# Non-interpenetration of rods derived by $\Gamma$-limits

Barbora Benešová, Daniel Campbell, Stanislav Hencl  Department of Mathematical
Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská
83, 186 00 Prague 8, Czechia  {benesova, campbell, hencl} @karlin.mff.cuni.cz
and  Martin Kružík  Czech Academy of Sciences, Institute of Information Theory
and Automation, Pod vodárenskou věží 4, 182 00 Prague 8, Czechia, and
Department of Physics, Faculty of Civil Engineering, Czech Technical
University, Thákurova 7, 166 29 Prague 6, Czechia<EMAIL_ADDRESS>
###### Abstract.

Ensuring non-interpenetration of matter is a fundamental prerequisite when
modeling the deformation response of solid materials. In this contribution, we
thoroughly examine how this requirement, equivalent to the injectivity of
deformations within bulk structures, manifests itself in dimensional-reduction
problems. Specifically, we focus on the case of rods embedded in a two-
dimensional plane. Our results focus on $\Gamma$-limits of energy functionals
that enforce an admissible deformation to be a homeomorphism. These
$\Gamma$-limits are evaluated along a passage from the bulk configuration to
the rod arrangement. The proofs rely on the equivalence between the weak and
strong closures of the set of homeomorphisms from $\mathbb{R}$ to
$\mathbb{R}^{2}$, a result that is of independent interest and that we
establish in this paper, too.

Mathematics Subject Classification (2020): 49J45, 35Q74, 74B20
This work has been supported by the Czech Science Foundation through the
grants 23-04766S (BB & MK) and 21-01976S (DC & SH) as well as by the Ministry
of Education, Youth and Sport of the Czech Republic through the grant LL2105
CONTACT (BB), and 8J22AT017 (MK)

##  1\. Introduction

Mathematical modeling of (Cauchy) elastic solids aims at predicting the shape
of a specimen under the action of applied external forces or boundary
conditions. Hyperelasticity additionally assumes that the first Piola-
Kirchhoff stress tensor $T$ has a potential, and so it emphasizes the
conservative/nondissipative character of elasticity. This potential $W\geq 0$
has a physical meaning of the volume density of energy stored in the elastic
body. In the simplest setting, the stored energy depends only on the gradient
of the _deformation_ function $y:\bar{\Omega}\to\mathbb{R}^{n}$, where
$\Omega\subset\mathbb{R}^{n}$ is a reference configuration, i.e., the set
spanned by the specimen without applied loads. Typically, it is assumed to be
stressless and of zero stored energy, i.e., $T=0$ and $W=0$ if $y$ is the
identity map. The formula

(1.1) $\displaystyle T(x)=\frac{\partial W(\nabla y(x))}{\partial F}$

introduces the relationship of all mentioned quantities for $x\in\Omega$. Here
$F$ stands for a placeholder of $\nabla y$. As with any potential, $W$ can
only be identified up to an additive constant and it is typically assumed that
$W\geq 0$.

Given the hyperelasticity assumption, the _stable states_ of the elastic body
are the minimizers of the stored energy functional:

$y\mapsto\int_{\Omega}W(\nabla y)\mathrm{d}x-L(y)$

over the set of admissible deformations $\mathcal{A}$. Above, $L$ denotes a
functional representing work of external loads. In many occasions, it is
considered linear and continuous in the topology of deformations.

To ensure that the model is physically sound, several restrictions must be
imposed on the stored energy density $W$ and the set of deformations. Let
$n=3$ or $n=2$. Standard properties that can be found, e.g., in the book by
Ciarlet [13], include that every element in $\mathcal{A}$ is continuous,
_injective_ and orientation-preserving. Given that the reference configuration
is essentially arbitrary, it also makes sense to require that the inverse of a
physical deformations is also continuous. To fix terms, we define the set of
admissible deformations as

(1.2) $\displaystyle\mathcal{A}=\\{y:\Omega\to\mathbb{R}^{n}:y\in
W^{1,p}(\Omega;\mathbb{R}^{n}),y\text{ is a homeomorphism and
}\mathrm{det}\nabla y>0\text{ a.e.~{}in }\Omega\\},$

and $p>n$. Here $W^{1,p}(\Omega;\mathbb{R}^{n})$ stands for the usual Sobolev
space and the requirement that $\det\nabla y>0$ ensures orientation-
preservation. For the energy density $W$, we further impose frame-
indifference, i.e.,

$W(QF)=W(F)\quad\text{for all }Q\in\mathrm{SO}(n)\text{ and
}F\in\mathbb{R}^{n\times n}.$

It is also natural to impose a blow-up of the energy when $F$ is approaching
infinite extension or compression via

$W(F)\to\infty\quad\text{ when }|F|\to\infty\text{ or }\mathrm{det}\,F\to 0.$

We wish to emphasize that injectivity is intimately related to non-
interpenetration of matter and is thus of the uttermost importance for
physical processes. Indeed, an intact material can never undergo a deformation
that would map two points from the reference configuration to the same place.

Although it is important from the modeling point of view, this condition is
often disregarded in the mathematical literature. This is because many
difficult, and to-date open, questions; see e.g. [3, 5, 25] are connected with
the constraint on injectivity in variational problems. This is related not
only to static problems but also quasistatic and dynamic [25] because these
are often analyzed by means of time-discrete variational schemes. We highlight
particularly that it is largely unknown whether such functions can be
approximated by smooth injective ones or what are traces of maps from
$\mathcal{A}$ on $\partial\Omega$ although partial progress has been achieved
in the last years. We refer e.g. to [10, 23, 22].

Non-interpenetration of the matter seems to be almost completely unexplored in
_dimension reduction_ that is used to deduce models for a specimen whose one
dimension becomes negligible in comparison with other ones. This includes thin
films in the three dimensional space or rods in the plane. We give more
background on how a dimension reduction is usually performed in Section 2,
however the essence is to perform an appropriate limit when starting from a
bulk model. Up until today, to the best of the authors’ knowledge, no such
limit has been performed _once starting from a bulk model fully enforcing
injectivity_ (and thus ensuring non-interpenetration of matter) neither in
dimension $n=2$ or $n=3$ unless the limiting model is near to rigid; cf. [7,
8]. In some cases, we can recover the limit only in some constrained occasions
[17].

On the top of that, there seems not to be a common agreement how to define
non-interpenetration for lower dimensional objects. Clearly, in lower
dimensional structures, non-interpenetration cannot be set equal to
injectivity in this geometry as it is easy to imagine admissible deformations
violating injectivity like folding a thin sheet of paper. On the other hand,
any crossings in the thin film or rod must correspond to interpenetration. In
this sense, performing rigorously the above-mentioned limit could shed light
on this question, as the class of valid, _admissible and thus non-
interpenetrative_ deformations should also be a result of the limiting
process.

Within this contribution, we restrict our attention to dimension $n=2$ and
perform, to our best knowledge for the first time, a $\Gamma$-limit of bulk
energy functionals that take finite values _only_ on $\mathcal{A}$ (and are
infinite otherwise) when one dimension of the specimen approaches zero. We
perform the limit in the _membrane regime_ and thus the most flexible and thus
challenging one. In doing so, we arrive at energy functionals that only take
finite values on the _closure of homeomorphic functions_ in
$W^{1,p}((0,1);\mathbb{R}^{2})$ so that this set can be identified as the set
of non-interpenetrative deformations of rods. This corresponds to the idea of
Pantz [34] who proposed to take admissible deformations to be the closure of
embeddings isotopic to the reference configuration in the bulk or thin film
setting but did not perform a $\Gamma$-Limit. Of course, the topology of the
closure is important and a crucial step in our proof is to show that the weak
and strong closure of injective functions in $W^{1,p}((0,1);\mathbb{R}^{2})$
is the same and equals also the one given by the $C^{0}$-topology. This result
can be shown by an adaptation of the technique of DePhilippis and Pratelli
[18].

This paper is built up as follows: In Section 2 we start with providing some
background information on dimension reduction, in Section 3 we state then our
main results that are then proved in the last two sections of the paper.

##  2\. Background on dimension reduction

Many objects in mechanics can be treated in lower-dimensional geometry when
one dimension of the object is negligible compared to the other one or two.
For such objects, simplified models can be designed that are often simpler to
use in computational modeling [27, 28]. Analytically, on the static level,
this is built up on the fundamental results of Friesecke, James and Müller
[19, 20] that showed that it is possible to obtain such models as
$\Gamma$-limits [6] of bulk models.

Let us, at this point, give a short overview of the procedure and the results
obtained to date. As we consider a static situation here (and this is also the
setting of [19, 20]), the analysis starts from a bulk energy of the form111The
original results on [19, 20] are in dimension $n=3$ with the thin deformed
film described by a map from $\mathbb{R}^{2}$ to $\mathbb{R}^{3}$ but we alter
them to the planar situation considered here.

(2.1)
$\mathcal{E}_{h}(\tilde{y})=\int_{\Omega_{h}}W(\nabla\tilde{y}(\tilde{x}))\mathrm{d}\tilde{x}\qquad\text{
with }\Omega_{h}:=(0,1)\times(-h/2,h/2).$

It is more convenient to work on a fixed domain
$\Omega=(0,1)\times(-1/2,1/2)$, therefore we introduce for
$(x_{1},x_{2})\in\Omega$ and $y:\Omega\to\mathbb{R}^{2}$ the rescaling

(2.2) $\displaystyle y(x_{1},x_{2}):=\tilde{y}(x_{1},hx_{2})\ ,$

for $\tilde{y}:\Omega_{h}\to\mathbb{R}^{2}$ so that (2.1) takes the form

(2.3)
$\mathcal{E}_{h}(y)=h\int_{\Omega}W(\partial_{x_{1}}y(\tilde{x})|h^{-1}\partial_{x_{2}}y(\tilde{x}))\mathrm{d}\tilde{x}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\qquad\text{
with }\Omega:=(0,1)\times(-1/2,1/2).$

As the volume of the bulk object vanishes with the shrinking thickness, it is
natural that the same happens for its energy (2.1), which is even more
apparent in (2.3). Hence, we rescale the energy $\mathcal{E}_{h}$ just to
compensate for it. This scaling is the so-called a _membrane_ regime in which
one considers the limit $\mathcal{E}_{h}/h$ and thus only balances the
vanishing volume. This allows the incorporation of stretching or rotations of
the midplane. When taking $\alpha>1$ and considering
$\mathcal{E}_{h}/h^{\alpha}$ one obtains a whole hierarchy of more restrictive
models corresponding to non-linear bending, plate or shell theories (see
[20]); in this work however, we stick to the case $\alpha=1$.

To summarize, the thin-film model shall be obtained as an appropriate limit of
$\mathcal{E}_{h}/h$ as $h\to 0$. As we work in a variational setting, the
limiting process should also transfer information on minimizers and minimizing
sequences, which can be ensured by considering the $\Gamma$-convergence (see,
e.g., [6, 16]). Before we state the definition of the $\Gamma$-convergence
tailored to our setting we define for $p\geq 1$ an averaging operator
$\pi:L^{p}(\Omega;\mathbb{R}^{2})\to L^{p}((0,1);\mathbb{R}^{2})$ such that
for a.e. $0<t<1$

$\pi(u)(t):=\int_{-1/2}^{1/2}u(t,x)\mathrm{d}x.$

######  Definition 2.1.

Take $p\in(1,\infty)$. We say that a sequence of functionals
$\\{J_{h}\\}_{h}$, where $J_{h}:W^{1,p}(\Omega;\mathbb{R}^{2})\to\mathbb{R}$
$\pi$-$\Gamma$-converges to $J:W^{1,p}((0,1);\mathbb{R}^{2})\to\mathbb{R}$ if

  * •

for every sequence $\\{y_{h}\\}_{h>0}\subset W^{1,p}(\Omega;\mathbb{R}^{2})$
such that $\pi(y_{h})\to y$ in $L^{p}((0,1);\mathbb{R}^{2})$

$J(y)\leq\liminf_{h\to 0}J_{h}(y_{h});$

  * •

for every $u\in W^{1,p}((0,1);\mathbb{R}^{2})$ there exists a sequence of
functions $\\{u_{h}^{\mathrm{rc}}\\}_{h>0}\subset
W^{1,p}(\Omega,\mathbb{R}^{2})$ such that $\pi(u_{h}^{\mathrm{rc}})\to u$ in
$L^{p}((0,1);\mathbb{R}^{2})$ and

$J(u)\geq\limsup_{h\to 0}J_{h}(u_{h}^{\mathrm{rc}}).$

Hence, the definition of $\Gamma$-convergence includes two parts: a lower-
semicontinuity part and the construction of a so-called recovery sequence
$u_{h}^{\mathrm{rc}}$.

Under the crucial assumption of $p$-growth (for $p\in(1,\infty)$), i.e.,

$C_{1}(1+|F|^{p})\leq W(F)\leq C_{2}(1+|F|^{p})\qquad\text{ for some
}C_{1},C_{1}>0,$

it is proved in [19, 20] that $\frac{1}{h}\mathcal{E}_{h}$
$\pi$-$\Gamma$-converges to $\mathcal{E}$ with

$\mathcal{E}(y)=\int_{0}^{1}C\min_{\xi\in\mathbb{R}^{2}}W(y^{\prime}(x)|\xi)\mathrm{d}x,$

where $C$ denotes the convex hull which appears in the limit energy as
$\Gamma$-limits are always weakly lower semicontinuous, or, in other words,
the $\Gamma$ limiting process always involves a relaxation process, too.

######  Remark 2.2 (Higher dimensions).

Let us note that the original results in [19, 20] hold in a more general
setting than presented here. Indeed, besides investigating a larger class of
convergence regimes, there the bulk energy was defined for functions in $n=3$;
i.e. $\mathcal{E}_{h}:W^{1,p}(\omega\times(-1/2)\times(1/2);\mathbb{R}^{3})$
with $\omega\subset\mathbb{R}^{2}$. Then the limiting energy differs in the
relaxation means; i.e.

$\mathcal{E}(y)=\int_{\omega}Q\min_{\xi\in\mathbb{R}^{3}}W(\nabla_{p}y(x)|\xi)\mathrm{d}x,$

$Q$ is the quasiconvex hull, that is the largest quasiconvex function (in the
sense of Morrey [31, 32], see also [15]) below
$A\mapsto\min_{\xi\in\mathbb{R}^{3}}W(A|\xi)$.

The result has been further generalized by Anza-Hafsa and Madallena [1] (see
also [2]), building on the results of Ben Belgacem [4] and most importantly
the approximation results of Gromov and Eliashberg [21] in order to
incorporate local injectivity so that they could generalize the growth of $W$
to

$\displaystyle W(F)=+\infty\text{ if and only if }\operatorname{det}F\leq 0;$
for every $\delta>0$, there exists $c_{\delta}>0$ such that for all
$F\in\mathbb{R}^{2\times 2}$, if $\operatorname{det}F\geqslant\delta$ then
$\displaystyle W(F)\leq c_{\delta}\left(1+|F|^{p}\right),$

so that the local injectivity and the growth of the energy for infinite
compression could be included. Here, again the $\Gamma$-limit is given by
$\mathcal{E}$ as above, so remains unchanged. This may seem surprising at
first but the lower dimensionality of the object allows to approximate
(strongly) any function in $W^{1,p}((0,1);\mathbb{R}^{2})$ 222The result is
not limited to the plane and holds in higher dimensions, too. by locally
injective functions (see [21]) so that, in other words, local injectivity is
not a strong restriction in this context.

