\|\mathrm{D}^{3}\psi\|_{\mathrm{L}^{q}(\Omega)}\leq\theta\Big{\\}}.\end{split}$

In view of Theorem 2 (that will be established in Section 3), it suffices to
show that $\mathcal{K}^{\prime}$ is a (strongly) compact set in
$\mathcal{W}^{2,\infty}_{0}(\Omega)$. Let $(\psi_{m})_{1}^{\infty}$ be a
sequence in $\mathcal{K}^{\prime}$. Then, we have
$\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi_{m}\|_{\mathrm{L}^{\infty}(\Omega)}=1$
and $\|\mathrm{D}^{3}\psi_{m}\|_{\mathrm{L}^{q}(\Omega)}\leq\theta$, for all
$m\in\mathbb{N}$. By the compactness of the imbedding
$\mathrm{W}^{3,p}(\Omega)\Subset\mathrm{W}^{2,\infty}(\Omega)$, there exists
$\psi\in\mathrm{W}^{3,p}(\Omega)$ such that $\psi_{m_{k}}\longrightarrow\psi$
in $\mathrm{W}^{2,\infty}(\Omega)$ along a subsequence as $k\to\infty$. In
particular, $\psi_{m_{k}}\longrightarrow\psi$ in
$\mathcal{W}^{2,\infty}_{0}(\Omega)$, as $k\to\infty$. Since
$\mathrm{D}^{3}\psi_{m_{k}}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup\mathrm{D}^{3}\psi$
in $\mathrm{L}^{q}(\Omega;\mathbb{R}^{n^{3}})$ as well, the weak lower
semicontinuity of $\|\cdot\|_{\mathrm{L}^{q}(\Omega)}$ implies
$\psi\in\mathcal{K}^{\prime}$.

(ii) Given $\theta>0$ and finitely many disjoint open sets $\\{\Omega_{i}:i\in
I\\}$ which have Lip- schitz boundaries
$\partial\Omega_{i}\in\mathrm{C}^{0,1}$, and whose closures cover $\Omega$, we
define the set

$\begin{split}\mathcal{K}^{\prime\prime}&\,:=\,\mathbb{S}^{\mathcal{W}^{2,\infty}_{0}(\Omega)}\bigcap\Big{\\{}\psi\in\underset{i\in
I}{\oplus}\mathrm{C}^{2,\alpha}(\overline{\Omega}_{i})\,:\ \max_{i\in
I}\,[\mathrm{D}^{2}\psi]_{\mathrm{C}^{\alpha}(\overline{\Omega}_{i})}\leq\theta\Big{\\}}.\end{split}$

It is then easy to see that $\mathcal{K}^{\prime\prime}$ is a (strongly)
compact set in $\mathcal{W}^{2,\infty}_{0}(\Omega)$ (the proof follows similar
lines to those of part (i), and therefore the details are omitted). ∎

###  2.5. $\mathrm{L}^{p}$-limits of $\mathrm{L}^{\infty}$ constrained
minimisers

The proofs of Theorems 4 and 7 are based on the following standalone result of
independent interest, which establishes the existence and
$\mathrm{L}^{p}$-approximation of narrow (and global) minimisers of (1.1)
through penalised functionals, treated in a unified fashion as constrained
minimisers.

######  Theorem 16 (Existence and $\mathrm{L}^{p}$-approximation of
$\mathrm{L}^{\infty}$ constrained minimisers).

Let $F$ satisfy (1.6)-(1.7) and (1.15), and let
$\bar{u}\in\mathcal{W}^{2,\infty}(\Omega)$ be fixed. Suppose that
$\\{\mathcal{E}^{p}_{\bar{u}}(\Omega)\\}_{n\leq p\leq\infty}$ are classes of
functions, which satisfy:

(2.18) $\left\\{\ \
\begin{split}&\mathcal{E}^{p}_{\bar{u}}(\Omega)\subseteq\mathrm{W}^{2,p}_{\bar{u}}(\Omega),\text{
when }\,n\leq p<\infty,\\\ &\mathcal{E}^{p}_{\bar{u}}(\Omega)\text{ is weakly
closed, when }\,p<\infty,\\\
&\mathcal{E}^{p}_{\bar{u}}(\Omega)\subseteq\mathcal{E}^{q}_{\bar{u}}(\Omega),\text{
for }n\leq q\leq p\leq\infty,\\\
&\mathcal{E}^{\infty}_{\bar{u}}(\Omega)\subseteq\mathcal{W}^{2,\infty}_{\bar{u}}(\Omega),\text{
when }\,p=\infty,\\\
&\mathcal{E}^{\infty}_{\bar{u}}(\Omega)\,=\bigcap_{n<p<\infty}\mathcal{E}^{p}_{\bar{u}}(\Omega),\\\
&\bar{u}\in\mathcal{E}^{\infty}_{\bar{u}}(\Omega).\end{split}\right.$

Then, the following are true:

_(i)_ For any $\epsilon\geq 0$ and $p\geq n+1/c^{3}$, the functional
$\mathrm{F}_{p,\epsilon,\bar{u}}$ (defined in (1.19)) has a minimiser $u_{p}$
over the admissible class $\mathcal{E}^{p}_{\bar{u}}(\Omega)$.

_(ii)_ For any $\epsilon\geq 0$, the functional
$\mathrm{F}_{\infty}+\frac{\epsilon}{2}\|\cdot-\bar{u}\|^{2}_{\mathrm{L}^{2}(\Omega)}$
has a minimiser $u_{\infty}$ over the admissible class
$\mathcal{E}^{\infty}_{\bar{u}}(\Omega)$, where $\mathrm{F}_{\infty}$ is as in
(1.1). Additionally, there exists a sequence of indices $(p_{j})_{1}^{\infty}$
satisfying $p_{j}\to\infty$ as $j\to\infty$, along which we have

(2.19) $\left\\{\ \
\begin{split}&\mathrm{F}_{p,\epsilon,\bar{u}}(u_{p})\longrightarrow\mathrm{F}_{\infty}(u_{\infty})+\frac{\epsilon}{2}\|u_{\infty}-\bar{u}\|^{2}_{\mathrm{L}^{2}(\Omega)},\\\
&u_{p}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{\infty},\hskip
20.0pt\text{ in }W^{2,q}(\Omega),\ \forall\ q\in(1,\infty),\\\
&u_{p}\longrightarrow u_{\infty},\hskip 21.0pt\text{ in
}\mathrm{C}^{1,\gamma}(\overline{\Omega}),\ \forall\
\gamma\in(0,1).\end{split}\right.$

_(iii)_ If finally $\bar{u}$ minimises $\mathrm{F}_{\infty}$ over the
admissible class $\mathcal{E}^{\infty}_{\bar{u}}(\Omega)$ and also
$\epsilon>0$, then we have that $\bar{u}=u_{\infty}$. In this case, we also
have that the modes of convergence in (2.19) are in fact full as $p\to\infty$.

The main technical step for the proof of Theorem 16 is the following
coercivity estimate for (1.19).

######  Lemma 17 (Coercivity).

Let $\delta>0$ and let $w\in\mathcal{E}_{\bar{u}}^{p}(\Omega)$ be a
$\delta$-approximate minimiser of $\mathrm{F}_{p,\epsilon,\bar{u}}$ in
$\mathcal{E}_{\bar{u}}^{p}(\Omega)$, namely

$\mathrm{F}_{p,\epsilon,\bar{u}}(w)\,\leq\,\delta+\inf_{\mathcal{E}_{\bar{u}}^{p}(\Omega)}\mathrm{F}_{p,\epsilon,\bar{u}}.$

Then, for any $q\in[n,p]$, we have the estimates

(2.20) $\left\\{\ \
\begin{split}\|w\|_{\mathrm{W}^{2,q}(\Omega)}\,&\leq\,\frac{K(\Omega,q)}{c(1-\alpha)}\Big{(}M+\delta+\mathrm{F}_{\infty}(\bar{u})\Big{)},\
\ \ &\mathrm{(i)}\\\ \ \ \
\|\mathrm{A}\\!:\\!\mathrm{D}^{2}w\|_{\mathrm{L}^{q}(\Omega)}\,&\leq\,\frac{1}{c}\Big{(}1+\delta+\mathrm{F}_{\infty}(\bar{u})+\|w\|_{\mathrm{W}^{1,q}(\Omega)}^{\alpha}\Big{)},\
\ \ &\mathrm{(ii)}\end{split}\right.$

where $K(\Omega,q)>0$ is the constant appearing in Lemma 9, and $M>0$ is a
constant depending only on $\bar{u},\alpha,c,K(q,\Omega),\mathrm{A}$.