Imposing global injectivity in dimension reduction, however, stays largely
open to date. This issue has been addressed only in a couple of works that
obtained partial results. Olbermann and Runa [33] propose a definition of
interpenetration based on the Brouwer degree in the thin film setting. They
prove that in several scalings (but not the membrane scaling considered here),
interpenetrative deformations in the thin film can only be limits of
interpenetrative bulk deformations but do not give a full characterization of
the $\Gamma$-Limit. This full $\Gamma$-Limit has been found by Bresciani [7]
(see also [8]), however only in a very rigid scaling corresponding to the
linear von Kárman plate theory.

In some other works [30], non-interpenetration of the thin film is
characterized by introducing an “artificial thickness” so the bulk theory on
non-interpenetration is applicable. In the language of $\Gamma$-convergence
this means that a construction of a recovery sequence is prescribed and the
deformation of the thin film is said to be non-interpenetrative if this
prescription yields an injective bulk deformation.

##  3\. Main Results and Discussion

As in Section 2, we take the reference configuration to be
$\Omega_{h}:=(0,1)\times(-h/2,h/2)$, where $h>0$ denotes the thickness of the
specimen. Moreover, $\Omega:=\Omega_{1}$. We then introduce a family of
functionals

$I_{h}(\tilde{y}_{h}):=\begin{cases}\int_{\Omega_{h}}W(\nabla\tilde{y}(\tilde{x}))\,\mathrm{d}\tilde{x}&\quad\text{
if }\tilde{y}_{h}\in\mathcal{A}_{h},\\\ +\infty&\quad\text{ else,
}\end{cases}$

that enforces injectivity, i.e. the functional can only be finite on injective
deformations, and our aim is to pass to the limit $h\to 0^{+}$. Here
$\mathcal{A}_{h}$ is defined as in (1.2) with $\Omega_{h}$ instead of
$\Omega$. We perform the change of variables

$\varphi_{h}:\Omega\to\Omega_{h},\quad\varphi_{h}(x):=(x_{1},hx_{2})\quad\text{for
a.e. }(0,1)\times(-1/2;1/2).$

Setting $y_{h}:=\tilde{y}_{h}\circ\varphi_{h}$, and

$\nabla_{h}v:=\Big{(}\partial_{1}v\big{|}\frac{\partial_{2}v}{h}\Big{)}$

for every for every $v\in W^{1,p}(\Omega;\mathbb{R}^{2})$ yields
$\frac{1}{h}I_{h}$ leads to

(3.1) $\displaystyle
J_{h}(y_{h}):=\frac{1}{h}I_{h}(\tilde{y}_{h}):=\begin{cases}\int_{\Omega}W(\nabla_{h}y_{h}(x))\,\mathrm{d}{x}&\quad\text{
if }y_{h}\in\mathcal{A},\\\ +\infty&\quad\text{ else, }\end{cases}$

where we already introduced the _membrane scaling_ of the energy and
$\mathcal{A}$ is defined in (1.2) with $\Omega:=(0,1)\times(-1/2;1/2)$.

######  Remark 3.1 (Existence of minimizers in the bulk).

Let us stress that we do not investigate within this paper the existence of
minimizers if $J_{h}$ in (3.1), but just concentrate on the $\Gamma$-limit.
Hence, we do not put any convexity (or generalized convexity) assumptions on
$W$ and thus completely avoid the hard questions of the existence of
minimizers on injective function (see, e.g., [3]) outside the class of
polyconvex functionals. As the $\Gamma$-limit procedure, however, always also
contains relaxation [6], we know that the obtained lower dimensional object
will possess minimizers.

We will now investigate the $\Gamma$-limit as $h\to 0$ of the sequence of
functionals in (3.1) as introduced in Section 2. What we expect is that it is
particularly the _global non-interpenetration_ (or in other words global
injectivity) that needs to translate to the lower-dimensional object. The
intuition is that a thin bulk object can touch itself on the boundary upon
deformation but it cannot intersect. Once the thinning process continues ad
infinitum, the object consists only of a boundary that can touch. Thus,
injectivity as a characterizer of non-interpenetration has to be given up. On
the other hand, the property that intersections are not possible should
withstand the limiting procedure. As we shall see later, in Subsection 3.1 an
explicit characterization of this ”non-crossing” property is not easy, but the
following definition seems straightforward:

######  Definition 3.2.

Let $y\in W^{1,p}((0,1);\mathbb{R}^{2})$ with $p>1$. Then, we say that $y$ is
a _interpenetrative_ if for every bounded sequence $\\{y_{k}\\}\subset
W^{1,p}((0,1);\mathbb{R}^{2})$ such that
$\lim_{k\to\infty}\|y_{k}-y\|_{C^{0}((0,1);\mathbb{R}^{2})}=0$ there is
$K\in\mathbb{N}$ such that $\forall k\geq K$ $y_{k}$ is not a homeomorphisms.

In all other instances we say that $y\in W^{1,p}((0,1);\mathbb{R}^{2})$ with
$p>1$ is _non-interpenetrative_. In particular, for any such $y$ a bounded
sequence $\\{\tilde{y}_{k}\\}\subset W^{1,p}((0,1);\mathbb{R}^{2})$ of
homeomorphisms can be found such that
$\lim_{k\to\infty}\|\tilde{y}_{k}-y\|_{C^{0}((0,1);\mathbb{R}^{2})}=0$.

Indeed, the definition formalizes the intuitive idea that if the rod
intersected itself - as opposed to mere touching - then this needs to be the
case also for any other function in $W^{1,p}((0,1);\mathbb{R}^{2})$ that is
close enough to the original intersecting map. To prove, however, that this
definition of non-interpenetration is physically correct, we need to show that
$\Gamma$-limits of functional $J_{h}$ can only be finite on the class non-
interpenetrative functions.

We will show that this is indeed the fact and claim that the $\Gamma$-limit of
(3.1) is the following functional

(3.2)
$J(y(x)):=\begin{cases}\int_{0}^{1}C\min_{\xi\in\mathbb{R}^{2}}W(y^{\prime}(x)|\xi)\mathrm{d}{x}&\quad\text{
if }{y}\text{ is non-interpenetrative},\\\ +\infty&\quad\text{ else,
}\end{cases}$

where $C$ is the convex envelope.

This is formalized through the following theorem:

######  Theorem 3.3.

Let
$p\in\color[rgb]{0.46875,0,0.46875}\definecolor[named]{pgfstrokecolor}{rgb}{0.46875,0,0.46875}[\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}2,\infty)$
and suppose that $W$ in (3.1) is a continuous function on $\mathbb{R}^{2\times
2}$ with $\det(\cdot)>0$ while

$W(F)=+\infty\text{ if and only if }\operatorname{det}F\leq 0$

and satisfies the following growth condition: there exits a $c>0$ and for
every $\delta>0$, there exists $c_{\delta}>0$ such that for all
$F\in\mathbb{R}^{2\times 2}$ with $\operatorname{det}F\geqslant\delta$ it
holds

$\displaystyle c|F|^{p}\leq W(F)\leq c_{\delta}\left(1+|F|^{p}\right).$

Then, the functionals $J_{h}$ from (3.1) $\pi-\Gamma$ converge (in the sense
of Definition 2.1) to $J$ given in (3.2).

The proof of Theorem 3.3 relies crucially on the fact that the set non-
interpenetrative deformations can characterized as the closure of
homeomorphisms in $W^{1,p}((0,1);\mathbb{R}^{2})$ with respect to the weak and
strong topologies and that these are the same. Indeed, we shall use the
availability of the first closure while proving the ”liminf”-condition and the
latter to provide a recovery sequence. We summarize this statement in the
following theorem:

######  Theorem 3.4.

Let $p\in(1,\infty)$ and $y\in W^{1,p}((0,1),\mathbb{R}^{2})$. The following
conditions are equivalent

  1. (1)

The map $y\in W^{1,p}((0,1),\mathbb{R}^{2})$ is non-interpenetrative.

  2. (2)

There exists a bounded sequence of homeomorphisms $\\{y_{k}\\}\in
W^{1,p}((0,1);\mathbb{R}^{2})$ such that $y_{k}\rightharpoonup y\in
W^{1,p}((0,1);\mathbb{R}^{2})$.

  3. (3)

There exists a bounded sequence of homeomorphisms $\\{\tilde{y}_{k}\\}\in
W^{1,p}((0,1);\mathbb{R}^{2})$ such that $\tilde{y}_{k}\to y\in
W^{1,p}((0,1);\mathbb{R}^{2})$. Moreover, the approximating sequence can be
chosen in such a way that all $y_{k}$ are piecewise affine.

###  3.1. Discussion of non-interpenetration

The definition of non-interpenetration in Definition 3.2 is, albeit natural,
quite implicit. Indeed, it relies on constructing an approximating sequence,
which can be a challenging task in general situations. So, it is a natural
question if a more explicit and an easy-to-check condition can be given as a
characterization of non-interpenetration. Hence, we formulate the following
open problem:

######  Problem 3.5.

Is there a characterization of non-interpenetration (as defined in Definition
3.2) that can be easily checked be found?

Within this paper, we do not solve Problem 3.5, but we shall at least explore
some ideas that might look tempting but turn out to be insufficient. The
discussion thus will indicate that still non-interpenetration on lower
dimensional objects will need more exploration. We do, however, refer the
interested reader to the paper [9] for a discussion on related topics in the
$\mathbb{R}^{2}\to\mathbb{R}^{2}$ case.

One obvious direction that one could try is to look for inspiration in the
bulk setting. There, the so-called Ciarlet-Nečas condition (see [14]) that
reads as follows

(3.3) $\displaystyle\int_{\Omega_{h}}{\rm
det}\,\nabla\tilde{y}_{h}(x)\,\mathrm{d}x\leq\mathcal{L}^{n}(y_{h}(\Omega_{h}))$

can be imposed on $\tilde{y}_{h}\in\mathcal{A}_{h}$ in order to ensure
injectivity of deformations almost everywhere in the deformed configuration
$\tilde{y}_{h}(\Omega_{h})$. This condition, however, provides only scant
information if we pass to the lower-dimensional objects. Indeed, recall that
$\Omega=(0,1)\times(-1/2;1/2)$. As outlined in Section above, we perform the
change of variables

$\varphi_{h}:\Omega\to\Omega_{h},\quad\varphi_{h}(x):=(x_{1},hx_{2})\quad\text{for
a.e. }(0,1)\times(-1/2;1/2).$

Setting $y:=y_{h}\circ\varphi_{h}$, and

$\nabla_{h}v:=\Big{(}\partial_{1}v\big{|}\frac{\partial_{2}v}{h}\Big{)}$

for every for every $v\in W^{1,p}(\Omega;\mathbb{R}^{2})$. The limit passage
for $h\to 0$ in (3.3) leads to

$\displaystyle\int_{0}^{1}(\partial_{1}y_{h})^{\perp}\cdot b\,{\rm
d}S\leq\lim_{h\to 0}\frac{\mathcal{L}^{2}(y_{h}(\Omega_{h})}{h}$

where $b$ is a Cosserat vector obtained as $b=\lim_{h\to
0}h^{-1}\partial_{2}y_{h}$ in $W^{1,p}(\Omega;\mathbb{R}^{2})$ and for any
$a\in\mathbb{R}^{2}$, we have that $(a_{1},a_{2})^{\perp}=(-a_{2},a_{1})$. In
particular, if $b=(\partial_{1}y)^{\perp}/|(\partial_{1}y)^{\perp}|$, i.e., it
is the unit normal vector to the film in the deformed configuration, and if
$\lim_{h\to 0}\mathcal{L}^{2}(y_{h}(\Omega_{h}))/h=\mathcal{H}^{1}(y((0,1))$
we get

(3.4) $\displaystyle\int_{\omega}|(\partial_{1}y)^{\perp}|\,{\rm
d}S\leq\mathcal{H}^{1}(y((0,1))).$

The left-hand side of (3.4) is the length of the film calculated by the
change-of-variables formula while the right-hand side is the measured length.
Hence, (3.4) is violated by a folding deformation, which should be admissible
among the family of realistic thin-film deformations, while (3.4) is satisfied
if the film crosses itself, which violates non-self-interpenetration of matter
and is hence not admissible.

Another tempting path may be to characterize non-interpenetration by using
degree theory at least in the case when the limit deformation $y$ satisfies
that $y(0)=y(1)$ so we can assume that $y$ is defined on the unit circle in
$\mathbb{R}^{2}$ instead of $(0,1)$. Now, every injective continuous embedding
$y$ of the circle into the plane, decomposes the plane into two domains: a
bounded domain where the degree with respect to $\varphi$ is one (or minus
one) and the unbounded domain where the degree is zero. Since the degree is
stable under uniform convergence, one could ask whether the set of non-
interpenetrative functions is (at least in this special setting) not identical
to

${\mathcal{D}}:=\bigl{\\{}y\in W^{1,p}((0,1),\mathbb{R}^{2}):\ y(0)=y(1)\text{
and }\deg(x,y)\in\\{0,1\\}\text{ for a.e. }x\in\mathbb{R}^{2}\bigr{\\}}.$

But this is not the case as we show in Example 3.6:

######  Example 3.6.

(Horned devil) Imagine a sequence of (non-injective) mappings $y_{k}$ that
converge to $y$ as depicted in Fig. 1. It is not difficult to check that
$\deg(\cdot,y)=1$ both in the rectangle in the target and in both ”horns”.

a $k\to\infty$$deg(\cdot,y)=1$

Figure 1. Example of $y$ in ${\mathcal{D}}$ which cannot be approximated by
injective curves.

The function $y$ cannot be approximated by injective curves due to its
”horns”. However, (as Figure 1 shows) the approximating sequence can be chosen
such that it is injective in the left horn, but then necessarily a point of
non-injectivity has to appear in the right horn. Similarly, by symmetry, we
can approximate the right horn, but then we cannot approximate the left horn.
Therefore no simple condition for $y\in{\mathcal{D}}$ can fix this as locally
everything is fine and we run in the problem much later on.

To close the discussion on Problem 3.5 let us point out that the study of the
intersection theory in algebraic topology seems to be very related. Indeed,
the concept of _weakly simple curves_ , i.e., those for which, for each
$\varepsilon>0$, there exists a simple curve whose Frechet distance to the
studied one is at most $\varepsilon$, is closely related to non-
interpenetration. A characterization of weakly simple curves, even in the
piecewise affine setting, is highly desirable and also the only possibilities
available up to know seem to be algorithmic [12].