Proof of Lemma 17. Note first that by (1.1), (1.19), minimality and Young’s
inequality, we have

$\inf_{\mathcal{E}_{\bar{u}}^{p}(\Omega)}\mathrm{F}_{p,\epsilon,\bar{u}}\leq\mathrm{F}_{p,\epsilon,\bar{u}}(\bar{u})\leq\mathrm{F}_{\infty}(\bar{u}).$

Further, by (1.7) we have that $F(x,\eta,\mathrm{p},\cdot)^{-1}(\\{0\\})$ is a
singleton set. Hence, if we have that if we symbolise

(2.21)
$F(x,\eta,\mathrm{p},\cdot)^{-1}(\\{0\\})\,=\,\big{\\{}\bar{\xi}(x,\eta,\mathrm{p})\big{\\}},$

again by (1.7), we can easily deduce the inequality

(2.22) $|F(x,\eta,\mathrm{p},\xi)|\geq
c\big{|}\xi-\bar{\xi}(x,\eta,\mathrm{p})\big{|}.$

By Lemma 9 and (1.15), since $w-\bar{u}\in\mathrm{W}^{2,q}_{0}(\Omega)$, we
can estimate

$\begin{split}\mathrm{F}_{p,\epsilon,\bar{u}}(w)\,&\geq\,\|F(\mathrm{J}^{2}w)\|_{\mathrm{L}^{q}(\Omega)}\\\
&\geq\,c\big{\|}\mathrm{A}\\!:\\!\mathrm{D}^{2}w-\bar{\xi}(\cdot,w,\mathrm{D}w)\big{\|}_{\mathrm{L}^{q}(\Omega)}\\\
&\geq\,c\big{\|}\mathrm{A}\\!:\\!\mathrm{D}^{2}(w-\bar{u})\big{\|}_{\mathrm{L}^{q}(\Omega)}-c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{u}\|_{\mathrm{L}^{q}(\Omega)}-c\big{\|}\bar{\xi}(\cdot,w,\mathrm{D}w)\big{\|}_{\mathrm{L}^{q}(\Omega)}\\\
&\geq\,c\frac{\|w-\bar{u}\|_{\mathrm{W}^{2,q}(\Omega)}}{K(q,\Omega)}-c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{u}\|_{\mathrm{L}^{\infty}(\Omega)}-\big{\|}\big{(}|w|+|\mathrm{D}w|\big{)}^{\alpha}\big{\|}_{\mathrm{L}^{q}(\Omega)}-1\\\
&\geq\,c\frac{\|w-\bar{u}\|_{\mathrm{W}^{2,q}(\Omega)}}{K(q,\Omega)}-c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{u}\|_{\mathrm{L}^{\infty}(\Omega)}-\|w\|^{\alpha}_{\mathrm{W}^{1,\alpha
q}(\Omega)}-1\\\
&\geq\,c\frac{\|w-\bar{u}\|_{\mathrm{W}^{2,q}(\Omega)}}{K(q,\Omega)}-c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{u}\|_{\mathrm{L}^{\infty}(\Omega)}-\|w\|^{\alpha}_{\mathrm{W}^{1,q}(\Omega)}-1.\end{split}$

Therefore,

$\begin{split}\mathrm{F}_{p,\epsilon,\bar{u}}(w)\,&\geq\,c\frac{\|w-\bar{u}\|_{\mathrm{W}^{2,q}(\Omega)}}{K(q,\Omega)}-c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{u}\|_{\mathrm{L}^{\infty}(\Omega)}-\|w-\bar{u}\|^{\alpha}_{\mathrm{W}^{1,q}(\Omega)}\\\
&\ \ \ \ -\|\bar{u}\|^{\alpha}_{\mathrm{W}^{1,\infty}(\Omega)}-1\\\
&\geq\,c\frac{\|w-\bar{u}\|_{\mathrm{W}^{2,q}(\Omega)}}{K(q,\Omega)}-c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{u}\|_{\mathrm{L}^{\infty}(\Omega)}-\|w-\bar{u}\|^{\alpha}_{\mathrm{W}^{1,q}(\Omega)}\\\
&\ \ \ \ -\|\bar{u}\|^{\alpha}_{\mathrm{W}^{1,\infty}(\Omega)}-1\\\
&\geq\,\frac{c}{K(q,\Omega)}\|w-\bar{u}\|_{\mathrm{W}^{2,q}(\Omega)}-\big{(}\|w-\bar{u}\|_{\mathrm{W}^{2,q}(\Omega)}\big{)}^{\alpha}\\\
&\ \ \ \
-\Big{(}c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{u}\|_{\mathrm{L}^{\infty}(\Omega)}+\|\bar{u}\|^{\alpha}_{\mathrm{W}^{1,\infty}(\Omega)}+1\Big{)}.\end{split}$

By using the elementary inequality

$Nt-t^{\alpha}\geq N(1-\alpha)\big{[}t-N^{1/(\alpha-1)}\big{]},\ \ \ t\geq 0,\
N>0,$

the previous estimate yields

$\begin{split}\mathrm{F}_{p,\epsilon,\bar{u}}(w)\,&\geq\,\frac{c(1-\alpha)}{K(q,\Omega)}\bigg{[}\|w-\bar{u}\|_{\mathrm{W}^{2,q}(\Omega)}-\Big{(}\frac{c}{K(q,\Omega)}\Big{)}^{1/(\alpha-1)}\bigg{]}\\\
&\ \ \ \
-\Big{(}c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{u}\|_{\mathrm{L}^{\infty}(\Omega)}+\|\bar{u}\|^{\alpha}_{\mathrm{W}^{1,\infty}(\Omega)}+1\Big{)}.\end{split}$

This implies that

$\begin{split}\mathrm{F}_{p,\epsilon,\bar{u}}(w)\,&\geq\,\frac{c(1-\alpha)}{K(q,\Omega)}\|w\|_{\mathrm{W}^{2,q}(\Omega)}\\\
&\ \ \ \
-\bigg{\\{}\frac{c(1-\alpha)}{K(q,\Omega)}\bigg{[}\|\bar{u}\|_{\mathrm{W}^{2,q}(\Omega)}+\Big{(}\frac{c}{K(q,\Omega)}\Big{)}^{1/(\alpha-1)}\bigg{]}\\\
&\ \ \ \ \ \ \ \
+c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{u}\|_{\mathrm{L}^{\infty}(\Omega)}+\|\bar{u}\|^{\alpha}_{\mathrm{W}^{1,\infty}(\Omega)}+1\bigg{\\}}\\\
&=:\,\frac{c(1-\alpha)}{K(q,\Omega)}\|w\|_{\mathrm{W}^{2,q}(\Omega)}-M,\end{split}$

from which (2.20)(i) follows. Similarly, to obtain (2.20)(ii), we have

$\begin{split}\mathrm{F}_{p,\epsilon,\bar{u}}(w)\,&\geq\,\|F(\mathrm{J}^{2}w)\|_{\mathrm{L}^{q}(\Omega)}\\\
&\geq\,c\big{\|}\mathrm{A}\\!:\\!\mathrm{D}^{2}w-\bar{\xi}(\cdot,w,\mathrm{D}w)\big{\|}_{\mathrm{L}^{q}(\Omega)}\\\
&\geq\,c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}w\|_{\mathrm{L}^{q}(\Omega)}-\big{\|}\big{(}|w|+|\mathrm{D}w|\big{)}^{\alpha}\big{\|}_{\mathrm{W}^{1,q}(\Omega)}-1\\\
&\geq\,c\|\mathrm{A}\\!:\\!\mathrm{D}^{2}w\|_{\mathrm{L}^{q}(\Omega)}-\|w\|^{\alpha}_{\mathrm{W}^{1,q}(\Omega)}-1.\end{split}$

The lemma has been established. ∎

Now we may continue with the proof the theorem.

Proof of Theorem 16. (i) This is a simple application of the direct method.
Fix an exponent $p>n+1/c^{3}$ and let
$(w_{j})_{1}^{\infty}\subseteq\mathcal{E}_{\bar{u}}^{p}(\Omega)\subseteq
W_{\bar{u}}^{2,p}(\Omega)$ be a minimising sequence of
$\mathrm{F}_{p,\epsilon,\bar{u}}$ in $\mathcal{E}_{\bar{u}}^{p}(\Omega)$. By
Lemma 17, $(w_{j})_{1}^{\infty}$ is bounded in
$\mathrm{W}_{\bar{u}}^{2,p}(\Omega)$, hence by the sequential weak compactness
of bounded sets in $\mathrm{W}^{2,p}(\Omega)$, there exists
$u_{p}\in\mathcal{E}_{\bar{u}}^{p}(\Omega)$ and a subsequence of indices
$({j_{k}})_{1}^{\infty}$ such that
$w_{j_{k}}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{p}$ in
$\mathrm{W}_{\bar{u}}^{2,p}(\Omega)$, as $k\to\infty$. By assumption (2.18),
$\mathcal{E}_{\bar{u}}^{p}(\Omega)$ is weakly closed, therefore
$u_{p}\in\mathcal{E}_{\bar{u}}^{p}(\Omega)$. Since we have assumed
$p>n+1/c^{3}$, by (1.15) and (2.2) tested against rank-one matrices of the
form $(0,0,0,\xi)\otimes(0,0,0,\xi)$, it follows that
$\partial^{2}_{\xi\xi}\big{(}|F(x,\eta,\mathrm{p},\cdot)|^{p}\big{)}\geq 0$,
which implies that $|F(x,\eta,\mathrm{p},\cdot)|^{p}$ is convex. By (1.7) and
(1.15), we have the lower bound

$F(x,\eta,\mathrm{p},\xi)\geq
c\big{(}\xi-\bar{\xi}(x,\eta,\mathrm{p})\big{)}\geq\,c\hskip
1.0pt\xi-\big{(}|\eta|+|\mathrm{p}|\big{)}^{\alpha}-1,$

and hence the functional (1.19) is sequentially weakly lower semi-continuous
(see e.g. [19, Th. 3.23, p. 96]) on the weakly closed set
$\mathcal{E}_{\bar{u}}^{p}(\Omega)$. Hence, $u_{p}$ minimises the functional
$\mathrm{F}_{p,\epsilon,\bar{u}}$ over $\mathcal{E}_{\bar{u}}^{p}(\Omega)$.

(ii) Let $\\{u_{p}\\}_{n<p<\infty}\subseteq\mathrm{W}^{2,n}_{\bar{u}}(\Omega)$
be the family of minimisers from part (i). By applying Lemma 17, for each
fixed $q\in(1,\infty)$, $\\{u_{p}\\}_{n<p<\infty}$ is bounded in
$\mathrm{W}^{2,q}(\Omega)$, as a result of (2.20)(i). By sequential weak
compactness and a standard diagonal argument, there exists a sequence of
indices $(p_{j})_{1}^{\infty}$ and
$u_{\infty}\in\mathbb{\mathrm{missing}}W^{2,\infty}(\Omega)$ (recall (2.9))
such that $u_{p_{j}}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{\infty}$
in $\mathrm{W}^{2,q}(\Omega)$, as $j\to\infty$. By Morrey’s theorem and the
Lipschitz regularity of $\partial\Omega$, we also have that
$u_{p_{j}}\longrightarrow u_{\infty}$ in
$\mathrm{C}^{1,\gamma}(\overline{\Omega})$, as $j\to\infty$. Again by Lemma
17, for any $q\in(1,\infty)$ fixed and $p\geq q$, by (2.20)(ii) we have that

$\|\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{p}\|_{\mathrm{L}^{q}(\Omega)}\,\leq\,\frac{1+\delta+\mathrm{F}_{\infty}(\bar{u})+\|u_{p}\|^{\alpha}_{\mathrm{W}^{1,\infty}(\Omega)}}{c}.$

By the sequential weak lower semi-continuity of the $\mathrm{L}^{q}$ norm, and
the strong convergence of $u_{p_{j}}\longrightarrow u_{\infty}$ in
$\mathrm{C}^{1,\gamma}(\overline{\Omega})$ as $j\to\infty$, we have

$\|\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{\infty}\|_{\mathrm{L}^{q}(\Omega)}\leq\liminf_{j\to\infty}\|\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{p_{j}}\|_{\mathrm{L}^{q}(\Omega)}\,\leq\,\frac{1+\delta+\mathrm{F}_{\infty}(\bar{u})+\|u_{\infty}\|^{\alpha}_{\mathrm{W}^{1,\infty}(\Omega)}}{c}.$

Then, by letting $q\to\infty$ we infer that
$\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{\infty}\in\mathrm{L}^{\infty}(\Omega)$,
therefore $u_{\infty}\in\mathcal{W}^{2,\infty}_{\bar{u}}(\Omega)$.
Additionally, by (2.18), for any fixed $q\in(1,\infty)$, the family
$(u_{p})_{p\geq q}$ is contained in $\mathcal{E}_{\bar{u}}^{q}(\Omega)$. Since
$\mathcal{E}_{\bar{u}}^{q}(\Omega)$ is weakly closed, by letting
$p_{j}\to\infty$ we infer that
$u_{\infty}\in\mathcal{E}_{\bar{u}}^{q}(\Omega)$ for all $q\in(1,\infty)$. By
invoking (2.18) once again, we deduce that
$u_{\infty}\in\mathcal{E}_{\bar{u}}^{\infty}(\Omega)$. Now we show that
$u_{\infty}$ minimises
$\mathrm{F}_{\infty}+\frac{\epsilon}{2}\|\cdot-\bar{u}\|^{2}_{\mathrm{L}^{2}(\Omega)}$,
and also show the convergence of the energies. For any
$v\in\mathcal{E}_{\bar{u}}^{\infty}(\Omega)$ fixed and $p\geq q$, we have

$\begin{split}\big{\|}F(\mathrm{J}^{2}u_{p})\big{\|}_{\mathrm{L}^{q}(\Omega)}+\frac{\epsilon}{2}\|u_{p}-\bar{u}\|^{2}_{\mathrm{L}^{2}(\Omega)}&\leq\big{\|}F(\mathrm{J}^{2}u_{p})\big{\|}_{\mathrm{L}^{p}(\Omega)}+\frac{\epsilon}{2}\|u_{p}-\bar{u}\|^{2}_{\mathrm{L}^{2}(\Omega)}\\\
&=\mathrm{F}_{p,\epsilon,\bar{u}}(u_{p})\\\
&\leq\mathrm{F}_{p,\epsilon,\bar{u}}(v)\\\
&\leq\mathrm{F}_{\infty}(v)+\frac{\epsilon}{2}\|v-\bar{u}\|^{2}_{\mathrm{L}^{2}(\Omega)}.\end{split}$

By the sequential weak lower semi-continuity of the functional, we deduce

(2.23)
$\begin{split}\big{\|}F(\mathrm{J}^{2}u_{\infty})\big{\|}_{\mathrm{L}^{q}(\Omega)}+\frac{\epsilon}{2}\|u_{\infty}-\bar{u}\|^{2}_{\mathrm{L}^{2}(\Omega)}&\leq\liminf_{p_{j}\to\infty}\mathrm{F}_{p,\epsilon,\bar{u}}(u_{p})\\\
&\leq\limsup_{p_{j}\to\infty}\mathrm{F}_{p,\epsilon,\bar{u}}(u_{p})\\\
&\leq\mathrm{F}_{\infty}(v)+\frac{\epsilon}{2}\|v-\bar{u}\|^{2}_{\mathrm{L}^{2}(\Omega)},\end{split}$

for any $v\in\mathcal{E}_{\bar{u}}^{\infty}(\Omega)$. By letting $q\to\infty$
in inequality (2.23), the choice $v:=u_{\infty}$ completes the proof of
(2.19).

(iii) If we have $\epsilon>0$ and $\bar{u}$ minimises $\mathrm{F}_{\infty}$
over $\mathcal{E}_{\bar{u}}^{\infty}(\Omega)$, choosing $v:=\bar{u}$ in
inequality (2.23) and letting $q\to\infty$ yields $\bar{u}=u_{\infty}$. ∎

###  2.6. A lemma of strong compactness in $\mathrm{L}^{1}$

The following simple result, which is an extension of a lemma proved in the
appendix of [33], will be used in an essential fashion in Section 3. (Let us
recall here our notation $\tau_{z}$ for the translation operator.)

######  Lemma 18.

Let $\mathrm{A}\in\mathbb{R}^{n\times n}$ be a (strictly) positive matrix,
$\Omega\Subset\mathbb{R}^{n}$, $\sigma\in\mathrm{L}^{1}(\Omega)$ and
$\Sigma\in\mathrm{L}^{1}(\Omega;\mathbb{R}^{n})$. Suppose that
$f\in\mathrm{L}^{1}(\Omega)$ solves the PDE

$\ \ \ \ \ \ \
\mathrm{div}\big{(}\mathrm{A}\mathrm{D}f-\Sigma\big{)}+\sigma=0\ \ \text{ in
}\big{(}\mathrm{C}^{\infty}_{c}(\Omega)\big{)}^{\\!*},$

namely in the sense of distributions. Then, for any
$\Omega^{\prime}\Subset\Omega$, there exists a modulus of continuity
$\omega\in\mathrm{C}_{0^{+}}^{\nnearrow}[0,\infty)$ (depending only on
$n,\Omega^{\prime},\Omega,\mathrm{A}$), such that,

$\|\tau_{z}f-f\|_{\mathrm{L}^{1}(\Omega^{\prime})}\,\leq\,\omega(|z|)\Big{(}\|f\|_{\mathrm{L}^{1}(\Omega)}+\|\sigma\|_{\mathrm{L}^{1}(\Omega)}+\|\Sigma\|_{\mathrm{L}^{1}(\Omega)}\Big{)},$

for all $z\in\mathbb{R}^{n}$ with
$|z|<\mathrm{dist}(\Omega^{\prime},\partial\Omega)$. Further, for any
$q\in(1,n^{\prime})$, where $n^{\prime}=\frac{n}{n-1}$, there exists a
constant $K=K(q,n,\Omega^{\prime},\Omega,A)>0$ such that

$\|f\|_{\mathrm{L}^{q}(\Omega^{\prime})}\leq
K\Big{(}\|f\|_{\mathrm{L}^{1}(\Omega)}+\|\sigma\|_{\mathrm{L}^{1}(\Omega)}+\|\Sigma\|_{\mathrm{L}^{1}(\Omega)}\Big{)}.$

As a consequence, any set of distributional solutions which is bounded in
$\mathrm{L}^{1}(\Omega)$, is also strongly precompact in
$\mathrm{L}^{q}_{\mathrm{loc}}(\Omega)$, for all $q\in(1,n^{\prime})$.

Proof of Lemma 18. Without loss of generality, by a change of variables
argument as in Lemma 9, we may assume that $\mathrm{A}$ is the identity
matrix, and therefore we have the Poisson equation $\Delta
f=\mathrm{div}(\Sigma)-\sigma$ in $\Omega$. Fix
$\Omega^{\prime}\Subset\Omega$, and let fix also
$\Omega^{\prime\prime},\Omega^{\prime\prime\prime}$ such that
$\Omega^{\prime}\Subset\Omega^{\prime\prime}\Subset\Omega^{\prime\prime\prime}\Subset\Omega$.
By the interior nature of the desired estimate and the linearity of the PDE,
without loss of generality, a mollification argument allows us to assume that
$u,\sigma\in\mathrm{C}^{\infty}(\overline{\Omega^{\prime\prime\prime}})$,
$\Sigma\in\mathrm{C}^{\infty}(\overline{\Omega^{\prime\prime\prime}};\mathbb{R}^{n})$
$\partial\Omega^{\prime\prime\prime}$ is piecewise $\mathrm{C}^{\infty}$ and
the PDE holds classically in $\Omega^{\prime\prime\prime}$. Let
$\zeta\in\mathrm{C}^{\infty}_{c}(\mathbb{R}^{n})$ be a cut off function which
satisfies that
$\chi_{\Omega^{\prime\prime}}\leq\zeta\leq\chi_{\Omega^{\prime\prime\prime}}$.
By setting $\bar{\sigma}:=\zeta\sigma$ and $\bar{\Sigma}:=\zeta\Sigma$ the PDE
implies that $\Delta f=\mathrm{div}(\bar{\Sigma})-\bar{\sigma}$ in
$\Omega^{\prime\prime}$, and we have that
$\bar{\sigma}\in\mathrm{C}^{\infty}_{c}(\Omega)$,
$\bar{\Sigma}\in\mathrm{C}^{\infty}_{c}(\Omega;\mathbb{R}^{n})$, and also
$\bar{\sigma}|_{\Omega^{\prime\prime}}\equiv\sigma$ and
$\bar{\Sigma}|_{\Omega^{\prime\prime}}\equiv\Sigma$. Let now $R=R_{\Omega}>0$
be the radius of the smallest open ball such that
$\Omega-\Omega+\mathbb{B}_{1}\subseteq\mathbb{B}_{R}$. This means that
$x-y+z\in\mathbb{B}_{R}$, for all $x,y\in\Omega$ and all $z\in\mathbb{B}_{1}$.
Let $\eta\in\mathrm{C}^{\infty}_{c}(\mathbb{R}^{n})$ be a cut off function
such that $\eta\equiv 1$ on $\mathbb{B}_{R}$. By Green’s formula (see e.g.
[25, Ch. 2]), we can represent the solution $f$ on $\Omega^{\prime\prime}$ as