Finally, let us remark that while it seems plausible that non-interpenetration
in lower-dimensional objects will always be characterized by the possibility
of approximating the deformation by injective ones, the methods presented in
this paper only apply to the case of rods in the plane and deformations in
$W^{1,p}((0,1);\mathbb{R}^{2})$. However, as this setting does not penalize
rapid oscillations upon relaxation, it cannot be avoided (by any growth of
energy as $\mathrm{det}F\to 0$) that the null function lies in the set of
acceptable deformations for the rod. This paradox could most probably be only
prevented by including higher gradients of deformation in the energy or by
using a different scaling regime. Yet, it seems to be unknown if an equivalent
of Theorem 3.4 can hold in a smoother setting, i.e. we have the following:

######  Problem 3.7.

Does it hold that the weak and strong closure of homeomorphisms are the same
in $W^{k,p}((0,1);\mathbb{R}^{2})$ with $k\geq 2$?

##  4\. Proof of Theorem 3.3

In this section, we give the detailed proof of Theorem 3.3. It requires a few
auxiliary results. In the first one, we show that convexity is a crucial
property for sequential weak lower semicontinuity of integral functionals
along sequences of injective maps.

######  Lemma 4.1.

Suppose that $f:\mathbb{R}^{2}\to\mathbb{R}\cup\\{+\infty\\}$ is finite
everywhere except perhaps at zero. Then the integral functional

$I(y)=\int_{0}^{1}f(y^{\prime}(x))\mathrm{d}x$

with $I:W^{1,p}((0,1);\mathbb{R}^{2})\to\mathbb{R}$ is weakly lower
semicontinuous along sequences of homeomorphisms if and only if $f$ is convex
on $\mathbb{R}^{2}$, i.e., finite everywhere.

###### Proof.

One implication of the proof is automatic; it is well-known that the convexity
of $f$ ensures weak lower semicontinuity even in larger sets than those of
homeomorphisms.

Let us thus turn to proving necessity of the convexity. Let us take two
vectors $a,b\in\mathbb{R}^{2}$ and $t\in(0,1)$.

Let us suppose at first that $a$ and $b$ are not co-linear. Then for every
$n$, we may define the piecewise affine function $y_{n}$ through $y_{n}(0)=0$
and further inductively for $k=1\ldots n$ via

$y_{n}(x)=\begin{cases}y_{n}(\frac{k-1}{n})+a(x-\frac{k-1}{n})&\text{ for
}x\in(\frac{k-1}{n},\frac{k-1}{n}+\frac{t}{n}]\\\
y_{n}(\frac{k-1}{n}+\frac{t}{n})+b(x-\frac{k-1}{n}-\frac{t}{n})&\text{ for
}x\in(\frac{k-1}{n}+\frac{t}{n},\frac{k}{n}]\end{cases}$

As defined, the functions $y_{n}$ are continuous, piecewise affine, and
injective. Moreover, $y_{n}\rightharpoonup y$ where $y$ is the affine function
$y=\big{(}ta+(1-t)b\big{)}x$. Thus, if $I$ is weakly lower semicontinuous
along the sequence $\\{y_{n}\\}$, we have that

$\displaystyle\liminf_{n\to\infty}I(y_{n})$
$\displaystyle=\liminf_{n\to\infty}\sum_{k=1}^{n}\int_{\frac{k-1}{n}}^{\frac{k-1}{n}+\frac{t}{n}}f(a)\mathrm{d}x+\int_{\frac{k-1}{n}+\frac{t}{n}}^{\frac{k}{n}}f(b)\mathrm{d}x$
(4.1) $\displaystyle=tf(a)+(1-t)f(b)\geq f(ta+(1-t)b),$

the convexity condition for $a,b$ not co-linear.

It is well-known that a convex function is continuous (see e.g. [15]).
However, even the weakened condition in (4) assures that $f$ is continuous in
all points except for zero. We give the proof of this statement here for self-
containment, even if it follows the lines of the standard argument:

Take some point $0\neq a\in\mathbb{R}^{2}$ and let us assume that $f$ is not
continuous at $a$ so that there exists a sequence
$\\{a_{n}\\}_{n\in\mathbb{N}}$ such that $a_{n}\to a$ but
$f(a_{k})\nrightarrow f(a)$. As $a\neq 0$ at least one of the components of
$a$ needs to be non-zero. Let us assume that it is the first one. Without any
loss of generality (and upon passing to a subsequence if necessary) we may
assume that the sequences of the components, denoted as $\\{a^{1}\\}_{n}$ and
$\\{a^{2}\\}_{n}$, are monotone and at least one being strictly monotone.
Here, and in the following, we will always indicate the _component_ by the
upper index to avoid confusion with the sequence index.

Further, we may ask that exactly one of the following conditions holds

(4.2) $f(a_{n})-f(a)\geq\varepsilon\quad\text{or}\quad
f(a)-f(a_{n})\geq\varepsilon\quad\forall n\in\mathbb{N}.$

for some $\varepsilon>0$

Let us suppose that the first condition is met. We take some fixed element
$\tilde{a}\subset\\{a\\}_{n}$ such that $\tilde{a}^{1}\neq 0$ and
$\tilde{a}^{1}\neq a^{1}$ 333 If no such element can found, it means that for
$n$ big enough $a^{1}=a^{1}_{n}$ so that we instead rewrite
$a_{n}=t_{1}\begin{pmatrix}a^{1}\\\ \tilde{a}^{2}\\\
\end{pmatrix}+(1-t_{1})\begin{pmatrix}a^{1}\\\ a^{2}\\\ \end{pmatrix},$ which
makes the proof even easier.  and decompose elements $a_{n}$ (for $n$ big
enough) as

$a_{n}=r_{n}\begin{pmatrix}\tilde{a}^{1}\\\ a_{n}^{2}\\\
\end{pmatrix}+(1-r_{n})\begin{pmatrix}a^{1}\\\ a_{n}^{2}\\\ \end{pmatrix}$

with $r_{n}\in[0,1]$ which has to satisfy that $r_{n}\to 0$ as $n\to\infty$.
As the above convex combinations contains two vectors that are not co-linear,
we may use the above convexity property to show that

$f(a_{n})\leq r_{n}f\left(\begin{pmatrix}\tilde{a}^{1}\\\ a_{n}^{2}\\\
\end{pmatrix}\right)+(1-r_{n})f\left(\begin{pmatrix}a^{1}\\\ a_{n}^{2}\\\
\end{pmatrix}\right)$

Decomposing further (if $a^{2}\neq\tilde{a}^{2}$, otherwise this step is not
necessary)

$\begin{pmatrix}a^{1}\\\ a_{n}^{2}\\\
\end{pmatrix}=s_{n}\begin{pmatrix}a^{1}\\\ \tilde{a}^{2}\\\
\end{pmatrix}+(1-s_{n})\begin{pmatrix}a^{1}\\\ a^{2}\\\ \end{pmatrix}\text{
and }\begin{pmatrix}\tilde{a}^{1}\\\ a_{n}^{2}\\\
\end{pmatrix}=t_{n}\begin{pmatrix}\tilde{a}^{1}\\\ \tilde{a}^{2}\\\
\end{pmatrix}+(1-t_{n})\begin{pmatrix}\tilde{a}^{1}\\\ a^{2}\\\ \end{pmatrix}$

with the numbers $s_{n},t_{n}\in[0,1]$ converging to 0 as $n\to\infty$. Thus,
we get

$\displaystyle f(a_{n})$ $\displaystyle\leq
r_{n}f\begin{pmatrix}\tilde{a}^{1}\\\ a_{n}^{2}\\\
\end{pmatrix}+(1-r_{n})f\begin{pmatrix}a^{1}\\\ a_{n}^{2}\\\ \end{pmatrix}$
$\displaystyle\leq r_{n}\left(s_{n}f\begin{pmatrix}\tilde{a}^{1}\\\
\tilde{a}^{2}\\\ \end{pmatrix}+(1-s_{n})f\begin{pmatrix}\tilde{a}^{1}\\\
a^{2}\\\ \end{pmatrix}\right)+(1-r_{n})\left(t_{n}f\begin{pmatrix}a^{1}\\\
\tilde{a}^{2}\\\ \end{pmatrix}+(1-t_{n})f\begin{pmatrix}a^{1}\\\ a^{2}\\\
\end{pmatrix}\right)$ $\displaystyle\leq f(a)+C\max\\{r_{n},s_{n},t_{n}\\},$

where $C$ is a fixed number given by the values of the function $f$ in the
fixed vectors constructed as above. Combining with (4.2) we see that

$f(a)+\varepsilon\leq f(a_{n})\leq f(a)+C\max\\{r_{n},s_{n},t_{n}\\},$

which yields a contradiction for $n$ large enough because
$\max\\{r_{n},s_{n},t_{n}\\}\to 0$ as $n\to\infty$.

If the second condition in (4.2) holds, we show the claim in the case when
$a^{1},a^{2}>0$ and both sequences $\\{a^{1}\\}_{n}$ and $\\{a^{2}\\}_{n}$ a
strictly monotone decreasing to $a^{1}$ and $a^{2}$ with modification to the
other cases being straightforward. Then we decompose

$a=t_{n}\begin{pmatrix}1/2a^{1}\\\ a^{2}\\\
\end{pmatrix}+(1-t_{n})\begin{pmatrix}a_{n}^{1}\\\ a^{2}\\\
\end{pmatrix}\text{ and }\begin{pmatrix}a_{n}^{1}\\\ a^{2}\\\
\end{pmatrix}=r_{n}\begin{pmatrix}a_{n}^{1}\\\ 1/2a^{2}\\\
\end{pmatrix}+(1-r_{n})\begin{pmatrix}a_{n}^{1}\\\ a_{n}^{2}\\\ \end{pmatrix}$

Now, we analogously to above use again the convexity property to show that

$f(a_{n})+\varepsilon\leq f(a)\leq f(a_{n})+C\max\\{r_{n},t_{n}\\},$

which is again a contradiction for $n$ large enough.

Having the continuity property at our disposal we may show the convexity
inequality even for co-linear non-zero vectors $a$ and $b$ by defining

$a_{n}=\begin{pmatrix}a^{1}+1/n\\\ a^{2}\\\ \end{pmatrix}$

and taking the sequence

$y_{n}(x)=\begin{cases}y_{n}(\frac{k-1}{n})+a_{n}(x-\frac{k-1}{n})&\text{ for
}x\in(\frac{k-1}{n},\frac{k-1}{n}+\frac{t}{n}]\\\
y_{n}(\frac{k-1}{n}+\frac{t}{n})+b(x-\frac{k-1}{n}-\frac{t}{n})&\text{ for
}x\in(\frac{k-1}{n}+\frac{t}{n},\frac{k}{n}].\end{cases}$

Now, all $y_{n}$ are piecewise affine and injective and converge weakly to the
affine function $y(x)=(ta+(1-t)b)x$, which now may even be the zero function.
Using the definition of weak lower semicontinuity again shows that

$\liminf_{n\to\infty}tf(a_{n})+(1-t)f(b)\geq f(ta+(1-t)b)$

and thus relying on the continuity of $f$ in $a$, the claim. At this point, we
already know that $f$ is finite in 0 and analogously to above, we show that it
is continuous there, too. Now, we may again iterate and show the convexity
property for all vectors when possibly replacing the zero vector by
$\begin{pmatrix}1/n\\\ 0\\\ \end{pmatrix}$ in the construction.

∎

As, even on the set of homeomorphisms, weak-lower semicontinuity of an
integral functional on $W^{1,p}((0,1);\mathbb{R}^{2})$ translates to the
convexity of its density, it is natural to expect that the convex hull will
also be in the relaxation of functionals taking infinite values outside the
set of homeomorphisms. This is formalized as follows

######  Lemma 4.2 (Relaxation).

Let $p\in(1,\infty)$. Suppose that
$f:\mathbb{R}^{2}\to\mathbb{R}\cup\\{+\infty\\}$, $f\geq 0$, is finite and
continuous everywhere except at zero, where it is infinite. Moreover, suppose
that for every $\delta>0$, there exists $c_{\delta}>0$ such that it holds for
all $a\in\mathbb{R}^{2}$, $|a|\geqslant\delta$ that

$f(a)\leq c_{\delta}\left(1+|a|^{p}\right).$

Then the relaxation of the integral functional

$\mathcal{I}_{f}(y)=\begin{cases}\int_{0}^{1}f(y^{\prime}(x))\mathrm{d}{x}&\quad\text{
if }{y}\text{ is a homeomorphism},\\\ +\infty&\quad\text{ else, }\end{cases}$

with respect to the weak topology in $W^{1,p}((0,1);\mathbb{R}^{2})$, that is
the functional

(4.3)
$\mathcal{I}_{f}^{\text{rel}}(y):=\inf\\{\liminf_{k\to\infty}\mathcal{I}_{f}(y_{k});\\{y_{k}\\}_{k\in\mathbb{N}}\subset
W^{1,p}((0,1),\mathbb{R}^{2}),y_{k}\rightharpoonup y\text{ in
}W^{1,p}((0,1);\mathbb{R}^{2})\\}$

is given by

$\mathcal{I}_{f}^{\text{C}}(y)=\begin{cases}\int_{0}^{1}Cf(y^{\prime}(x))\mathrm{d}{x}&\quad\text{
if }{y}\text{ is non-interpenetrative},\\\ +\infty&\quad\text{ else,
}\end{cases}$

where $Cf$ is the convex envelope of $f$, which is defined as

$Cf=\sup\\{g:\mathbb{R}^{2}\to\mathbb{R}\textrm{ convex}:g\leq f\\}.$

Moreover, the sequence of homeomorphisms realizing the infimum in (4.3) can be
chosen as a piecewise affine.

######  Remark 4.3.

Notice that the relaxation corresponds to the weakly lower semicontinuous
envelope of the functional $\mathcal{I}_{f}$. It is well known that a
relaxation is a special case of the $\Gamma$-limit with a constant sequence of
functionals [6]. Thus, as in the general case, we know that the relaxed
functional has a minimizer which can be reached by infimizing sequences of the
original functional.

######  Remark 4.4 (Attaining the convex envelope).

Let us remind the reader that, since have that $f:\mathbb{R}^{2}\to\mathbb{R}$
is continuous away from zero ($f(0)=+\infty$, and that
$\lim_{|a|\to\infty}f(a)=+\infty$) a combination of three vectors is
sufficient to calculate the convex envelope up to a small error of size
$\gamma$. To be more precise, for a given $\xi\in\mathbb{R}^{2}$, and
$\gamma>0$ there are some $t^{\gamma}_{i}\geq 0$ with
$\sum_{i=1}^{3}t^{\gamma}_{i}=1$, and $a_{i}^{\gamma}\in\mathbb{R}^{2}$ such
that $\xi=\sum_{i=1}^{3}t_{i}a_{i}^{\gamma}$, and the convex envelope $Cf$
evaluated at $\xi$ satisfies (see [15, Thm. 2.35])

$Cf(\xi)+\gamma>\sum_{i=1}^{3}t_{i}^{\gamma}f(a_{i}^{\gamma})\ .$

###### Proof of Lemma 4.2.

Let us choose $y\in W^{1,p}((0,1);\mathbb{R}^{2})$ and an arbitrary sequence
$\\{y_{k}\\}_{k\in\mathbb{N}}\subset W^{1,p}((0,1);\mathbb{R}^{2})$ with
$y_{k}\rightharpoonup y$. Let us distinguish two cases: first, assume that $y$
is interpenetrative. Then, according to Theorem 3.4, there cannot exist a
sequence of homeomorphisms converging weakly to $y$, for otherwise, $y$ would
be non-interpenetrative. So, in this case

$\mathcal{I}_{f}^{C}(y)=\mathcal{I}_{f}^{\text{rel}}(y)=+\infty.$

Let us thus concentrate on the case when $y$ is non-interpenetrative and
$\\{y_{k}\\}$ is a sequence of homeomorphisms converging weakly converges to
$y$. We then have

$\liminf_{k\to\infty}\mathcal{I}_{f}(y_{k})=\liminf_{k\to\infty}\int_{0}^{1}f(y_{k}^{\prime})\,\mathrm{d}x\geq\liminf_{k\to\infty}\int_{0}^{1}Cf(y_{k}^{\prime})\,\mathrm{d}x\geq\mathcal{I}_{f}^{C}(y^{\prime}),$

and since this is independent of the particular sequence, we obtain
$\mathcal{I}_{f}^{\text{rel}}(y)\geq\mathcal{I}_{f}^{C}(y)$.

Now, we need to show the opposite inequality. To this end, we construct a
sequence $\\{\bar{y}_{k}^{n}\\}\subset W^{1,p}((0,1);\mathbb{R}^{2})$ such
that

$\mathcal{I}_{f}^{C}(y)\geq\lim_{k\to\infty}\mathcal{I}(\bar{y}_{k}).$

First, as $y$ is non-interpenetrative, we may appeal to Theorem 3.4 and choose
a _piece-wise affine_ approximating (in the strong topology) sequence
$\\{\bar{y}_{k}\\}\subset W^{1,p}((0,1);\mathbb{R}^{2})$ such that all
$\bar{y}_{k}$ are homeomorphisms. Now, because $Cf$ is a convex function (and
thus finite and continuous), we can fix $\delta>0$ and find a constant
$\tilde{c}$ such that $Cf(a)\leq\tilde{c}$ if $|a|\leq\delta$. Moreover, for
$|a|\geq\delta$ the function $Cf$ inherits the growth condition of $f$ so that
in sum

$Cf(a)\leq c(1+|a|^{p})\qquad\text{ on }\mathbb{R}^{2}.$

Thus, we may use the continuity of Nemytskii mappings (see e.g. [36]) to
assert that

$\lim_{k\to\infty}\mathcal{I}_{f}^{C}(\bar{y}_{k})=\mathcal{I}_{f}^{C}(y),$

so that when choosing a subsequence of $k$’s if necessary, we have that

$\mathcal{I}_{f}^{C}(y)\geq\mathcal{I}_{f}^{C}(\bar{y}_{k})-\frac{1}{k}.$

Let us now fix $k$ and one $\bar{y}_{k}$. We can assume $\bar{y}_{k}$ is
affine on the intervals $\left[x_{k}^{l},x_{k}^{l+1}\right)$, $l=0\ldots
N(k)$. We may find a numbers $\beta(k)$, $n(k)$ such that

$|x_{k}^{l+1}-x_{k}^{l}|\gg\beta(k)\text{ for all }l=0\ldots N(k),$

and, moreover, for all
$x\in\left(x_{k}^{l}+\beta(k),x_{k}^{l+1}-\beta(k)\right)$ with $l=0\ldots
N(k)$ we have that

$B\Big{(}\bar{y}_{k}(x),\frac{1}{n(k)}\Big{)}\cap\bar{y}_{k}(\tilde{x})=\emptyset\text{
for all }\tilde{x}\in(x_{k}^{\ell},x_{k}^{\ell+1})\text{ with }l\neq\ell.$

In other words, as long as we are at a $\beta(k)$-distance from the points
$x_{k}^{l}$ we can find a neighborhood around the piecewise affine function,
where we can perturb it without interfering with injectivity. That is exactly
what we will do and we will construct a function $\tilde{y}_{k}^{n}$ by
replacing $\bar{y}_{k}$ by suitable oscillations in order to attain the convex
envelope.

Let us detail the particular construction on one interval
$(x_{k}^{l},x_{k}^{l+1})$, where we denote $\xi:=\bar{y}_{k}^{\prime}$. Then
we know from Remark 4.4 that for a given $\gamma>0$ there are some
$t_{\gamma,i}\geq 0$ and $a_{\gamma,i}\in\mathbb{R}^{2}$ satisfying
$\sum_{i=1}^{3}t_{\gamma,i}=1$ and
$\xi=\sum_{i=1}^{3}t_{\gamma,i}a_{\gamma,i}$ such that

$Cf(\xi)+\gamma>\sum_{i=1}^{3}t_{\gamma,i}f(a_{\gamma,i})\ .$

Let us define for $x\in[0,1]$ the auxiliary function

$y^{\gamma}(x)=\begin{cases}a_{\gamma,1}x&\text{ if }x\in[0,t_{\gamma,1}],\\\
a_{\gamma,2}(x-t_{\gamma,1})+a_{\gamma,1}t_{\gamma,1}&\text{ if
}x\in[t_{\gamma,1},t_{\gamma,1}+t_{\gamma,2}],\\\
a_{\gamma,3}(x-t_{\gamma,1}-t_{\gamma,2})+a_{\gamma,1}t_{\gamma,1}+a_{\gamma,3}t_{\gamma,2}&\text{
if }x\in[t_{\gamma,1}+t_{\gamma,2},1],\end{cases}$

Now, as defined, $y^{\gamma}$ might not be a homeomorphism at this point. This
could happen if either one of the vectors $a_{\gamma,i}$ is the zero vector,
or if two (or all three) of the vectors are mutually co-linear. As $f$ is
infinite at 0, we may exclude the first case, but the second might still
happen. If so, we may assume, without loss of generality, that $a_{\gamma,1}$
and $a_{\gamma,2}$ are co-linear and $a_{\gamma,3}$ is either non-colinear
with either or aligned with $a_{\gamma,2}$ (which means that
$a_{\gamma,3}=\alpha\widetilde{a}_{\gamma,2}$ with $\alpha>0$). As
$a_{\gamma,1}\neq 0$ we can find one component of $a_{\gamma,1}$ which is non-
zero and without loss of generality suppose it is the first one. Then we can
use the alternative definition

$y^{\gamma}(x)=\begin{cases}\begin{pmatrix}a_{\gamma,1}^{1}+a_{\gamma,1}^{1}/2\\\
a_{\gamma,1}^{2}\\\ \end{pmatrix}x&\text{ if }x\in[0,t_{\gamma,i1}/2],\\\
\begin{pmatrix}a_{\gamma,1}^{1}+a_{\gamma,1}^{1}/2\\\ a_{\gamma,1}^{2}\\\
\end{pmatrix}\frac{t_{\gamma,1}}{2}+\begin{pmatrix}a_{\gamma,1}^{1}-a_{\gamma,1}^{1}/2\\\
a_{\gamma,1}^{2}\\\ \end{pmatrix}(x-\frac{t_{\gamma,1}}{2})&\text{ if
}x\in[t_{\gamma,1}/2,t_{\gamma,1}]\\\
a_{\gamma,2}(x-t_{\gamma,1})+a_{\gamma,1}t_{\gamma,1}&\text{ if
}x\in[t_{\gamma,1},t_{\gamma,1}+t_{\gamma,2}],\\\
a_{\gamma,3}(x-t_{\gamma,1}-t_{\gamma,2})+a_{\gamma,1}t_{\gamma,1}+a_{\gamma,2}t_{\gamma,2}&\text{
if }x\in[t_{\gamma,1}+t_{\gamma,2}],\end{cases}$

where we, as before, denoted vector components by upper indices.

Now, $y^{\gamma}$ is again a homeomorphism that equals to $\xi\dot{x}$ at the
boundary of $(0,1)$.

We now scale the found functions $y^{\gamma}$ and set and for $n\in\mathbb{N}$
and $1\leq i\leq n$

$y^{\gamma}_{n}(x)=\frac{1}{n}y^{\gamma}(nx)$

if $0\leq x\leq 1/n$ and if $i>1$

$\displaystyle y^{\gamma}_{n}(x)$
$\displaystyle=y^{\gamma}_{n}(x-(i-1)/n)+y^{\gamma}_{n}((i-1)/n),\
x\in((i-1)/n,i/n)\ $

Then $y^{\gamma}_{n}\rightharpoonup\xi\cdot x$ and $y^{\gamma}_{n}$ equals to
$\xi\cdot x$ at the end points of the interval $(0,1)$.

Now, a calculation analogous to (4), the fact that $Cf\leq f$, and Lemma 4.1
show that

$Cf(\xi)=\int_{0}^{1}Cf(\xi)\,\mathrm{d}x\leq\lim_{n\to\infty}\int_{0}^{1}f((y^{\gamma}_{n})^{\prime})\,\mathrm{d}x<\int_{0}^{1}Cf(\xi)\,\mathrm{d}x+\gamma=Cf(\xi)+\gamma.$

As $\gamma>0$ was arbitrary a uniform bound on $(y^{\gamma}_{n})^{\prime}$
implied by coercivity of $f$ allows us to extract a diagonal sequence for
$\gamma\to 0$ and $n\to\infty$ such that

$Cf(\xi)=\int_{0}^{1}Cf(\xi)\,\mathrm{d}x=\lim_{n\to\infty}\int_{0}^{1}f((y^{\gamma(n)}_{n})^{\prime})\,\mathrm{d}x,$

as well as

$Cf(\xi)\geq\int_{0}^{1}f((y^{\gamma(n)}_{n})^{\prime})\,\mathrm{d}x-\frac{1}{n}.$

We now set

$\bar{y}_{k}^{n}(x)=\begin{cases}\bar{y}_{k}(x)&\text{ if
}x\in[x_{k}^{l},x_{k}^{l}+\beta(k)]\text{ or
}[x_{k}^{l+1}-\beta(k),x_{k}^{l+1})\\\ \bar{y}_{k}(x_{k}^{l}+\beta(k))\\\
\quad+(x_{k}^{l+1}-x_{k}^{l}-2\beta(k))y^{\gamma(n)}_{n}\big{(}\frac{x-x_{k}^{l}-\beta(k)}{x_{k}^{l+1}-x_{k}^{l}-2\beta(k)}\big{)}&\text{
if }x\in(x_{k}^{l}+\beta(k),x_{k}^{l+1}-\beta(k))\end{cases}$

and by a change of variables we obtain that

$\displaystyle\int_{x_{k}^{l}+\beta(k)}^{x_{k}^{l+1}-\beta(k)}Cf(\bar{y}_{k}^{\prime})\mathrm{d}x$
$\displaystyle=\int_{x_{k}^{l}+\beta(k)}^{x_{k}^{l+1}-\beta(k)}Cf(\xi)\mathrm{d}x=(x_{k}^{l+1}-x_{k}^{l}-2\beta(k))Cf(\xi)$
$\displaystyle\geq(x_{k}^{l+1}-x_{k}^{l}-2\beta(k))\int_{0}^{1}f((y^{\gamma(n)}_{n})^{\prime})\,\mathrm{d}x-\frac{(x_{k}^{l+1}-x_{k}^{l}-2\beta(k))}{n}$
$\displaystyle=\int_{x_{k}^{l}+\beta(k)}^{x_{k}^{l+1}-\beta(k)}f([\bar{y}_{k}^{n}]^{\prime})\mathrm{d}x-\frac{(x_{k}^{l+1}-x_{k}^{l}-2\beta(k))}{n}.$

Then $\bar{y}_{k}^{n}$ is the sought recovery sequence because

$\displaystyle\mathcal{I}_{f}^{C}(y)$
$\displaystyle\geq\mathcal{I}_{f}^{C}(\bar{y}_{k}^{\prime})-\frac{1}{k}=\sum_{l=0}^{N(k)}\int_{x_{k}^{l}}^{x_{k}^{l+1}}Cf(\bar{y}_{k}^{\prime})\mathrm{d}x-\frac{1}{k}\geq\sum_{l=0}^{N(k)}\int_{x_{k}^{l}+\beta(k)}^{x_{k}^{l+1}-\beta(k)}Cf(\bar{y}_{k}^{\prime})\mathrm{d}x-\frac{1}{k}$
$\displaystyle\geq\sum_{l=0}^{N(k)}\int_{x_{k}^{l}+\beta(k)}^{x_{k}^{l+1}-\beta(k)}f([\bar{y}_{k}^{n}]^{\prime})\mathrm{d}x-\frac{1-2\beta(k)N(k)}{n}-\frac{1}{k}$
$\displaystyle\geq\sum_{l=0}^{N(k)}\bigg{(}\int_{x_{k}^{l}}^{x_{k}^{l+1}}f([\bar{y}_{k}^{n}]^{\prime})\mathrm{d}x-\beta(k)f(y_{k}^{\prime}(x_{k}^{l}))-\beta(k)f(y_{k}^{\prime}(x_{k}^{l+1}))\bigg{)}-\frac{1-2\beta(k)N(k)}{n}-\frac{1}{k}$
$\displaystyle\geq\sum_{l=0}^{N(k)}\bigg{(}\int_{x_{k}^{l}}^{x_{k}^{l+1}}f([\bar{y}_{k}^{n}]^{\prime})\mathrm{d}x-2\beta(k)\int_{x_{k}^{l}}^{x_{k}^{l+1}}f(y_{k}^{\prime}(x))\mathrm{d}x\bigg{)}-\frac{1-2\beta(k)N(k)}{n}-\frac{1}{k}$
$\displaystyle\geq\mathcal{I}_{f}(\bar{y}_{k}^{n})-\beta(k)\mathcal{I}_{f}(y_{k})-\frac{1}{n}-\frac{1}{k}.$

Now letting $k$ to infinity, $\mathcal{I}_{f}(y_{k})$ may diverge but as we
have the freedom to choose $\beta(k)$ small enough, we can still assure that
$\beta(k)\mathcal{I}_{f}(y_{k})$ converges to zero, which proves the claim. ∎

######  Lemma 4.5 (Tubular lemma).

Let $y\in W^{1,p}((0,1);\mathbb{R}^{2})$ be injective and bi-Lipschitz. Let
$\mathbf{b}\in C^{1}((0,1);\mathbb{R}^{2})$ be such that
$\mathrm{det}(y^{\prime}|\mathbf{b})\geq\delta$. Then, we can find a number
$h=h(L,\delta,\|\mathbf{b}\|_{C^{1}((0,1);\mathbb{R}^{2})})$ such that

$\displaystyle\tilde{y}_{h}:(0,1)\times[-h/2,h/2]\to\mathbb{R}^{2}\ni(x^{1},x^{2})\mapsto
y(x^{1})+x^{2}\mathbf{b}(x^{1})$

is a homeomorphism.

###### Proof.

Clearly, from its definition, the function $y_{h}$ is Lipschitz and locally
injective because

$\mathrm{det}(\nabla
y_{h}(x^{1},x^{2}))=\mathrm{det}(y^{\prime}+x^{2}\mathbf{b}^{\prime}|\mathbf{b})\geq\delta/2$

for all $h\leq h_{0}$ with $h_{0}$ small enough.