$f=h-\Phi*\bar{\sigma}+\Phi*\mathrm{div}(\bar{\Sigma}),$

for some harmonic function $h$ on $\Omega^{\prime\prime}$, where $\Phi$ is the
fundamental solution of the Laplace equation. Then, for any fixed
$z\in\mathbb{R}^{n}$ with
$|z|<\mathrm{dist}(\Omega^{\prime},\partial\Omega^{\prime\prime})$, we have
the estimate

$\begin{split}\big{\|}\tau_{z}f-f\big{\|}_{\mathrm{L}^{1}(\Omega^{\prime})}&\leq\big{\|}\tau_{z}(\Phi*\mathrm{div}(\bar{\Sigma}))-\Phi*\mathrm{div}(\bar{\Sigma})\big{\|}_{\mathrm{L}^{1}(\Omega^{\prime})}\\\
&\ \ \
\,+\big{\|}\tau_{z}(\Phi*\bar{\sigma})-\Phi*\bar{\sigma}\big{\|}_{\mathrm{L}^{1}(\Omega^{\prime})}+\|\tau_{z}h-h\|_{\mathrm{L}^{1}(\Omega^{\prime})}.\end{split}$

We will obtain the desired estimate by estimating each of the three terms in
the above inequality separately. By the identities

$\tau_{z}(\Phi*(\mathrm{div}(\bar{\Sigma}))=\tau_{z}(\mathrm{D}\Phi*\bar{\Sigma})=(\tau_{z}\mathrm{D}\Phi)*\bar{\Sigma},$

(where we interpret the convoluted inner product of vector functions in the
above identities as
$\mathrm{D}\Phi*\bar{\Sigma}=\sum_{i=1}^{n}\mathrm{D}_{i}\Phi*\bar{\Sigma}_{i}$),
we estimate by using Young’s inequality

$\begin{split}\Big{\|}\tau_{z}(\Phi*(\mathrm{div}(\bar{\Sigma}))-\Phi*(\mathrm{div}(\bar{\Sigma}))\Big{\|}_{\mathrm{L}^{1}(\Omega^{\prime})}\,&=\,\Big{\|}\big{(}\tau_{z}(\mathrm{D}\Phi)-\mathrm{D}\Phi\big{)}*\bar{\Sigma}\Big{\|}_{\mathrm{L}^{1}(\Omega^{\prime})}\\\
&=\,\Big{\|}\big{[}\big{(}\tau_{z}(\eta\mathrm{D}\Phi)-\eta\mathrm{D}\Phi\big{)}\chi_{\mathbb{B}_{R}}\big{]}*\bar{\Sigma}\Big{\|}_{\mathrm{L}^{1}(\mathbb{R}^{n})}\\\
&\leq\,\big{\|}\tau_{z}(\eta\mathrm{D}\Phi)-\eta\mathrm{D}\Phi\big{\|}_{\mathrm{L}^{1}(\mathbb{B}_{R})}\|\bar{\Sigma}\|_{\mathrm{L}^{1}(\mathbb{R}^{n})}\\\
&\leq\,\big{\|}\tau_{z}(\eta\mathrm{D}\Phi)-\eta\mathrm{D}\Phi\big{\|}_{\mathrm{L}^{1}(\mathbb{R}^{n})}\|\Sigma\|_{\mathrm{L}^{1}(\Omega)}\end{split}$

Similarly, we have the estimate

$\begin{split}\big{\|}\tau_{z}(\Phi*\bar{\sigma})-\Phi*\bar{\sigma}\big{\|}_{\mathrm{L}^{1}(\Omega^{\prime})}\,\leq\,\big{\|}\tau_{z}(\eta\Phi)-\eta\Phi\big{\|}_{\mathrm{L}^{1}(\mathbb{R}^{n})}\|\sigma\|_{\mathrm{L}^{1}(\Omega)}.\end{split}$

Further, by the mean value theorem for harmonic functions and interior
derivative estimates, we have

$\|\tau_{z}h-h\|_{\mathrm{L}^{1}(\Omega^{\prime})}\leq|z|\|\mathrm{D}h\|_{\mathrm{L}^{\infty}(\Omega^{\prime\prime})}\leq|z|\|h\|_{\mathrm{L}^{1}(\Omega)}.$

Further, by using that $h=f+\Phi*\bar{\sigma}-\mathrm{D}\Phi*\bar{\Sigma}$,
Young’s inequality implies

$\begin{split}\|h\|_{\mathrm{L}^{1}(\Omega)}\,&\leq\|f\|_{\mathrm{L}^{1}(\Omega)}+\big{\|}\Phi*\bar{\sigma}\big{\|}_{\mathrm{L}^{1}(\Omega)}+\big{\|}\mathrm{D}\Phi*\bar{\Sigma}\big{\|}_{\mathrm{L}^{1}(\Omega)}\\\
&=\|f\|_{\mathrm{L}^{1}(\Omega)}+\big{\|}(\Phi\chi_{\mathbb{B}_{R}})*\bar{\sigma}\big{\|}_{\mathrm{L}^{1}(\Omega)}+\big{\|}(\mathrm{D}\Phi\chi_{\mathbb{B}_{R}})*\bar{\Sigma}\big{\|}_{\mathrm{L}^{1}(\Omega)}\\\
&\leq\|f\|_{\mathrm{L}^{1}(\Omega)}+\|\Phi\|_{\mathrm{L}^{1}(\mathbb{B}_{R})}\|\sigma\|_{\mathrm{L}^{1}(\Omega)}+\|\mathrm{D}\Phi\|_{\mathrm{L}^{1}(\mathbb{B}_{R})}\|\Sigma\|_{\mathrm{L}^{1}(\Omega)}.\end{split}$

Therefore, we have

$\|\tau_{z}h-h\|_{\mathrm{L}^{1}(\Omega^{\prime})}\leq|z|\big{(}1+\|\Phi\|_{\mathrm{W}^{1,1}(\mathbb{B}_{R})}\big{)}\Big{(}\|f\|_{\mathrm{L}^{1}(\Omega)}+\|\sigma\|_{\mathrm{L}^{1}(\Omega)}+\|\Sigma\|_{\mathrm{L}^{1}(\Omega)}\Big{)}.$

Finally, we conclude by putting the pieces together. Since
$\eta\Phi\in\mathrm{L}^{1}(\mathbb{R}^{n})$ and also
$\eta\mathrm{D}\Phi\in\mathrm{L}^{1}(\mathbb{R}^{n};\mathbb{R}^{n})$, by the
continuity of the translation operation, there exists $\omega\in
C^{\nnearrow}_{0}[0,\infty)$ such that

$\big{\|}\tau_{z}(\eta\Phi)-\eta\Phi\big{\|}_{\mathrm{L}^{1}(\mathbb{R}^{n})}+\big{\|}\tau_{z}(\eta\mathrm{D}\Phi)-\eta\mathrm{D}\Phi\big{\|}_{\mathrm{L}^{1}(\mathbb{R}^{n})}\leq\omega(|z|).$

By the above estimates, the desired $\mathrm{L}^{1}$ equi-continuity estimate
ensues. Now we establish the $\mathrm{L}^{q}_{\mathrm{loc}}$ bound. Fix
$q\in(1,n^{\prime})$, and fixed also $\zeta\in\mathrm{C}^{\infty}_{c}(\Omega)$
with $\chi_{\Omega^{\prime}}\leq\zeta\leq\chi_{\Omega^{\prime\prime\prime}}$.
Let $\phi\in\mathrm{C}^{\infty}(\Omega)$ be arbitrary, and test the PDE
against $\zeta\phi\in\mathrm{C}^{\infty}_{c}(\Omega)$, to obtain

$\begin{split}\int_{\Omega}\zeta
f(\mathrm{A}\\!:\\!\mathrm{D}^{2}\phi)\,\mathrm{d}\mathcal{L}^{n}&=-\int_{\Omega^{\prime\prime\prime}}f\mathrm{A}\\!:\\!\Big{(}\mathrm{D}\zeta\otimes\mathrm{D}\phi+\mathrm{D}\phi\otimes\mathrm{D}\zeta+\phi\mathrm{D}^{2}\zeta\Big{)}\,\mathrm{d}\mathcal{L}^{n}\\\
&\ \ \
-\int_{\Omega^{\prime\prime\prime}}\Sigma\cdot\Big{(}\phi\mathrm{D}\zeta+\zeta\mathrm{D}\phi\Big{)}\,\mathrm{d}\mathcal{L}^{n}-\int_{\Omega^{\prime\prime\prime}}\sigma\zeta\phi\,\mathrm{d}\mathcal{L}^{n}.\end{split}$