Moreover, we can find find $h_{1}\leq h_{0}$ (if necessary) and a number
$\gamma$ such that for all $h\leq h_{1}$ and all $\xi\in(0,1)$ we have that

$|y_{h}(x_{1})-y_{h}(x_{2})|\geq\frac{\delta}{4}|x_{1}-x_{2}|\quad\text{
whenever }|x_{1}^{1}-\xi|\leq\gamma\text{ and }|x_{2}^{1}-\xi|\leq\gamma.$

Indeed, taking $x_{1}$ and $x_{2}$ in $(0,1)\times[-h/2,h/2]$, and letting
$\varphi(t)=x_{1}+t(x_{2}-x_{1})$, we may write

$\displaystyle|y_{h}(x_{1})-y_{h}(x_{2})|$
$\displaystyle=\Big{|}\int_{0}^{1}\nabla
y_{h}(\varphi(s))\varphi^{\prime}(s)\mathrm{d}s\Big{|}$
$\displaystyle=\Big{|}\int_{0}^{1}\nabla
y_{h}(\xi,0)(x_{2}-x_{1})+\nabla\big{(}y_{h}(\varphi(s))-y_{h}(\xi,0)\big{)}(x_{2}-x_{1})\mathrm{d}s\Big{|}$
$\displaystyle\geq\Big{(}\frac{\delta}{2}-\|\nabla
y_{h}\|_{L^{\infty}((0,1)\times[-h/2,h/2];\mathbb{R}^{2}}\sqrt{h_{1}^{2}+4\gamma^{2}}\Big{)}|x_{2}-x_{1}|$
$\displaystyle\geq\frac{\delta}{4}|x_{2}-x_{1}|,$

where the last inequality is by choosing $h_{1}$ and $\gamma$ small enough.

Particularly, we have a kind of uniform local injectivity implying that
whenever $x_{1},x_{2}$ are such that $|x_{1}^{1}-x_{2}^{1}|\leq 2\gamma$,
$y_{h}(x_{1})\neq y_{h}(x_{2})$ for all $h\leq h_{1}$.

Finally, let us ensure that $y_{h}$ is also globally injective which will
ensure the claim. To this end, we may need to choose $h$ once again smaller if
necessary. Indeed, as $y$ is a _globally_ injective mapping, there has to be a
number $\alpha>0$ such that for any two $x_{1},x_{2}$ with
$|x_{1}^{1}-x_{2}^{1}|\geq 2\gamma$ we have that
$|y(x_{1})-y(x_{2})|\geq\alpha$. Thus, choosing $h_{3}\leq h_{2}$ and
$h_{3}<\frac{\alpha}{4\|\mathbf{b}\|_{C^{0}((0,1);\mathbb{R}^{2})}}$ yields
that $y_{h}$ is a global homeomorphism for all $h\leq h_{3}$.

∎

######  Remark 4.6 (Smoothing and the positivity of the Jacobian).

Let us assume that we have a piece-wise affine homeomorphism $y_{k}(x)$ such
that $y_{k}^{\prime}(x)=a_{k}^{l}$ on $(x_{k}^{l},x_{k}^{l+1})$, $l=0\ldots
N(k)$, with $a_{k}^{l}$ some given vectors. Assume moreover that we are given
the piece-wise constant function $b_{k}=\xi_{a_{k}^{l}}$ on
$(x_{k}^{l},x_{k}^{l+1})$, $l=0\ldots N(k)$ such that

$\mathrm{det}(a_{k}^{l}|\xi_{a_{k}^{l}})\geq 2\delta,$

then we can find a smooth functions ${\bf b}_{k,i}$ that converge strongly (as
$i\to\infty$) in $L^{p}((0,1);\mathbb{R}^{2})$ to the piece-wise constant
function $b_{k}$ and a number $\varepsilon=\varepsilon(y_{k},\delta)$

(4.4)
$\mathrm{det}(y_{k}^{\prime}(x)|\mathbf{b}_{k,i})\geq\varepsilon.\qquad\text{
for a.a. $x\in(0,1)$},$

Let us first notice that applying the standard mollifier by convex averaging
may fail in providing the right approximants. Indeed, take the situation as
depicted in Figure 2. Then, applying the standard convex averaging will result
in the vector $\mathbf{b}_{k,i}$ corresponding to the dashed arrow somewhere
near the corner, and thus, the orientation will change.

Figure 2. An illustration of how convex averaging might change the
orientation.

However, we can fix the smoothing process by introducing the following
procedure: We introduce the intermediate vector $\zeta_{k}^{l}$ defined as

$\zeta_{k}^{l}=\begin{cases}-a_{k}^{l}+a_{k}^{l+1}&\text{ if
}\mathrm{det}(a_{k}^{l}|a_{k}^{l+1})>0,\\\ a_{k}^{l}-a_{k}^{l+1}&\text{ if
}\mathrm{det}(a_{k}^{l}|a_{k}^{l+1})<0\\\ \xi_{a_{k}^{l}}&\text{ if
}\mathrm{det}(a_{k}^{l}|a_{k}^{l+1})=0\end{cases}$

Now, we define the vector

$b_{k,i}(x)=\begin{cases}\zeta_{k}^{l}&\text{ if
}x\in(x_{k}^{l+1}-\frac{1}{i},x_{k}^{l+1}+\frac{1}{i}),l=0\ldots N(k)-2\\\
\xi_{a_{k}^{l}}&\text{ if
}x\in(x_{k}^{l}+\frac{1}{i},x_{k}^{l+1}-\frac{1}{i})\text{ for
}l=1...N(k)-1\\\ \xi_{a_{k}^{0}}&\text{ for }x\in(0,x_{k}^{0}-\frac{1}{i})\\\
\xi_{a_{k}^{N(k)}}&\text{ for }x\in(x_{k}^{N(k)}+\frac{1}{i},1),\end{cases}$

so that basically we only alter the original piece-wise constant function
$b_{k}$ near the corners. At this point, for the construction to make sense,
we need to assume $i$ large enough (as compared to
$\mathrm{min}_{l}|x_{k}^{l+1}-x_{k}^{l}|$) and eventually, as $i\to\infty$, we
will have changed it only on a negligible set.

This alternation has the purpose to now be able to apply the standard
mollifier with the diameter of its kernel smaller than $\frac{1}{2i}$ because
$\det(a_{k}^{l}|\zeta_{k}^{l})>2\varepsilon$ and
$\det(a_{k}^{l+1}|\zeta_{k}^{l})>2\varepsilon$. Indeed, if
$\mathrm{det}(a_{k}^{l}|a_{k}^{l+1})>0$ we have that

$\displaystyle\det(a_{k}^{l}|\zeta_{k}^{l})$
$\displaystyle=\det(a_{k}^{l}|-a_{k}^{l}+a_{k}^{l+1})=\det(a_{k}^{l}|a_{k}^{l+1})$
$\displaystyle\det(a_{k}^{l+1})|\zeta_{k}^{l})$
$\displaystyle=\det(a_{k}^{l+1})|-a_{k}^{l}+a_{k}^{l+1})=\det(a_{k}^{l}|a_{k}^{l+1}),$

and the calculation is analogous in the case when
$\mathrm{det}(a_{k}^{l}|a_{k}^{l+1})<0$ while if this determinant is zero
$a_{k}^{l}$ and $a_{k}^{l+1}$ are aligned and thus
$\det(a_{k}^{l+1}|\xi_{k}^{l})$ is also positive. As $k$ is finite, we can
find a minimum value (that only depends on $y_{k}$ and $\delta$) so that we
see that

$\det(a_{k}^{l}|\zeta_{k}^{l})>\tilde{\varepsilon}\qquad\text{for all
}l=0\ldots k-1.$

After this modification, the convex averaging translates to the determinant
because the determinant is multi-linear (and so convex averaging of the second
column translates to convex averaging of the determinant) and thus

$\mathrm{det}(y_{k}^{\prime}(x)|\mathbf{b}_{k,i})\geq\mathrm{min}(\tilde{\varepsilon},\delta).$

###### Proof of Theorem 3.3.

To prove the veracity of the first point of Definition 2.1, let us take a
sequence $\\{y_{h}\\}_{h>0}\subset W^{1,p}(\Omega,\mathbb{R}^{2})$ such that
$\pi(y_{h})\to y$ in $L^{p}((0,1);\mathbb{R}^{2})$. If $\liminf_{h\to
0}J_{h}(y_{h})=+\infty$, the statement is automatic. Thus, we may, without
loss of generality restrict ourselves to a subsequence of $h$’s (not
relabelled) such that $J_{h}(y_{h})\leq C$. Thus all $y_{h}$’s are
homeomorphisms on $\Omega$.

As the energy $W$ in (3.1) satisfies the coercivity

$W(F)\geq c|F|^{p},$

we know that the sequence $\\{y_{h}\\}$ under question is actually bounded in
$W^{1,p}(\Omega;\mathbb{R}^{2})$ and thus, we may select a non-relabeled sub-
sequence such that

$y_{h}\rightharpoonup\tilde{y}\text{ in }W^{1,p}(\Omega;\mathbb{R}^{2}).$

However, we can even show a stronger statement: indeed, due to the scaling in
the second variable, we have that

$\int_{\Omega}|\partial_{1}y_{h}|^{p}+\frac{1}{h^{p}}|\partial_{2}y_{h}|^{p}\mathrm{d}x\leq
C,$

so that $\partial_{2}y_{h}\to 0$ strongly in $L^{p}(\Omega;\mathbb{R}^{2})$.
This, in particular, means that $\tilde{y}$ is constant in the second variable
so that we can identify

$\tilde{y}(x,0)=\lim_{h\to 0}\pi(y_{h})=y(x).$

In the case $p>2$, we know that the functions $\\{y_{h}\\}$ in the sequence
are uniformly Hölder-continuous, and thus the same holds for the restrictions
$y_{h}^{\mathrm{r}}=y_{h}(x,0)$. In the case $p=2$ we apply [24, Corllary
7.5.1] to get a uniform modulus of continuity for the sequence and its limit.
Therefore, the limiting function $y$ is non-interpenetrative by definition.

Now we have that

$\displaystyle\liminf_{h\to 0}J_{h}(y_{h})$ $\displaystyle=\liminf_{h\to
0}\int_{\Omega}W(\partial_{1}y_{h}(x)|h^{-1}\partial_{2}y_{h}(x))\mathrm{d}{x}$
$\displaystyle\geq\int_{0}^{1}\int_{-1/2}^{1/2}\liminf_{h\to
0}W_{0}(\partial_{1}y_{h}(x))\mathrm{d}x\geq J(y),$

where $W_{0}(a)=C\min_{\xi\in\mathbb{R}^{2}}W(a|\xi)$, the first inequality is
due to Fatou’s lemma, the pointwise inequality $W(a|\zeta)\geq W_{0}(a)$
satisfied for all $\zeta\in\mathbb{R}^{2}$. The last inequality is by then by
the uniform convergence mentioned above; recall here also that the the rod
energy functional $J(y)$ is given in (3.2).

Now, we need to construct the recovery sequence. To this end, let us fix $y\in
W^{1,p}((0,1);\mathbb{R}^{2})$. If $y$ is interpenetrative $J(y)=+\infty$ so
that any sequence approximating $y$ (e.g. as in standard dimension reduction
arguments) would do as a recovery sequence.

Let us thus concentrate on the case when $y$ is non-interpenetrative. In this
case, want to appeal to Lemma 4.2 and thus define the auxiliary function

(4.5) $f(a)=\min_{\xi}W(a|\xi).$

Notice that, as $W$ is p-coercive and continuous, the minimum can always be
found. Notice also that, as for every $0\neq a\in\mathbb{R}^{0}$, the matrix

$\begin{pmatrix}a_{1}&-a_{2}\\\ a_{2}&a_{1}\\\ \end{pmatrix}$

has positive determinant, $f$ is finite everywhere in $\mathbb{R}^{2}$ except
in $0$, where it is infinite. Then, owing to Lemma 4.2, we find a sequence of
_piece-wise_ affine homeomorphisms $\\{\bar{y}_{k}\\}\subset
W^{1,p}((0,1);\mathbb{R}^{2})$ such that

(4.6) $J(y)\geq\int_{0}^{1}f([\bar{y}_{k}]^{\prime})\mathrm{d}x-\frac{1}{k}.$

Let us now fix $k$ and one $\bar{y}_{k}$. We can assume $\bar{y}_{k}$ is
affine on the intervals $(x_{k}^{l},x_{k}^{l+1})$, $l=0\ldots N(k)$. Then, we
can further continue rewriting (4.6) as

$\int_{0}^{1}\min_{\xi\in\mathbb{R}^{2}}W([\bar{y}_{k}]^{\prime}|\xi)\mathrm{d}x-\frac{1}{k}=\sum_{l=0}^{N(k)}\int_{x_{k}^{l}}^{x_{k}^{l+1}}\min_{\xi\in\mathbb{R}^{2}}W([a_{k}^{l}]|\xi)\mathrm{d}x-\frac{1}{k},$

where $a_{k}^{l}=[\bar{y}_{k}]^{\prime}$ on $(x_{k}^{l},x_{k}^{l+1})$. Let us
denote the minimizer in (4.5) as $\xi_{a_{k}^{l}}$ and take the piece-wise
constant function $b_{k}=\xi_{a_{k}^{l}}$ on $(x_{k}^{l},x_{k}^{l+1})$,
$l=0\ldots N(k)$. Notice that for some constant $\delta=\delta(k)$, we have
that

$\mathrm{det}(a_{k}^{l}|\xi_{a_{k}}^{l})\geq 2\delta$

for all $l=0\ldots N(k)$,

We now apply Remark 4.6 to $b_{k}$ in order to define a smooth function
$\mathbf{b}_{k,i}$ ($i\in\mathbb{N})$ such that
$\mathrm{det}([a_{k}^{l}]|\mathbf{b}_{k,i})\geq\varepsilon(k)$ for all
$x\in(0,1)$. Moreover, as the $W$ is $p$-coercive, it is evident that
$b_{k}\in L^{p}((0,1);\mathbb{R}^{2})$ and thus we can assume that
$\mathbf{b}_{k,i}\to b_{k}$ strongly in $L^{p}((0,1);\mathbb{R}^{2})$ (as
$i\to\infty$), whence, when appealing also to the $p$-growth of $W$ (with a
constant $C_{\varepsilon}$) on the set where $\det(F)\geq\varepsilon$ and thus
to Nemytskii continuity (see [36]), we may choose $i\geq i_{0}(k)$ such that

$\sum_{l=0}^{N(k)}\int_{x_{k}^{l}}^{x_{k}^{l+1}}\min_{\xi\in\mathbb{R}^{2}}W([a_{k}^{l}]|\xi)\mathrm{d}x-\frac{1}{k}\geq\sum_{l=0}^{N(k)}\int_{x_{k}{k}^{l}}^{x_{k}^{l+1}}W([\bar{y}_{k}]^{\prime}|\mathbf{b}_{k,i})\mathrm{d}x-\frac{2}{k}$

At this point, we can use Lemma 4.5 to define for $h\leq h_{0}(i,k)$ a
homeomorphism
$y_{h,k,i}(x_{1},x_{2})=\bar{y}_{k}(x_{1})+x_{2}\mathbf{b}_{k,i}$. Using the
$p$-growth of $W$ (with a constant $C_{\delta}$) on the set where
$\det(F)\geq\delta$ and thus to Nemytskii continuity again, we may choose
$h_{0}$ smaller if necessary to see that

$J(y)\geq\sum_{l=0}^{N(k)}\int_{x_{k}^{l}}^{x_{k}^{l+1}}W([\bar{y}_{k}]^{\prime}|\mathbf{b}_{k,i})\mathrm{d}x-\frac{2}{k}\geq\frac{1}{h}\int_{-h/2}^{h/2}W(\nabla
y_{h,k,i})\mathrm{d}x-\frac{3}{k},$

which yields the desired recovery sequence and thus the claim. ∎

##  5\. Proof of Theorem 3.4

In this Section, we prove Theorem 3.4. To this end, introduce some additional
notation to simplify the writing later on. We start by setting

$\mathcal{I}_{p}:=\bigl{\\{}\varphi:(0,1)\to\mathbb{R}^{2}:\ \varphi\in
W^{1,p}((0,1);\mathbb{R}^{2})\text{ such that }\varphi\text{ is continuous and
one-to-one}\bigr{\\}}$

and the weak and strong closures of the set

$\displaystyle\overline{\mathcal{I}_{p}}^{|\cdot|}$
$\displaystyle:=\bigl{\\{}\varphi:(0,1)\to\mathbb{R}^{2}:\
\exists\\{\varphi_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{I}_{p}\text{ such
that }\varphi_{k}\to\varphi\text{ strongly in
}W^{1,p}((0,1);\mathbb{R}^{2})\bigr{\\}},$
$\displaystyle\overline{\mathcal{I}_{p}}^{w}$
$\displaystyle:=\bigl{\\{}\varphi:(0,1)\to\mathbb{R}^{2}:\
\exists\\{\varphi_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{I}_{p}\text{ such
that }\varphi_{k}\rightharpoonup\varphi\text{ weakly in
}W^{1,p}((0,1),\mathbb{R}^{2})\bigr{\\}}.$

With this notation, we reformulate Theorem 3.4 as

Theorem 3.4 (reformulated).Let $p\in[1,\infty)$ and $\varphi\in
W^{1,p}((0,1);\mathbb{R}^{2})$. The following conditions are equivalent

  1. (1)

$\varphi\in\overline{\mathcal{I}_{p}}^{|\cdot|}$.

  2. (2)

$\varphi\in\overline{\mathcal{I}_{p}}^{w}$.

  3. (3)

There are $\tilde{\varphi}_{k}\in\mathcal{I}_{p}$ that are equicontinuous and
$\tilde{\varphi}_{k}\rightrightarrows\varphi$.

Moreover, the closure stays $\overline{\mathcal{I}_{p}}^{|\cdot|}$ stays the
same even if we only admit piece-wise affine sequences.

######  Remark 5.1.

In Theorem 3.4, we allow for the weak convergence
$\varphi_{k}\mathop{\rule[3.0pt]{16.0pt}{0.4pt}\rule[3.05pt]{0.05pt}{0.4pt}\rule[3.1pt]{0.05pt}{0.4pt}\rule[3.15pt]{0.05pt}{0.4pt}\rule[3.2pt]{0.05pt}{0.4pt}\rule[3.25pt]{0.05pt}{0.4pt}\rule[3.3pt]{0.05pt}{0.4pt}\rule[3.35pt]{0.05pt}{0.4pt}\rule[3.4pt]{0.05pt}{0.4pt}\rule[3.45pt]{0.05pt}{0.4pt}\rule[3.5pt]{0.05pt}{0.4pt}\rule[3.55pt]{0.05pt}{0.4pt}\rule[3.6pt]{0.05pt}{0.4pt}\rule[3.65pt]{0.05pt}{0.4pt}\rule[3.7pt]{0.05pt}{0.4pt}\rule[3.75pt]{0.05pt}{0.4pt}\rule[3.8pt]{0.05pt}{0.4pt}\rule[3.85pt]{0.05pt}{0.4pt}\rule[3.9pt]{0.05pt}{0.4pt}\rule[3.95pt]{0.05pt}{0.4pt}\rule[4.0pt]{0.05pt}{0.4pt}\rule[4.05pt]{0.05pt}{0.4pt}\rule[4.1pt]{0.05pt}{0.4pt}\rule[4.15pt]{0.05pt}{0.4pt}\rule[4.2pt]{0.05pt}{0.4pt}\rule[4.25pt]{0.05pt}{0.4pt}\rule[4.3pt]{0.05pt}{0.4pt}\rule[4.35pt]{0.05pt}{0.4pt}\rule[4.4pt]{0.05pt}{0.4pt}\rule[4.45pt]{0.05pt}{0.4pt}\rule[4.5pt]{0.05pt}{0.4pt}\rule[4.55pt]{0.05pt}{0.4pt}\rule[4.6pt]{0.05pt}{0.4pt}\rule[4.65pt]{0.05pt}{0.4pt}\rule[4.7pt]{0.05pt}{0.4pt}\rule[4.75pt]{0.05pt}{0.4pt}\rule[4.8pt]{0.05pt}{0.4pt}\rule[4.85pt]{0.05pt}{0.4pt}\rule[4.9pt]{0.05pt}{0.4pt}\rule[4.95pt]{0.05pt}{0.4pt}\rule[5.0pt]{0.05pt}{0.4pt}\rule[5.05pt]{0.05pt}{0.4pt}\rule[5.1pt]{0.05pt}{0.4pt}\rule[5.15pt]{0.05pt}{0.4pt}\rule[5.2pt]{0.05pt}{0.4pt}\rule[5.25pt]{0.05pt}{0.4pt}\rule[5.3pt]{0.05pt}{0.4pt}\rule[5.35pt]{0.05pt}{0.4pt}\rule[5.4pt]{0.05pt}{0.4pt}\rule[5.45pt]{0.05pt}{0.4pt}\rule[5.5pt]{0.05pt}{0.4pt}\rule[5.55pt]{0.05pt}{0.4pt}\rule[5.6pt]{0.05pt}{0.4pt}\rule[5.65pt]{0.05pt}{0.4pt}\rule[5.7pt]{0.05pt}{0.4pt}\rule[5.75pt]{0.05pt}{0.4pt}\rule[5.8pt]{0.05pt}{0.4pt}\rule[5.85pt]{0.05pt}{0.4pt}\rule[5.9pt]{0.05pt}{0.4pt}\rule[5.95pt]{0.05pt}{0.4pt}\rule[6.0pt]{0.05pt}{0.4pt}~{}}\limits\varphi$
in $W^{1,1}((0,1);\mathbb{R}^{2})$ but we assume that $\varphi\in
W^{1,1}((0,1);,\mathbb{R}^{2})$ and hence we obtain the uniform integrability
of $\varphi_{k}^{\prime}$. A similar characterization for $w^{*}$, strict or
area-strict limits in $\mathrm{BV}((0,1);\mathbb{R}^{2})$ would be more
complicated as the limit class is not contained in
$\mathcal{C}^{0}((0,1);\mathbb{R}^{2})$ which we illustrate in Example 5.2.
Continuity is a natural assumption for us as discontinuous curves may pass
”through” themselves quite often.

######  Example 5.2.

We define the continuous piece-wise linear injective functions
$h_{k}:(0,1)\to\mathbb{R}^{2}$ as follows. Let $h_{k}(0)=0$ and

$h_{k}^{\prime}(x)=\begin{cases}\frac{1}{k}&\text{ for
}x\in[0,\tfrac{1}{2}-\tfrac{1}{k}]\cup[\tfrac{1}{2}+\tfrac{1}{k},1],\\\
\frac{k}{2}&\text{ for
}x\in[\tfrac{1}{2}-\tfrac{1}{k},\tfrac{1}{2}+\tfrac{1}{k}].\\\ \end{cases}$

Then we define $\varphi_{k}(x)=[h_{k}(x),0]$. This sequence is bounded in
$W^{1,1}(((0,1);\mathbb{R}^{2})$ and so $\varphi_{k}\mathop{\hskip
5.2pt{}^{*}\rule[3.0pt]{16.0pt}{0.4pt}\rule[3.05pt]{0.05pt}{0.4pt}\rule[3.1pt]{0.05pt}{0.4pt}\rule[3.15pt]{0.05pt}{0.4pt}\rule[3.2pt]{0.05pt}{0.4pt}\rule[3.25pt]{0.05pt}{0.4pt}\rule[3.3pt]{0.05pt}{0.4pt}\rule[3.35pt]{0.05pt}{0.4pt}\rule[3.4pt]{0.05pt}{0.4pt}\rule[3.45pt]{0.05pt}{0.4pt}\rule[3.5pt]{0.05pt}{0.4pt}\rule[3.55pt]{0.05pt}{0.4pt}\rule[3.6pt]{0.05pt}{0.4pt}\rule[3.65pt]{0.05pt}{0.4pt}\rule[3.7pt]{0.05pt}{0.4pt}\rule[3.75pt]{0.05pt}{0.4pt}\rule[3.8pt]{0.05pt}{0.4pt}\rule[3.85pt]{0.05pt}{0.4pt}\rule[3.9pt]{0.05pt}{0.4pt}\rule[3.95pt]{0.05pt}{0.4pt}\rule[4.0pt]{0.05pt}{0.4pt}\rule[4.05pt]{0.05pt}{0.4pt}\rule[4.1pt]{0.05pt}{0.4pt}\rule[4.15pt]{0.05pt}{0.4pt}\rule[4.2pt]{0.05pt}{0.4pt}\rule[4.25pt]{0.05pt}{0.4pt}\rule[4.3pt]{0.05pt}{0.4pt}\rule[4.35pt]{0.05pt}{0.4pt}\rule[4.4pt]{0.05pt}{0.4pt}\rule[4.45pt]{0.05pt}{0.4pt}\rule[4.5pt]{0.05pt}{0.4pt}\rule[4.55pt]{0.05pt}{0.4pt}\rule[4.6pt]{0.05pt}{0.4pt}\rule[4.65pt]{0.05pt}{0.4pt}\rule[4.7pt]{0.05pt}{0.4pt}\rule[4.75pt]{0.05pt}{0.4pt}\rule[4.8pt]{0.05pt}{0.4pt}\rule[4.85pt]{0.05pt}{0.4pt}\rule[4.9pt]{0.05pt}{0.4pt}\rule[4.95pt]{0.05pt}{0.4pt}\rule[5.0pt]{0.05pt}{0.4pt}\rule[5.05pt]{0.05pt}{0.4pt}\rule[5.1pt]{0.05pt}{0.4pt}\rule[5.15pt]{0.05pt}{0.4pt}\rule[5.2pt]{0.05pt}{0.4pt}\rule[5.25pt]{0.05pt}{0.4pt}\rule[5.3pt]{0.05pt}{0.4pt}\rule[5.35pt]{0.05pt}{0.4pt}\rule[5.4pt]{0.05pt}{0.4pt}\rule[5.45pt]{0.05pt}{0.4pt}\rule[5.5pt]{0.05pt}{0.4pt}\rule[5.55pt]{0.05pt}{0.4pt}\rule[5.6pt]{0.05pt}{0.4pt}\rule[5.65pt]{0.05pt}{0.4pt}\rule[5.7pt]{0.05pt}{0.4pt}\rule[5.75pt]{0.05pt}{0.4pt}\rule[5.8pt]{0.05pt}{0.4pt}\rule[5.85pt]{0.05pt}{0.4pt}\rule[5.9pt]{0.05pt}{0.4pt}\rule[5.95pt]{0.05pt}{0.4pt}\rule[6.0pt]{0.05pt}{0.4pt}~{}}\limits\varphi$
in $BV(((0,1);\mathbb{R}^{2})$ where

$\varphi(x)=\begin{cases}[0,0]&\text{ for }x\in[0,\tfrac{1}{2}],\\\
[1,0]&\text{ for }x\in[\tfrac{1}{2},1].\\\ \end{cases}$

but $\varphi$ is not continuous.

The rest of this section will be devoted to proving Theorem 3.4, with the main
difficulty stemming from the construction part. This means that, being given a
function $\varphi\in W^{1,p}((0,1);\mathbb{R}^{2})$ such that it can be
approximated by homeomorphisms in the $C^{0}$-norm, we need to show that we
can construct piece-wise affine injective approximations of $\varphi$ in the
strong topology of $W^{1,p}((0,1);\mathbb{R}^{2})$. Before embarking on the
technical details of the proof of Theorem 3.4, we give here (heuristically)
the main ideas of the proof:

First, we introduce a grid, as is common in constructing piece-wise affine
approximations. Importantly, however, the grid is chosen in the _image_ , so
in a sense we partition $\varphi((0,1))$. This gives us better control over
the possible non-injectivities. Also, we will need to choose the grid
carefully (and call it then a good arrival grid, see Definition 5.4), so that,
near each crossing point of $\varphi((0,1))$ and the grid, the function
$\varphi$ behaves essentially like an affine function that passes from one
rectangle of the grid to another. See Figure 3, where one possible grid is
shown. Also, we refer to Figure 5, where the choice of the good arrival grid
is illustrated in more detail.

Figure 3. An illustration of the partitioning for a given functions $\varphi$.
The partitioning is performed in the deformed configuration by means of a so-
called _good arrival grid_.

Second, we have a closer look at the intersection points of $\varphi((0,1))$
with the grid, we call one such point $z$. Because we chose the grid well, we
know that the set $\varphi^{-1}(z)$ will have at most _finitely_ many points.
Our first goal will be to tear these possible non-injectivities apart on a
sufficiently small neighborhood of $z$; indeed, due to the choice of the grid,
we know that $\varphi$ cannot oscillate too much in the vicinity of $z$, so
that in particular, there is some distance to another cross-point or vertex of
the grid in order to ”fix” the non-injectivities. To do that, we use the
approximating sequence of homeomorphisms. In fact, we have to shift the points
on the grid everywhere, not just when non-injectivities happen, to obtain an
injective function later. But we omit this in the illustration in Figure 4.
The construction is detailed in _Step 1 of the Proof of Theorem 3.4._

Figure 4. An illustration of injectifying on the grid in one point and piece-
wise affine approximation which is illustrated in grey.

Once torn apart, we just connect the individual intersecting points by a
piecewise affine function. Here we need to be careful that this does not
interfere with injectivity. But it will come in handy that the injectivity
discussion can actually be separated into the individual rectangles where we
then can use topological arguments. This is detailed in _Step 2 of the Proof
of 3.4._

Let us now formalize the arguments. As outlined, we start the discussion with
some preliminary results and the definition of a good arrival grid. First, we
recall that in dimension one the weak convergence in
$W^{1,1}((0,1);\mathbb{R}^{2})$ translates to uniform convergence. This is
well-known (see e.g. the results in [26]) but we re-prove it here for the sake
of completeness.