This identity implies the estimate

$\begin{split}\left|\int_{\Omega}\zeta
f(\mathrm{A}\\!:\\!\mathrm{D}^{2}\phi)\,\mathrm{d}\mathcal{L}^{n}\right|&\leq
C(n)\big{(}|\mathrm{A}|+1\big{)}\|\zeta\|_{\mathrm{W}^{2,\infty}(\Omega)}\\\
&\ \ \
\cdot\Big{(}\|f\|_{\mathrm{L}^{1}(\Omega)}+\|\sigma\|_{\mathrm{L}^{1}(\Omega)}+\|\Sigma\|_{\mathrm{L}^{1}(\Omega)}\Big{)}\|\phi\|_{\mathrm{W}^{1,\infty}({\Omega^{\prime\prime\prime}})},\end{split}$

for some universal $C=C(n)>0$. Fix now any
$\psi\in\mathrm{C}^{\infty}_{c}(\Omega)$. By standard $\mathrm{L}^{p}$ theory
for elliptic equations and bootstrap regularity (see e.g. [25, Ch. 6 & 9],
there exists a unique
$\phi\in\mathrm{C}^{\infty}(\Omega)\cap\mathrm{C}^{1}_{0}(\Omega)$ such that
$\mathrm{A}\\!:\\!\mathrm{D}^{2}\phi=\psi$ on $\Omega$, and also

$\|\phi\|_{\mathrm{W}^{1,\infty}({\Omega^{\prime\prime\prime}})}\leq
C\|\mathrm{A}\\!:\\!\mathrm{D}^{2}\phi\|_{\mathrm{L}^{q^{\prime}}({\Omega^{\prime\prime\prime}})},$

for some $C=C(q,n,\Omega)>0$, which is possible by Morrey’s embedding theorem
because $q^{\prime}>n$ (since by assumption $q\in(1,n^{\prime})$), and also
$\Omega^{\prime\prime\prime}\Subset\Omega$ has piecewise smooth boundary.
Therefore, there exists a $K=K(q,n,\Omega^{\prime},\Omega,\mathrm{A})>0$ such
that

$\begin{split}\left|\int_{\Omega}(\zeta
f)\psi\,\mathrm{d}\mathcal{L}^{n}\right|&\leq
K\Big{(}\|f\|_{\mathrm{L}^{1}(\Omega)}+\|\sigma\|_{\mathrm{L}^{1}(\Omega)}+\|\Sigma\|_{\mathrm{L}^{1}(\Omega)}\Big{)}\|\psi\|_{\mathrm{L}^{q^{\prime}}(\Omega)},\end{split}$

for any $\psi\in\mathrm{C}^{\infty}_{c}(\Omega)$. By standard $\mathrm{L}^{p}$
theory (see e.g. [35, Ch. 14]), this implies

$\begin{split}\|f\|_{\mathrm{L}^{q}(\Omega^{\prime})}\leq\|\zeta
f\|_{\mathrm{L}^{q}(\Omega)}\leq
K\Big{(}\|f\|_{\mathrm{L}^{1}(\Omega)}+\|\sigma\|_{\mathrm{L}^{1}(\Omega)}+\|\Sigma\|_{\mathrm{L}^{1}(\Omega)}\Big{)},\end{split}$

as claimed. The last claim of the statement regarding strong compactness
follows by the Féchet-Kolmogorov and the Vitali convergence theorems. ∎

###  2.7. Nodal sets and non-trivial solutions

The following result lists some properties of solutions to equations of the
form (1.9), but with general $L^{\infty}$ coefficients. It holds true much
more generally, but we specialise the statement only to the case of interest
in this paper.

######  Proposition 19.

Let $\mathrm{A}\in\mathbb{R}^{n\times n}$ be a (strictly) positive matrix,
$\Omega\Subset\mathbb{R}^{n}$, $\mathrm{K}\in\mathrm{L}^{\infty}(\Omega)$ and
$\mathrm{L}\in\mathrm{L}^{\infty}(\Omega;\mathbb{R}^{n})$. Suppose that
$f\in\mathrm{L}^{1}(\Omega)$ solves the PDE

(2.24) $\ \ \ \ \ \
\mathrm{div}\big{(}\mathrm{A}\mathrm{D}f-\mathrm{L}f\big{)}+\mathrm{K}f=0\
\,\text{ in }\Omega,$

in the sense of distributions, namely in
$\big{(}\mathrm{C}^{\infty}_{c}(\Omega)\big{)}^{\\!*}$. Then:

_(i)_ Any solution $f\in\mathrm{L}^{1}(\Omega)$ to (2.24) satisfies
$f\in{\mathrm{W}}^{1,p}_{\mathrm{loc}}(\Omega)$, for all $p\in(1,\infty)$, and
thus (the precise representative of) $f$ is in
$\mathrm{C}^{0,\alpha}(\Omega)$, for all $\alpha\in(0,1)$.

_(ii)_ There exist a non-trivial solution $f^{*}\not\equiv 0$ to (2.24).

_(iii)_ For any non-constant solution $f$ to (2.24), the nodal set $\\{f=0\\}$
is closed and countably $(n-1)$-rectifiable. Thus, the
$\mathcal{H}^{n-1}$-Hausdorff measure of $\\{f=0\\}$ is locally finite, and in
particular it is an $\mathcal{L}^{n}$-nullset in $\Omega$.

Proof of Proposition 19. (i) Since the coefficients $\mathrm{K},\mathrm{L}$ of
(2.24) are essentially bounded and $\mathrm{A}$ is a constant positive matrix,
by the result [11, Corollary 1.2.8, p. 10] of Bogachev-Krylov-Röckner, it
follows that $f\in\mathrm{W}^{1,p}_{\mathrm{loc}}(\Omega)$ for all
$p\in(1,\infty)$. Therefore, upon identifying $f$ with its precise
representative, we have $f\in\mathrm{C}^{0,\gamma}(\Omega)$ for all
$\gamma\in(0,1)$, as a consequence of Morrey’s imbedding theorem. In
particular, $\\{f=0\\}$ is a relatively closed set in $\Omega$.

(ii) For any $\mu\in\mathbb{R}$, we define

$\mathbb{B}_{\mu}[f,g]:=\int_{\Omega}\Big{(}\big{(}\mathrm{A}\mathrm{D}f-\mathrm{L}f\big{)}\cdot\mathrm{D}g+(\mu-\mathrm{K})fg\Big{)}\,\mathrm{d}\mathcal{L}^{n}.$

Then,
$\mathbb{B}_{\mu}:\mathrm{W}^{1,2}_{0}(\Omega)\times\mathrm{W}^{1,2}_{0}(\Omega)\longrightarrow\mathbb{R}$
is a continuous bilinear form, which is also coercive for all
$\mu\geq\mu^{*}$, where the value $\mu^{*}>0$ depends only on the
coefficients. By the Lax-Milgram theorem, the operator
$\mathfrak{L}_{\mu}:\mathrm{W}^{1,2}_{0}(\Omega)\longrightarrow\mathrm{W}^{-1,2}(\Omega)$
given by

$\mathfrak{L}_{\mu}f:=\mathrm{div}\big{(}\mathrm{A}\mathrm{D}f-\mathrm{L}f\big{)}+\mathrm{K}f-\mu
f,$

is a linear isomorphism. Therefore, the inverse
$(\mathfrak{L}_{\mu})^{-1}:\mathrm{W}^{-1,2}(\Omega)\longrightarrow\mathrm{W}^{-1,2}(\Omega)$
is a compact operator, as a consequence of the compactness of the canonical
inclusions
$\mathrm{W}^{1,2}_{0}(\Omega)\Subset\mathrm{L}^{2}(\Omega)\Subset\mathrm{W}^{-1,2}(\Omega)$.
By the Fredholm alternative, the equation

(2.25) $f+(\mu(\mathfrak{L}_{\mu})^{-1})f=(\mathfrak{L}_{\mu})^{-1}\phi$

either it has a solution $f\not\equiv 0$ when $\phi\equiv 0$, or else it is
solvable for all $\phi\in\mathrm{W}^{-1,2}(\Omega)$. By noting that (2.25) can
be rewritten as $\mathfrak{L}_{\mu}f=\phi-\mu f$, which is equivalent to
$\mathfrak{L}_{0}f=\phi$, it follows that either $\mathfrak{L}_{0}f=0$ has a
solution $f^{*}\in\mathrm{W}^{1,2}_{0}(\Omega)\setminus\\{0\\}$, or else the
problem $\mathfrak{L}_{0}h=\mathrm{K}-\mathrm{div}\,\mathrm{L}$ has a solution
$h\in\mathrm{W}^{1,2}_{0}(\Omega)$. The latter is equivalent to the
solvability of the boundary value problem $\mathfrak{L}_{0}f=0$ in $\Omega$
with $f=1$ on $\partial\Omega$ by some
$f^{*}\in\mathrm{W}^{1,2}(\Omega)\setminus\\{0\\}$. In either case, it follows
that (2.24) has a non-trivial solution $f^{*}\not\equiv 0$.