######  Lemma 5.3.

Let $\varphi,\varphi_{k}\in W^{1,1}((0,1);\mathbb{R}^{2})$ be such that
$\varphi_{k}\mathop{\rule[3.0pt]{16.0pt}{0.4pt}\rule[3.05pt]{0.05pt}{0.4pt}\rule[3.1pt]{0.05pt}{0.4pt}\rule[3.15pt]{0.05pt}{0.4pt}\rule[3.2pt]{0.05pt}{0.4pt}\rule[3.25pt]{0.05pt}{0.4pt}\rule[3.3pt]{0.05pt}{0.4pt}\rule[3.35pt]{0.05pt}{0.4pt}\rule[3.4pt]{0.05pt}{0.4pt}\rule[3.45pt]{0.05pt}{0.4pt}\rule[3.5pt]{0.05pt}{0.4pt}\rule[3.55pt]{0.05pt}{0.4pt}\rule[3.6pt]{0.05pt}{0.4pt}\rule[3.65pt]{0.05pt}{0.4pt}\rule[3.7pt]{0.05pt}{0.4pt}\rule[3.75pt]{0.05pt}{0.4pt}\rule[3.8pt]{0.05pt}{0.4pt}\rule[3.85pt]{0.05pt}{0.4pt}\rule[3.9pt]{0.05pt}{0.4pt}\rule[3.95pt]{0.05pt}{0.4pt}\rule[4.0pt]{0.05pt}{0.4pt}\rule[4.05pt]{0.05pt}{0.4pt}\rule[4.1pt]{0.05pt}{0.4pt}\rule[4.15pt]{0.05pt}{0.4pt}\rule[4.2pt]{0.05pt}{0.4pt}\rule[4.25pt]{0.05pt}{0.4pt}\rule[4.3pt]{0.05pt}{0.4pt}\rule[4.35pt]{0.05pt}{0.4pt}\rule[4.4pt]{0.05pt}{0.4pt}\rule[4.45pt]{0.05pt}{0.4pt}\rule[4.5pt]{0.05pt}{0.4pt}\rule[4.55pt]{0.05pt}{0.4pt}\rule[4.6pt]{0.05pt}{0.4pt}\rule[4.65pt]{0.05pt}{0.4pt}\rule[4.7pt]{0.05pt}{0.4pt}\rule[4.75pt]{0.05pt}{0.4pt}\rule[4.8pt]{0.05pt}{0.4pt}\rule[4.85pt]{0.05pt}{0.4pt}\rule[4.9pt]{0.05pt}{0.4pt}\rule[4.95pt]{0.05pt}{0.4pt}\rule[5.0pt]{0.05pt}{0.4pt}\rule[5.05pt]{0.05pt}{0.4pt}\rule[5.1pt]{0.05pt}{0.4pt}\rule[5.15pt]{0.05pt}{0.4pt}\rule[5.2pt]{0.05pt}{0.4pt}\rule[5.25pt]{0.05pt}{0.4pt}\rule[5.3pt]{0.05pt}{0.4pt}\rule[5.35pt]{0.05pt}{0.4pt}\rule[5.4pt]{0.05pt}{0.4pt}\rule[5.45pt]{0.05pt}{0.4pt}\rule[5.5pt]{0.05pt}{0.4pt}\rule[5.55pt]{0.05pt}{0.4pt}\rule[5.6pt]{0.05pt}{0.4pt}\rule[5.65pt]{0.05pt}{0.4pt}\rule[5.7pt]{0.05pt}{0.4pt}\rule[5.75pt]{0.05pt}{0.4pt}\rule[5.8pt]{0.05pt}{0.4pt}\rule[5.85pt]{0.05pt}{0.4pt}\rule[5.9pt]{0.05pt}{0.4pt}\rule[5.95pt]{0.05pt}{0.4pt}\rule[6.0pt]{0.05pt}{0.4pt}~{}}\limits\varphi$
in $W^{1,1}((0,1);\mathbb{R}^{2})$. Then $\varphi_{k}\rightrightarrows\varphi$
on $(0,1)$, where $\rightrightarrows$ denotes the uniform convergence.

###### Proof.

The proof relies on the representation

(5.1)
$\varphi(x_{1})=\varphi(x_{2})+\int_{x_{1}}^{x_{2}}\varphi^{\prime}(x)\mathrm{d}x$

which holds for functions in $W^{1,1}((0,1);\mathbb{R}^{2})$ for a.a.
$x_{1},x_{2}\in(0,1)$. Now, as $\varphi_{k}\rightharpoonup\varphi$, we have in
particular that the sequence is uniformly bounded in
$W^{1,1}((0,1);\mathbb{R}^{2})$. Further, we may assume (passing to a
subsequence if necessary) that $\varphi_{k}\to\varphi$ strongly in
$L^{1}((0,1);\mathbb{R}^{2})$ and pointwise a.e., so we can pick
$x_{0}\in(0,1)$ with
$|\varphi_{k}(x_{0})-\varphi(x_{0})|\xrightarrow{k\to\infty}0$ and
$\varphi(x_{0})\leq R$ for some $R>0$ fixed.

Setting $x_{2}=x_{0}$ in (5.1) shows that the sequence
$\\{\varphi_{k}\\}_{k\in\mathbb{N}}$ is uniformly bounded. Further, as the
sequence $\\{\varphi_{k}^{\prime}\\}_{k\in\mathbb{N}}$ is weakly convergent in
$L^{1}((0,1);\mathbb{R}^{2})$ it is also equiintegrable by the Dunford-Pettis
theorem (see e.g. [35, Theorem A.12]) which gives the uniform integrability of
these functions, i.e. for every $\varepsilon>0$ there is a $\delta>0$ such
that

$\text{ for every }A\text{ with }\mathcal{L}^{1}(A)<\delta\text{ we have
}\sup_{k\in\mathbb{N}}\int_{A}|\varphi^{\prime}_{k}|<\varepsilon.$

Especially this implies the equicontinuity of
$\\{\varphi_{k}\\}_{k\in\mathbb{N}}$. Thus, the convergence is uniform due the
Arzela-Ascoli theorem.

∎

We now introduce the concept of a good arrival grid that was actually
introduced in [18]. The following is an adaption to our context:

######  Definition 5.4 (Good arrival grid).

Let $\varphi\in W^{1,1}((0,1);\mathbb{R}^{2})$ be non-constant and let $R>0$
be such that $\varphi((0,1))$ is compactly contained in $Q(0,R)$ (the cube of
side-length $2R$ centered in $0$). Let $\delta>0$ be arbitrary and let the
numbers

$-R=w_{0}<w_{1}<\cdots<w_{N}<w_{N+1}=R\text{ satisfy }w_{n+1}-w_{n}<\delta$

for every $0\leq n\leq N$. We say that

(5.2) $\mathcal{G}=\bigcup_{n=0}^{N+1}\\{w_{n}\\}\times[-R,R]\ \cup\
\bigcup_{n=0}^{N+1}[-R,R]\times\\{w_{n}\\}\subset\overline{Q(0,R)}$

is a _$\delta$ -fine good arrival grid for $\varphi$_ if the set
$F:=\\{x\in(0,1):\varphi(x)\in\mathcal{G}\\}$ is finite and for every $p\in F$
it holds that

  * $a)$

$\varphi(p)\in\mathcal{G}\setminus\\{(w_{n},w_{m});0\leq n,m\leq N+1\\}$, i.e.
$\varphi(p)$ is not a cross point of the grid $\mathcal{G}$

  * $b)$

there exists $\varphi^{\prime}(p)$,

  * $c)$

$|\langle\varphi^{\prime}(p),\vec{n}\rangle|>0$ where $\vec{n}$ is a unit
vector perpendicular to the side of $\mathcal{G}$ containing $\varphi(p)$.

Figure 5. The grid above is a good arrival grid for $\varphi_{G}$. It is not a
good grid for $\varphi_{B}$ for several reasons: around $A$ it intersects the
grid in infinitely many points, it crosses a point $C$ which is the cross
point of the grid, it does not have a derivative at $D$ and its derivative at
$T$ is parallel to the grid.

The existence of many such arrival grids was proven in [18, Lemma 3.6] and the
following is a variant of that claim adapted to our context.

######  Lemma 5.5.

Let $\varphi\in W^{1,1}((0,1);\mathbb{R}^{2})$ be non-constant and let $R>0$
be such that $\overline{\varphi((0,1))}$ is compactly contained in $Q(0,R)$.
For every $\delta>0$ there exists a $\delta$-fine good arrival grid
$\mathcal{G}$ for $\varphi$.

In the proof we will use the classical area formula which we recall here for
convenience: Let $h:(0,1)\to\mathbb{R}$ be absolutely continuous and
$A\subset(0,1)$ be a Borel set, then (see e.g. [26, Theorem 3.65 for
$\psi=\chi_{A}$])

(5.3) $\int_{A}|h^{\prime}(t)|dt=\int_{\mathbb{R}}\mathcal{H}^{0}(\\{t\in A:\
h(t)=z\\})\,d(z).$

###### Proof of Lemma 5.5.

Call $\pi_{1}(x,y)=x,\pi_{2}(x,y)=y$ the projections to the axes. Let us
recall that each $\varphi\in W^{1,1}((0,1);\mathbb{R}^{2})$ is absolutely
continuous and thus $\pi_{i}\circ\varphi$ are absolutely continuous so that
the area formula applies to them and their derivatives exist a.e.. Denote

$N_{i}=\\{t\in(0,1);\text{ derivative }\varphi^{\prime}(t)\text{ does not
exist}\\}\cup\\{t\in(0,1);(\pi_{i}\circ\varphi)^{\prime}(t)=0\\}.$

We can clearly fix a Borel set $\tilde{N_{i}}\supset N_{i}$ of the same
measure. For a.e. $t\in\tilde{N_{i}}$ we have
$(\pi_{i}\circ\varphi)^{\prime}(t)=0$ and hence we can use the area formula
(5.3) to obtain

$\mathcal{L}^{1}(\pi_{i}\circ\varphi(N_{i}))\leq\mathcal{L}^{1}(\pi_{i}\circ\varphi(\tilde{N}_{i}))\leq\int_{\mathbb{R}}\mathcal{H}^{0}(\\{t\in\tilde{N}_{i}:\
\pi_{i}(\varphi(t))=z\\})\,d(z)=\int_{\tilde{N}_{i}}(\pi_{i}\circ\varphi)^{\prime}(t)\;dt=0.$

Further, since $\varphi\in W^{1,1}((0,1);\mathbb{R}^{2})$, the area formula
applied to $\pi_{i}\circ\varphi$ again gives

$\infty>\int_{0}^{1}|\varphi^{\prime}|d\mathcal{L}^{1}\geq\int_{0}^{1}|\pi_{i}\circ\varphi^{\prime}|d\mathcal{L}^{1}=\int_{-R}^{R}\mathcal{H}^{0}(\\{t:\
\pi_{i}(\varphi(t))=z\\})\,d\mathcal{L}^{1}(z)$

and so

$\mathcal{H}^{0}\Bigl{(}\varphi((0,1))\cap\bigl{(}\\{x\\}\times\mathbb{R}\bigr{)}\Bigr{)}<\infty\
\text{and}\
\mathcal{H}^{0}\Bigl{(}\varphi((0,1))\cap\bigl{(}\mathbb{R}\times\\{y\\}\bigr{)}\Bigr{)}<\infty$

for almost every $x,y\in\mathbb{R}$. We find an open set

$\displaystyle U\supset\,\big{(}\pi_{1}\circ\varphi(N_{1})\big{)}$
$\displaystyle\cup\,\big{(}\pi_{2}\circ\varphi(N_{2})\big{)}\,\cup\\{x\in\mathbb{R};\mathcal{H}^{0}(\varphi((0,1))\cap\\{x\\}\times\mathbb{R})=\infty\\}$
$\displaystyle\cup\\{y\in\mathbb{R};\mathcal{H}^{0}(\varphi((0,1))\cap\mathbb{R}\times\\{y\\})=\infty\\}$

such that $\mathcal{L}^{1}(U)<\delta/2$. Then every interval of length
$\delta/2$ contains a point of $\mathbb{R}\setminus U$. We separate $[-R,R]$
into $N$ intervals $I_{n}$ of length $\delta/2$ (there is no barrier to assume
that $R\delta^{-1}\in\mathbb{N}$). In each interval $I_{n}$ we choose a point
$w_{n}$ from $I_{n}\setminus U$ arbitrarily. The distance of $w_{n}$ from his
neighbors is at most $\delta$. By the choice of $U$ we satisfy the definition
of a good arrival grid for $\varphi$. ∎

We define the concept of the generalized segment, already introduced in [18],
which will be useful throughout the proof of Theorem 3.4.

######  Definition 5.6 (generalized segments).

Let $\mathcal{G}\subset Q(0,1)\subset\mathbb{R}^{2}$ be a grid (the finite
union of horizontal and vertical lines). Let $R$ be a rectangle of the grid
$\mathcal{G}$ (the closure of a component of $Q(0,1)\setminus\mathcal{G}$).
Let $X\neq Y$ and $X,Y\in\partial R\subset\mathcal{G}$.

Given $\xi>0$ a small parameter, the _generalized segment with parameter
$\xi$_ $[XY]$ between $X$ and $Y$ in $R$ is defined as the standard segment
$XY$ if the two points are not on the same side of $\partial R$; otherwise,
$[XY]$ is the union of two segments of the form $XM$ and $MB$ where $M$ is the
point inside $R$ whose distance from the side containing $X$ and $Y$ is
$\xi|X-Y|/2$ and the projection of $M$ on the segment $XY$ is the mid-point of
$XY$.

The following claim from [11] is easy.

######  Proposition 5.7.

Let $R\subset\mathbb{R}^{2}$ be a rectangle and let $a,b\in\partial R$. Let
$S$ be a generalized segment from $a$ to $b$ in $R$ with parameter $\xi>0$ and
let $\tilde{S}\subset S$ be a closed and connected subset of $S$. Then

$\mathcal{H}^{1}(\tilde{S})\leq(1+\xi)\operatorname{diam}(\tilde{S}),$

where $\operatorname{diam}(\tilde{S})$ is the shortest distance between the
two end points of the segment.

We are now at the position to prove Theorem 3.4:

###### Proof of Theorem 3.4.

Condition $(1)$ implies condition $(2)$ immediately. By Lemma 5.3, condition
$(2)$ implies condition $(3)$. Therefore we only need to prove that $(3)$
implies $(1)$. We proceed in several steps and prove at the same time that the
approximating sequence can be chosen as piece-wise affine.

Step 1 - Construction of an injective approximation $\hat{\varphi}_{\delta}$
using the grid $\mathcal{G}_{\delta}$:
Let $\varphi\in W^{1,p}((0,1),\mathbb{R}^{2})$ for some $p\in[1,\infty)$. By
condition $(3)$ we can find equicontinuous $\tilde{\varphi}_{\delta}$ so that
$\tilde{\varphi}_{\delta}\rightrightarrows\varphi$ as $\delta\to 0$. We first
construct some preliminary injective continuous maps $\hat{\varphi}_{\delta}$
using $\tilde{\varphi}_{\delta}$ to essentially create an injective
approximation in the grid $\mathcal{G}_{\delta}$. Later we will use these
functions to define $\varphi_{\delta}$ for which we will have
$\varphi_{\delta}\xrightarrow{\delta\to 0^{+}}\varphi$ in
$W^{1,p}((0,1),\mathbb{R}^{2})$.