(iii) We begin by considering the auxiliary boundary value problem

$\mathrm{div}(\mathrm{A}\mathrm{D}\psi)=\mathrm{div}\,\mathrm{L}\,\text{ on
}\Omega,\ \ \ \psi=0\,\text{ on }\partial\Omega.$

By the Lax-Milgram theorem, there exists a unique solution
$\psi\in\mathrm{W}^{1,2}_{0}(\Omega)$, which by a result of Di Fazio [21, Th.
3.1], it satisfies $\psi\in\mathrm{W}^{1,p}_{\mathrm{loc}}(\Omega)$ for all
$p\in(1,\infty)$. By Morrey’s theorem, (the precise representative of) $\psi$
is in $\mathrm{C}^{0,\alpha}(\Omega)$, for all $\alpha\in(0,1)$. We now set
$\phi:=e^{-\psi}f$, where $f\in\mathrm{L}^{1}(\Omega)$ is any given non-
constant solution to (2.24). By part (i), all three function $\phi,\psi,f$ lie
in $\mathrm{W}^{1,p}_{\mathrm{loc}}(\Omega)$ for all $p\in(1,\infty)$, whilst
$\mathrm{L}\in\mathrm{L}^{\infty}(\Omega;\mathbb{R}^{n})$. Therefore, by using
that $\mathrm{div}(\mathrm{A}\mathrm{D}\psi-\mathrm{L})=0$ in $\Omega$, we may
rigorously compute

$\begin{split}0&=\mathrm{div}\big{(}\mathrm{A}\mathrm{D}f-\mathrm{L}f\big{)}+\mathrm{K}f\\\
&=\mathrm{div}\Big{(}e^{\psi}\mathrm{A}\mathrm{D}\phi+e^{\psi}\phi\big{(}\mathrm{A}\mathrm{D}\psi-\mathrm{L}\big{)}\Big{)}+\mathrm{K}e^{\psi}\phi\\\
&=\mathrm{div}\big{(}(e^{\psi}\mathrm{A})\mathrm{D}\phi\big{)}+\big{(}\mathrm{A}\mathrm{D}\psi-\mathrm{L}\big{)}\cdot[e^{\psi}\mathrm{D}\phi+e^{\psi}\phi\mathrm{D}\psi]+(e^{\psi}\mathrm{K})\phi,\end{split}$

in $\Omega$. Hence, the function $\phi$ is a weak solution to the divergence
PDE

$\mathrm{div}\big{(}(e^{\psi}\mathrm{A})\mathrm{D}\phi\big{)}+\big{[}e^{\psi}\big{(}\mathrm{A}\mathrm{D}\psi-\mathrm{L}\big{)}\big{]}\cdot\mathrm{D}\phi+\big{[}e^{\psi}\big{(}\\!\big{(}\mathrm{A}\mathrm{D}\psi-\mathrm{L}\big{)}\cdot\mathrm{D}\psi+\mathrm{K}\big{)}\big{]}\phi=0\
\text{ in }\Omega.$

By the previous, the coefficients satisfy

$e^{\psi}\big{(}\mathrm{A}\mathrm{D}\psi-\mathrm{L}\big{)},\
e^{\psi}\big{(}\\!\big{(}\mathrm{A}\mathrm{D}\psi-\mathrm{L}\big{)}\cdot\mathrm{D}\psi+\mathrm{K}\big{)}\,\in\mathrm{L}^{p}_{\mathrm{loc}}(\Omega),\
\ e^{\psi}\mathrm{A}\in\mathrm{C}^{0,\alpha}(\Omega;\mathbb{R}^{n\times n}),$

for all $p\in(1,\infty)$, and all $\alpha\in(0,1)$. By the result [26, Th. 1.7
& Rem. 1.8(3)] of Hardt and Simon, it follows that for any point
$\bar{x}\in\Omega$ at which $\phi$ has finite order of vanishing, the set
$\\{\phi=0\\}\cap\mathbb{B}_{\rho}(\bar{x})$ is countably $(n-1)$-rectifiable,
for all sufficiently small $\rho>0$. Since $f=e^{\psi}\phi$, we have
$\\{f=0\\}=\\{\phi=0\\}$, and since
$e^{\pm\psi}\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\Omega)$, it follows that
$f$ and $\phi$ have the same sets of points with finite order of vanishing. On
the other hand, by the strong unique continuation principle applied to (2.24),
it follows that $f$ has finite order of vanishing everywhere on $\Omega$ (see
e.g. Koch-Tataru [37, 39]). Hence, $\\{f=0\\}$ is countably
$(n-1)$-rectifiable, and in particular it is a Lebesgue nullset. The
proposition ensues. ∎

##  3\. Proofs of the main results

We may now establish our first main result.

Proof of Theorem 2. Step 1. Let $u_{\infty}\in\mathcal{W}^{2,\infty}(\Omega)$
and $f_{\infty}\in\mathrm{L}^{1}(\Omega)$ be given, and suppose they satisfy
the system of equations (1.8)-(1.9). We assume first that

$f_{\infty}\not\equiv 0.$

Step 2. We define the functions

(3.1)
$\mathrm{K}_{\infty}:=\frac{\partial_{\eta}F(\mathrm{J}^{2}u_{\infty})}{\partial_{\xi}F(\mathrm{J}^{2}u_{\infty})},\
\ \ \
\mathrm{L}_{\infty}:=\frac{\partial_{\mathrm{p}}F(\mathrm{J}^{2}u_{\infty})}{\partial_{\xi}F(\mathrm{J}^{2}u_{\infty})},$

and note that $\mathrm{K}_{\infty},|\mathrm{L}_{\infty}|$ are in
$\mathrm{L}^{\infty}(\Omega)$: indeed, by (1.7) we have that
$\partial_{\xi}F(\mathrm{J}^{2}u_{\infty})\geq c$ a.e. on $\Omega$, and also

$\big{|}\partial_{\eta}F(\mathrm{J}^{2}u_{\infty})\big{|}+\big{|}\partial_{\mathrm{p}}F(\mathrm{J}^{2}u_{\infty})\big{|}\leq
C\Big{(}\|u_{\infty}\|_{\mathrm{W}^{1,\infty}(\Omega)}+\|\mathrm{A}\\!:\\!\mathrm{D}^{2}u_{\infty}\|_{\mathrm{L}^{\infty}(\Omega)}\Big{)},$

a.e. on $\Omega$.

Step 3. By (1.9) and (3.1), $f_{\infty}\in\mathrm{L}^{1}(\Omega)$ is a
distributional solution to

(3.2) $\ \ \ \
\mathrm{div}\big{(}\mathrm{A}\mathrm{D}f_{\infty}-\mathrm{L}_{\infty}f_{\infty}\big{)}+\mathrm{K}_{\infty}f_{\infty}\,=\,0\
\ \ \ \text{in }\Omega.$

By Proposition 19, we have $\mathcal{L}^{n}\big{(}\\{f_{\infty}=0\\}\big{)}=0$
and $f_{\infty}\in{\mathrm{W}}^{1,p}_{\mathrm{loc}}(\Omega)$ for all
$p\in(1,\infty)$. Additionally, upon identifying $f_{\infty}$ with its precise
representative, the nodal set $\\{f_{\infty}=0\\}$ is countably
$(n-1)$-rectifiable and closed.

Step 4. We will now establish that there exists some
$C\in\mathrm{C}^{\nnearrow}[0,\infty)$ such that

$\big{|}F(x,\eta,\mathrm{p},\xi)\big{|}\,\leq\,C\big{(}|\eta|+|\mathrm{p}|+|\xi|\big{)},$

for a.e. $x\in\Omega$ and all
$(\eta,\mathrm{p},\xi)\in\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}$. To
this aim, recalling (2.21) and assumption (1.7), it follows that

$\begin{split}\big{|}F(x,\eta,\mathrm{p},\xi)\big{|}\,&=\,\Big{|}F(x,\eta,\mathrm{p},\xi)-F(x,\eta,\mathrm{p},\bar{\xi}(x,\eta,\mathrm{p}))\Big{|}\\\
&=\,\left|\int_{\bar{\xi}(x,\eta,\mathrm{p})}^{\xi}\partial_{\xi}F(x,\eta,\mathrm{p},\cdot)\,\mathrm{d}\mathcal{L}^{1}\right|,\end{split}$

which yields

$\begin{split}\big{|}F(x,\eta,\mathrm{p},\xi)\big{|}\,&\leq\,\big{|}\bar{\xi}(x,\eta,\mathrm{p})-\xi\big{|}C\Big{(}|\bar{\xi}(x,\eta,\mathrm{p})|+|\xi|+|\eta|+|\mathrm{p}|\Big{)}\\\
&\leq\,\big{(}C\big{(}|\eta|+|\mathrm{p}|\big{)}+|\xi|\big{)}C\Big{(}C\big{(}|\eta|+|\mathrm{p}|\big{)}+|\xi|+|\eta|+|\mathrm{p}|\Big{)}.\end{split}$

Hence, the desired estimate ensues for the new function in
$\mathrm{C}^{\nnearrow}[0,\infty)$ given by
$t\mapsto\big{[}C(t)+t\big{]}C\big{(}C(t)+t\big{)}$.

Step 5. By assumption (1.7), we may fix $x\in\Omega$ (this is possible on a
subset of full measure) and apply Taylor’s theorem to
$|F(x,\cdot,\cdot,\cdot)|^{2}\in\mathrm{C}^{2}(\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R})$,
to deduce that

(3.3)
$\begin{split}\big{|}F(\mathrm{J}^{2}u_{\infty}&+\mathrm{J}^{2}\psi)\big{|}^{2}=\big{|}F(\mathrm{J}^{2}u_{\infty})\big{|}^{2}\,+\,2F(\mathrm{J}^{2}u_{\infty})\Big{\\{}\partial_{*}F(\mathrm{J}^{2}u_{\infty})\cdot\mathrm{J}^{2}_{*}\psi\Big{\\}}\\\
&+\bigg{(}\int_{0}^{1}(1-\lambda)\partial_{**}^{2}(|F|^{2})\big{(}\mathrm{J}^{2}u_{\infty}+\lambda\mathrm{J}^{2}\psi\big{)}\,\mathrm{d}\lambda\bigg{)}:\mathrm{J}^{2}_{*}\psi\otimes\mathrm{J}^{2}_{*}\psi,\end{split}$

a.e. on $\Omega$. By the identity (2.2) for $p=2$, assumption (1.7) and Step
4, we have the bound

$\big{|}\partial_{**}^{2}(|F|^{2})(x,\eta,\xi)\big{|}\,\leq\,C\big{(}|\eta|+|\mathrm{p}|+|\xi|\big{)}$

for some function $C\in\mathrm{C}^{\nnearrow}[0,\infty)$. By using this bound
and equation (1.8) (which implies
$\big{|}F(\mathrm{J}^{2}u_{\infty})\big{|}=\mathrm{F}_{\infty}(u_{\infty})$
a.e. on $\Omega$), (3.3) yields

(3.4)
$\begin{split}\big{|}F(\mathrm{J}^{2}u_{\infty}+\mathrm{J}^{2}\psi)\big{|}^{2}&\geq\,|\mathrm{F}_{\infty}(u_{\infty})|^{2}\\\
&+\,2\mathrm{F}_{\infty}(u_{\infty})\operatorname{sgn}(f_{\infty})\Big{\\{}\partial_{*}F(\mathrm{J}^{2}u_{\infty})\cdot\mathrm{J}^{2}_{*}\psi\Big{\\}}\\\
&-\,C\Big{(}\|\mathrm{J}^{2}_{*}u_{\infty}\|_{\mathrm{L}^{\infty}(\Omega)}+\|\mathrm{J}^{2}_{*}\psi\|_{\mathrm{L}^{\infty}(\Omega)}\Big{)}\|\mathrm{J}^{2}_{*}\psi\|^{2}_{\mathrm{L}^{\infty}(\Omega)},\end{split}$

a.e. on $\Omega$.

Step 6. Let us now define the functional
$\Theta_{\infty}:\mathcal{W}^{2,\infty}_{0}(\Omega)\longrightarrow\mathbb{R}$
by setting

(3.5)
$\Theta_{\infty}(\psi)\,:=\,\underset{\Omega}{\mathrm{ess}\sup}\Big{[}\operatorname{sgn}(f_{\infty})\Big{\\{}\partial_{*}F(\mathrm{J}^{2}u_{\infty})\cdot\mathrm{J}^{2}_{*}\psi\Big{\\}}\Big{]},$

and set

(3.6)
$N_{\infty}\,:=\,C\big{(}1+\|\mathrm{J}^{2}_{*}u_{\infty}\|_{\mathrm{L}^{\infty}(\Omega)}\big{)}.$

In view of (3.5)-(3.6), we may deduce from (3.4) that

(3.7)
$\begin{split}(\mathrm{F}_{\infty}(u_{\infty}+\psi))^{2}&\geq\,(\mathrm{F}_{\infty}(u_{\infty}))^{2}+\Big{\\{}2\mathrm{F}_{\infty}(u_{\infty})\Theta_{\infty}(\psi)-N_{\infty}\|\mathrm{J}^{2}_{*}\psi\|^{2}_{\mathrm{L}^{\infty}(\Omega)}\Big{\\}},\end{split}$

for all $\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)$ satisfying
$\|\mathrm{J}^{2}_{*}\psi\|_{\mathrm{L}^{\infty}(\Omega)}\leq 1$ (namely in
the closed unit ball of $\mathcal{W}^{2,\infty}_{0}(\Omega)$ with respect to
the $\|\cdot\|^{\prime}_{\mathcal{W}^{2,\infty}_{0}(\Omega)}$-norm in (2.13)).

Step 7. Let
$\mathcal{K}\subseteq\mathbb{S}^{\mathcal{W}^{2,\infty}_{0}(\Omega)}$ be a
compact set. We will regard the unit sphere of
$\mathcal{W}^{2,\infty}_{0}(\Omega)$ to be with respect to the
$\smash{\|\cdot\|^{\prime}_{\mathcal{W}^{2,\infty}_{0}(\Omega)}}$-norm in
(2.13). We aim at showing that there exists $\theta_{\infty}>0$ such that

(3.8) $\ \ \ \Theta_{\infty}(\psi)\geq\,\theta_{\infty},\ \ \
\forall\,\psi\in\mathcal{K}.$

Temporarily assuming that (3.8) holds true, we will now show how we can
complete the proof of the theorem. From (3.7)-(3.8), for any
$\psi\in\mathcal{K}$ and $0<t<1$, by noting that $\Theta_{\infty}$ is
$1$-homogeneous, we have

$(\mathrm{F}_{\infty}(u_{\infty}+t\psi))^{2}\geq\,(\mathrm{F}_{\infty}(u_{\infty}))^{2}+\,t\Big{\\{}2\mathrm{F}_{\infty}(u_{\infty})\theta_{\infty}-tN_{\infty}\Big{\\}},$

for all $\psi\in\mathcal{K}$. By restricting $t\in(0,\rho)$, where

$\rho:=\frac{2\mathrm{F}_{\infty}(u_{\infty})\theta_{\infty}}{N_{\infty}},$

the desired conclusion (1.10) ensues (having assumed that (3.8) holds true).

Step 8. The remaining of the proof is devoted to establishing the lower bound
(3.8) for the functional
$\Theta_{\infty}:\mathcal{W}^{2,\infty}_{0}(\Omega)\longrightarrow\mathbb{R}$
defined in (3.5). We will first show

(3.9) $\Theta_{\infty}(\psi)>0,\ \ \
\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)\setminus\\{0\\}.$

To this aim, fix $\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)\setminus\\{0\\}$.
By Lemma 11(ii), the divergence equation (1.9) is satisfies also weakly in
$(\mathcal{W}^{2,\infty}_{0}(\Omega))^{*}$, namely when testing against
functions in the space $\mathcal{W}^{2,\infty}_{0}(\Omega)$. Thus, (1.9)
implies

(3.10)
$\int_{\Omega}f_{\infty}\Big{(}\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi+\mathrm{L}_{\infty}\\!\cdot\mathrm{D}\psi+\mathrm{K}_{\infty}\psi\Big{)}\,\mathrm{d}\mathcal{L}^{n}\,=\,0.$

We now claim that

(3.11)
$\mathcal{L}^{n}\Big{(}\Big{\\{}f_{\infty}\Big{(}\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi+\mathrm{L}_{\infty}\\!\cdot\mathrm{D}\psi+\mathrm{K}_{\infty}\psi\Big{)}\neq
0\Big{\\}}\Big{)}>0.$

If for the sake of contradiction we suppose that

$\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi+\mathrm{L}_{\infty}\\!\cdot\mathrm{D}\psi+\mathrm{K}_{\infty}\psi=0,\
\text{ a.e.\ on }\Omega.$

Since $|\psi|=|\mathrm{D}\psi|=0$ on $\partial\Omega$ and $\psi\in
W_{0}^{2,q}(\Omega)$ for all $q\in(1,\infty)$, the extension $\bar{\psi}$ of
$\psi$ by zero to $\mathbb{R}^{n}\setminus\Omega$ is compactly supported in
$\mathbb{R}^{n}$ and satisfies that
$\bar{\psi}\in\mathrm{W}^{2,q}_{c}(\mathbb{R}^{n})$, solving strongly the PDE

$\mathrm{A}\\!:\\!\mathrm{D}^{2}\bar{\psi}+\bar{\mathrm{L}}_{\infty}\\!\cdot\mathrm{D}\bar{\psi}+\bar{\mathrm{K}}_{\infty}\bar{\psi}=0\
\ \ \text{ a.e.\ on }\mathbb{R}^{n},$

where
$\bar{\mathrm{K}}_{\infty},|\bar{\mathrm{L}}_{\infty}|\in\mathrm{L}^{\infty}_{c}(\mathbb{R}^{n})$
are the extensions of ${\mathrm{K}}_{\infty},{\mathrm{L}}_{\infty}$ by zero on
$\mathbb{R}^{n}\setminus\Omega$. In that case, by [26, Th. (1.7) and Rem.
(1.8)] we infer that $\mathcal{L}^{n}(\\{\bar{\psi}=0\\})=0$, contradicting
that we have selected $\bar{\psi}\equiv 0$ on
$\mathbb{R}^{n}\setminus\\{0\\}$. As a conclusion, (3.11) is true. Since we
already know from Step 3 that $f_{\infty}=0$ at most on an
$\mathcal{L}^{n}$-nullset, (3.11) follows. Further, we already know from
(3.10) that the average must vanish over $\Omega$. As a result, there must
exist Borel sets $\Omega^{\pm}\subseteq\Omega$ with
$\mathcal{L}^{n}(\Omega^{\pm})>0$ such that

$\frac{f_{\infty}}{|f_{\infty}|}\Big{(}\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi+\mathrm{L}_{\infty}\\!\cdot\mathrm{D}\psi+\mathrm{K}_{\infty}\psi\Big{)}\gtrless\,0\
\ \ \text{ a.e.\ on }\Omega^{\pm}.$

In view of (3.1), the above yields

$\ \ \ \ \
\frac{f_{\infty}}{|f_{\infty}|}\Big{(}\partial_{\xi}F(\mathrm{J}^{2}u_{\infty})\mathrm{A}\\!:\\!\mathrm{D}^{2}\psi+\partial_{\mathrm{p}}F(\mathrm{J}^{2}u_{\infty})\cdot\mathrm{D}\psi+\partial_{\eta}F(\mathrm{J}^{2}u_{\infty})\psi\Big{)}>\,0,\
\ \ \text{ a.e.\ on }\Omega^{+}.$

Using our notation for jets, we can rewrite the above as

$\operatorname{sgn}(f_{\infty})\Big{\\{}\partial_{*}F(\mathrm{J}^{2}u_{\infty})\cdot\mathrm{J}^{2}_{*}\psi\Big{\\}}>\,0\
\ \ \text{ a.e.\ on }\Omega^{+},$

and $\mathcal{L}^{n}(\Omega^{+})>0$. In conclusion, we obtain (in view of
(3.5)) that

$\Theta_{\infty}(\psi)\,=\,\underset{\Omega}{\mathrm{ess}\sup}\Big{[}\operatorname{sgn}(f_{\infty})\Big{\\{}\partial_{*}F(\mathrm{J}^{2}u_{\infty})\cdot\mathrm{J}^{2}_{*}\psi\Big{\\}}\Big{]}>0,$

for any $\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)\setminus\\{0\\}$.

Step 9. Now we establish the bound (3.8) for the functional $\Theta_{\infty}$.
In view of Step 8, it suffices to show that $\Theta_{\infty}$ is continuous.
Indeed, on the one hand, for any
$\phi,\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)$ we have the estimate

$\Theta_{\infty}(\phi)=\,\Theta_{\infty}(\phi+\psi-\psi)\,\leq\,\Theta_{\infty}(\phi+\psi)\,+\,\|\mathrm{J}^{2}_{*}u_{\infty}\|_{\mathrm{L}^{\infty}(\Omega)}\|\mathrm{J}^{2}_{*}\psi\|_{\mathrm{L}^{\infty}(\Omega)}.$

On the other hand, by the convexity of $\Theta_{\infty}$, we have that

$\begin{split}\Theta_{\infty}(\phi+\psi)-\Theta_{\infty}(\phi)\,&\leq\bigg{(}\Theta_{\infty}\bigg{(}\phi+\frac{\psi}{\|\mathrm{J}^{2}_{*}\psi\|_{\mathrm{L}^{\infty}(\Omega)}}\bigg{)}-\Theta_{\infty}(\phi)\bigg{)}\|\mathrm{J}^{2}_{*}\psi\|_{\mathrm{L}^{\infty}(\Omega)}\\\
&\leq\,2\|\mathrm{J}^{2}_{*}u_{\infty}\|_{\mathrm{L}^{\infty}(\Omega)}\Big{(}1+\|\mathrm{J}^{2}_{*}\phi\|_{\mathrm{L}^{\infty}(\Omega)}\Big{)}\|\mathrm{J}^{2}_{*}\psi\|_{\mathrm{L}^{\infty}(\Omega)}.\end{split}$

Step 10. In conclusion, $\Theta_{\infty}$ is continuous and positive on the
compact set $\mathcal{K}$, yielding that

$\theta_{\infty}\,:=\,\inf_{\mathcal{K}}\Theta_{\infty}\,>\,0.$

The proof of Theorem 2 is now complete. ∎

We may now establish our second main result.

Proof of Theorem 3. Let $p\geq 2$ and fix
$\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)$. We continue our arguments from
the proof of Step 5 of Theorem 2. By Taylor’s theorem for
$|F(x,\cdot,\cdot)|^{p}$, we have

(3.12)
$\begin{split}\big{|}F(\mathrm{J}^{2}u_{\infty}+&\mathrm{J}^{2}\psi)\big{|}^{p}=\big{|}F(\mathrm{J}^{2}u_{\infty})\big{|}^{p}\\\
&+\,p\big{|}F(\mathrm{J}^{2}u_{\infty})\big{|}^{p-2}F(\mathrm{J}^{2}u_{\infty})\Big{\\{}\partial_{*}F(\mathrm{J}^{2}u_{\infty})\cdot\mathrm{J}^{2}_{*}\psi\Big{\\}}\\\
&+\bigg{(}\int_{0}^{1}(1-\lambda)\partial_{**}^{2}(|F|^{p})\big{(}\mathrm{J}^{2}u_{\infty}+\lambda\mathrm{J}^{2}\psi\big{)}\,\mathrm{d}\lambda\bigg{)}:\mathrm{J}^{2}_{*}\psi\otimes\mathrm{J}^{2}_{*}\psi,\end{split}$

a.e. on $\Omega$. We will now show that, under (LABEL:1.13)-(1.13), the last
term of (3.12) above is non-negative for all sufficiently large $p\geq 2$,
when $\psi$ is restricted appropriately. More precisely:

Case 1: Suppose that (LABEL:1.13) is satisfied. Then, the essential range of
$\mathrm{J}^{2}u_{\infty}$ satisfies
$\mathrm{ess}((\mathrm{J}^{2}u_{\infty})(\Omega))\subseteq\Omega\times L\times
R$. Since $L,R$ are open intervals and the essential range is a compact set,
for any $\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)$ and any $\lambda\in(0,1)$,
we may select $\rho>0$ small enough such that
$\|\mathrm{J}^{2}_{*}\psi\|_{\mathrm{L}^{\infty}(\Omega)}<\rho$, in order to
obtain

$\mathrm{ess}\big{(}(\mathrm{J}^{2}u_{\infty}+\lambda\mathrm{J}^{2}\psi)(\Omega)\big{)}\subseteq\Omega\times
L\times R.$

By choosing the corresponding $\bar{p}\geq 2$, it follows that
$\partial_{**}^{2}(|F|^{p})\big{(}\mathrm{J}^{2}u_{\infty}+\lambda\mathrm{J}^{2}\psi\big{)}\geq
0$ for all $p\geq\bar{p}$, when
$\psi\in\mathbb{B}_{\rho}^{\mathcal{W}^{2,\infty}_{0}(\Omega)}$.

Case 2: Suppose (LABEL:1.14) is satisfied. Then, the essential range satisfies
$\mathrm{ess}(u_{\infty}(\Omega))\subseteq L$. Since $L$ is an open interval
and the essential range is a compact set, for any
$\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)$ and any $\lambda\in(0,1)$, we may
select $\rho>0$ small enough such that
$\|\psi\|_{\mathrm{L}^{\infty}(\Omega)}<\rho$, and any sufficiently large
interval $R\subseteq\mathbb{R}$, in order to obtain

$\mathrm{ess}\big{(}(\mathrm{J}^{2}u_{\infty}+\lambda\mathrm{J}^{2}\psi)(\Omega)\big{)}\subseteq\Omega\times
L\times R.$

Then, for the corresponding $\bar{p}\geq 2$, it follows that
$\partial_{**}^{2}(|F|^{p})\big{(}\mathrm{J}^{2}u_{\infty}+\lambda\mathrm{J}^{2}\psi\big{)}\geq
0$ for all $p\geq\bar{p}$, when
$\psi\in\mathbb{B}_{\rho}^{\mathrm{L}^{\infty}(\Omega)}\cap\mathcal{W}^{2,\infty}_{0}(\Omega)$.

Case 3: Suppose (1.13) is satisfied. Fix any
$\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)$ and choose
$R,L\subseteq\mathbb{R}$ large enough and the corresponding $\bar{p}\geq 2$,
so that the essential range satisfies

$\mathrm{ess}\big{(}\big{(}\mathrm{J}^{2}u_{\infty}+\lambda\mathrm{J}^{2}\psi\big{)}(\Omega)\big{)}\subseteq\Omega\times
L\times R.$

It follows that
$\partial_{**}^{2}(|F|^{p})\big{(}\mathrm{J}^{2}u_{\infty}+\lambda\mathrm{J}^{2}\psi\big{)}\geq
0$ for all $p\geq\bar{p}$.

In conclusion, for any fixed $\psi\in\mathcal{W}^{2,\infty}_{0}(\Omega)$
appropriately restricted depending on the assumptions, we have

$\begin{split}\big{|}F(\mathrm{J}^{2}u_{\infty}+&\mathrm{J}^{2}\psi)\big{|}^{p}-\big{|}F(\mathrm{J}^{2}u_{\infty})\big{|}^{p}\\\
&\geq\,p\big{|}F(\mathrm{J}^{2}u_{\infty})\big{|}^{p-2}F(\mathrm{J}^{2}u_{\infty})\Big{\\{}\partial_{*}F(\mathrm{J}^{2}u_{\infty})\cdot\mathrm{J}^{2}_{*}\psi\Big{\\}},\end{split}$

a.e. on $\Omega$. In view of (1.8) and (1.1), the above yields

$\begin{split}\underset{\Omega}{\mathrm{ess}\sup}\big{|}F(\mathrm{J}^{2}u_{\infty}+&\mathrm{J}^{2}\psi)\big{|}^{p}-(\mathrm{F}_{\infty}(u_{\infty}))^{p}\\\
&\geq\,p(\mathrm{F}_{\infty}(u_{\infty}))^{p-1}\underset{\Omega}{\mathrm{ess}\sup}\left[\operatorname{sgn}(f_{\infty})\Big{\\{}\partial_{*}F(\mathrm{J}^{2}u_{\infty})\cdot\mathrm{J}^{2}_{*}\psi\Big{\\}}\right].\end{split}$

By (3.5) and (1.1), since powers commute with the essential supremum of non-
negative functions, the above inequality can be rewritten as

$\begin{split}\big{(}F_{\infty}(u_{\infty}+\psi)\big{)}^{p}-(\mathrm{F}_{\infty}(u_{\infty}))^{p}\,\geq\,p(\mathrm{F}_{\infty}(u_{\infty}))^{p-1}\Theta_{\infty}(\psi).\end{split}$