Let $0<\delta<\frac{1}{2}$ be fixed. Let $\mathcal{G}_{\delta}$ be a
$\delta$-fine arrival grid for $\varphi$ and let the numbers $w_{n}$ be
defined as in (5.2). By definition of the good arrival grid, we know that the
set

$F_{\delta}:=\\{t\in(0,1):\varphi(t)\in\mathcal{G}_{\delta}\\}$

is finite. We now start to show that we can find a sufficiently small
neighborhood $B(t,\eta)$ and $B(\varphi(t),\varepsilon)$ for $t\in F_{\delta}$
so that the intervals $B(t,\eta)$ are pair-wise disjoint and the balls
$B(\varphi(t),\varepsilon)$ are pair-wise disjoint and do not intersect the
corners $(w_{n},w_{m})$. Moreover, we want to take $\varepsilon$ so small
that, roughly speaking, an affine interpolation connecting any point in
$B(t_{1},\eta)$ and $B(t_{2},\eta)$ will be a good approximation (in the
$W^{1,p}((0,1),\mathbb{R}^{2})$-norm) for the original functions $\varphi$. Of
course, having then the freedom to select the points to construct the
approximation, we will do that carefully to respect injectivity.

At each point $t\in F_{\delta}$ it holds that $|\varphi^{\prime}(t)|>0$ and
$\varphi^{\prime}(t)$ is not parallel to the side of $\mathcal{G}_{\delta}$
containing $\varphi(t)$. Therefore we can find a $v>0$ an
$\eta=\eta(\delta)>0$ and for every $t\in F_{\delta}$ a unit vector
$\vec{n}_{t}$ perpendicular to $\mathcal{G}_{\delta}$ at $\varphi(t)$ such
that

(5.4) $\langle\varphi(t+h)-\varphi(t),\vec{n}_{t}\rangle\geq hv$

for all $h\in[-\eta,\eta]$. By the differentiability of $\varphi$ at $t$ we
can also assume that

$\bigl{\|}\varphi(t+h)-\varphi(t)-\varphi^{\prime}(t)h\bigr{\|}\leq|h|v$

for every $h\in[-\eta,\eta]$. Moreover, by choosing $\eta$ smaller if
necessary, we may assume that the intervals $[t-\eta,t+\eta]$ are pairwise
disjoint for $t\in F_{\delta}$. Call

$\displaystyle\varepsilon_{1}$
$\displaystyle=\varepsilon_{1}(\delta):=\min\\{\operatorname{dist}((w_{n},w_{m}),\varphi(t));\,t\in
F_{\delta},1\leq n,m\leq N\\},$ $\displaystyle\varepsilon_{2}$
$\displaystyle=\varepsilon_{2}(\delta):=\min\big{\\{}\varepsilon_{1},\min\big{\\{}|\varphi(t)-\varphi(s)|;\,s,t\in
F_{\delta},\varphi(t)\neq\varphi(s)\big{\\}}\big{\\}},$

where, by definition, $\varepsilon_{2}\leq\varepsilon_{1}$. Then the balls
$B(z,\varepsilon_{2})$, $z\in F_{\delta}$, are pairwise disjoint and do not
intersect corners of the grid. By further shrinking the balls at the
intersection of $\varphi((0,1))$ with $\mathcal{G}_{\delta}$, if necessary, we
can take

$\varepsilon_{3}<\min\\{\varepsilon_{2},\eta v\\},$

so it holds that

$\varphi(s)\in B(z,\varepsilon_{3})\text{ for some
}z\in\varphi(F_{\delta})\Leftrightarrow s\in[t-\eta,t+\eta]\text{ for some
}t\in F_{\delta}\text{ and }\varphi(t)=z.$

This means that inside the balls $B(z,\varepsilon_{3})$, we can only see the
images of the intervals $[t-\eta,t+\eta]$, with $t\in F_{\delta}$.

At last, we may still shrink the neighborhoods $B(\varphi(t),\varepsilon)$ so
that we obtain a good approximation also on the discrete derivatives: Thanks
to the fact that $F_{\delta}$ is finite, we can fix a number $\nu>0$ such that
for every $t_{1},t_{2}\in F_{\delta}$ with $\varphi(t_{1})\neq\varphi(t_{2})$
we have

(5.5)
$\|\varphi(t_{1})-\varphi(t_{2})\|+4\nu<(1+\delta)\|\varphi(t_{1})-\varphi(t_{2})\|.$

We can also assume that our $\nu$ is so small that for every distinct
$t_{1},t_{2}\in F_{\delta}$ we have

(5.6) $(t_{2}-t_{1})^{1-p}\nu^{p}\\#F_{\delta}\leq\delta,$

where $\\#F_{\delta}$ is the number of elements in $F_{\delta}$. Choosing now
$\varepsilon<\min\\{\varepsilon_{3},\nu,\frac{\delta}{\\#F_{\delta}}\\}$ (here
the last term will be needed in the $W^{1,1}$-case) yields balls sufficiently
small for our needs.

Now, we want to find a suitable equicontinuous injective approximant
$\tilde{\varphi}_{k}$, which we will later use to construct the strong
injective approximation. What we will need is to take this first approximative
function near enough to $\varphi$ so that we can assure that it crosses the
grid _only_ inside the constructed balls $B(z,\varepsilon)$. To ensure that,
we proceed as follows: For each $t\in F_{\delta}$ we define
$t^{-},t^{+}\in[t-\eta,t+\eta]$ as the smallest and largest argument
respectively such that $\varphi(t^{\pm})\in\partial B(\varphi(t),\varepsilon)$
(see Fig. 6). Notice by (5.4) that $\varphi(t^{-})$ and $\varphi(t^{+})$ lie
on different components of $\partial
B(\varphi(t),\varepsilon)\setminus\mathcal{G}_{\delta}$.

Figure 6. An illustration of the definition of the points
$\tilde{\varphi}_{\delta}(\tilde{t}^{\pm})$ and $\hat{\varphi}(t)$ in
$B(\varphi(t),\varepsilon)$. The curve $\varphi$ is drawn by a full line while
the curve and $\tilde{\varphi}_{\delta}$ are dash-dotted. The segment
$\tilde{\varphi}_{\delta}(t^{-})\tilde{\varphi}_{\delta}(t^{+})$ is also
indicated in the figure.

Now we choose

$\tilde{\varepsilon}:=\frac{1}{2}\operatorname{dist}\Bigl{(}\varphi\bigl{(}(0,1)\setminus\bigcup_{t\in
F_{\delta}}(t^{-},t^{+})\bigr{)},\mathcal{G}_{\delta}\Bigr{)}.$

By condition $(3)$, we may find a map
$\tilde{\varphi}_{\delta}:(0,1)\to\mathbb{R}^{2}$ that is injective,
continuous and
$\|\tilde{\varphi}_{\delta}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-\varphi\|_{C^{0}((0,1);\mathbb{R}^{2})}<\tilde{\varepsilon}$,
i.e. it crosses the grid exactly in the balls $B(z,\varepsilon)$.

We now use that function $\tilde{\varphi}_{\delta}$ to ”tear apart” the non-
injectivities on the grid: For each $t\in F_{\delta}$ we define
$\tilde{t}^{-},\tilde{t}^{+}\in[t-\eta,t+\eta]$ as the smallest and largest
argument respectively such that
$\tilde{\varphi}_{\delta}(\tilde{t}^{\pm})\in\partial
B(\varphi(t),\varepsilon)$ (see Fig. 6). By

$\|\tilde{\varphi}_{\delta}-\varphi\|_{C^{0}((0,1);\mathbb{R}^{2})}<\tilde{\varepsilon}\text{
we obtain
}|\tilde{\varphi}_{\delta}(\tilde{t}^{\pm})-\varphi(t^{\pm})|<\tilde{\varepsilon}.$

We know that $\varphi(t^{-})$ and $\varphi(t^{+})$ lie on different components
of $\partial B(\varphi(t),\varepsilon)\setminus\mathcal{G}_{\delta}$ and hence
by the definition of $\tilde{\varepsilon}$ we obtain that also
$\tilde{\varphi}_{\delta}(\tilde{t}^{-})$ and
$\tilde{\varphi}_{\delta}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\tilde{t}^{+})$
lie on different components of $\partial
B(\varphi(t),\varepsilon)\setminus\mathcal{G}_{\delta}$. Therefore the segment
$\tilde{\varphi}_{\delta}(t^{-})\tilde{\varphi}_{\delta}(t^{+})$ intersects
$\mathcal{G}_{\delta}$ exactly once, call this point
$\hat{\varphi}_{\delta}(t)$.

Note that we obviously have

(5.7) $\|\varphi(t)-\hat{\varphi}_{\delta}(t)\|<2\varepsilon<2\nu.$

Now we use $\tilde{\varphi}_{\delta}$ to define the intermediary injective
continuous $\hat{\varphi}_{\delta}:(0,1)\to\mathbb{R}^{2}$. On
$[\tilde{t}^{-},\tilde{t}^{+}]$ we define $\hat{\varphi}_{\delta}$ as the
2-linear map that sends $[\tilde{t}^{-},t]$ onto the segment
$\tilde{\varphi}_{\delta}(\tilde{t}^{-})\hat{\varphi}_{\delta}(t)$ and sends
$[t,\tilde{t}^{+}]$ onto the segment
$\hat{\varphi}_{\delta}(t)\tilde{\varphi}_{\delta}(\tilde{t}^{+})$. We do this
for all $t\in F_{\delta}$. On the set $(0,1)\setminus\bigcup_{t\in
F_{\delta}}[\tilde{t}^{-},\tilde{t}^{+}]$ we put
$\hat{\varphi}_{\delta}=\tilde{\varphi}_{\delta}$.

Figure 7. Definition of $\hat{\varphi}_{\delta}$ inside of ball
$B=B(\varphi(t),\varepsilon)$ such that $\varphi$ passes through $\varphi(t)$
three times. the curve $\varphi$ is denoted by a full line while
$\tilde{\varphi}_{\delta}$ is dash-dotted. On the left, we can see how
$\varphi$ passes through $\varphi(t)$ three times, and in the middle picture,
we can see in detail how three injective pieces of $\tilde{\varphi}_{\delta}$
approximate $\varphi$ inside $B$. On the right picture, we can see that three
pieces of $\tilde{\varphi}_{\delta}$ are injective inside $B$ and hence their
replacement by $\hat{\varphi}_{\delta}$ (see the three-dotted segments there)
are also injective there and do not intersect.

Clearly $\hat{\varphi}_{\delta}$ is continuous. Thanks to the injectivity of
$\ \tilde{\varphi}_{\delta}$ it is not hard to observe that
$\hat{\varphi}_{\delta}$ is injective. Indeed, we know that
$B(z,\varepsilon)$, $z\in F_{\delta}$, are pairwise disjoint and outside of
these balls our $\hat{\varphi}_{\delta}$ is injective since it agrees with
$\tilde{\varphi}_{\delta}$. Moreover, by the choice of the numbers
$\tilde{t}_{i}^{-},\tilde{t}_{i}^{+}$, we know that
$\tilde{\varphi}_{\delta}((0,1)\setminus\bigcup_{i}(\tilde{t}_{i}^{-},\tilde{t}_{i}^{+}))$
is disjoint with all the balls $B(z,\varepsilon)$. Thus, it is sufficient to
verify the injectivity on one fixed ball $B(z,\varepsilon)$, $z\in
F_{\delta}$. There, we have finitely many $t_{1},\ldots,t_{k}\in F_{\delta}$
such that $\varphi(t_{i})=z$ and each such $t_{i}$ gives us one segment
$\tilde{\varphi}_{\delta}(\tilde{t}_{i}^{-})\tilde{\varphi}_{\delta}(\tilde{t}_{i}^{+})$
inside $B(z,\varepsilon)$ (see Fig. 7). Since the curve
$\tilde{\varphi}_{\delta}$ was injective, it follows that parts of
$\tilde{\varphi}_{\delta}$ between points
$\tilde{\varphi}_{\delta}(\tilde{t}_{i}^{-})\tilde{\varphi}_{\delta}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\tilde{t}_{i}^{+})$
do not intersect and hence a simple topological argument valid in dimension
two gives that segments
$\tilde{\varphi}_{\delta}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\tilde{t}_{i}^{-})\tilde{\varphi}_{\delta}(\tilde{t}_{i}^{+})$
also do not intersect (see Fig. 7). Let us note that the argument can be drawn
upon similar lines as a possible proof of the Jordan curve theorem; we refer
to e.g. [29] where an analytical proof that is easily adaptable to our setting
is given. Thus, injectivity of $\hat{\varphi}_{\delta}$ follows. Note that
$\hat{\varphi}_{\delta}^{-1}(\mathcal{G}_{\delta})=F_{\delta}$ so the set
$F_{\delta}$ is the same for $\hat{\varphi}_{\delta}$ as it was for the
original $\varphi$.

Step 2 - Construction of injective ${\varphi}_{\delta}$ and approximation in
$W^{1,p}((0,1);\mathbb{R}^{2})$ for $p>1$:

Now we define $\varphi_{\delta}$ basically as an affine interpolation between
the points $\hat{\varphi}_{\delta}(t_{i}),t_{i}\in F_{\delta}$, when care is
needed if two $\hat{\varphi}_{\delta}(t)$’s lie on the same side. In more
detail, for each $t\in F_{\delta}$ we put
$\varphi_{\delta}(t)=\hat{\varphi}_{\delta}(t)$. For all neighboring pairs
$t_{1},t_{2}\in F_{\delta}$ such that
$\varphi_{\delta}(t_{1})=\hat{\varphi}_{\delta}(t_{1})$ and
$\varphi_{\delta}(t_{2})=\hat{\varphi}_{\delta}(t_{2})$ do not lie on the same
side of $\mathcal{G}_{\delta}$ we define $\varphi_{\delta}$ on $[t_{1},t_{2}]$
as the affine function which equals $\hat{\varphi}_{\delta}$ at $t_{1}$ and
$t_{2}$. It is not difficult to see that if we have two segment
$\varphi_{\delta}(t_{1})\varphi_{\delta}(t_{2})$ and
$\varphi_{\delta}(\tilde{t}_{1})\varphi_{\delta}(\tilde{t}_{2})$ of this type
that they do not intersect as the original curves
$\hat{\varphi}_{\delta}([t_{1},t_{2}])$ and
$\hat{\varphi}_{\delta}([\tilde{t}_{1},\tilde{t}_{2}])$ do not intersect as
$\hat{\varphi}$ is injective. Indeed, either the two segments lie in the same
rectangle of the grid, in which case we may rely on the same topological
argument as above. Or they lie in two different rectangles and thus are
disjoint. In case our neighboring pair $t_{3},t_{4}\in F_{\delta}$ lies on the
same segment in the grid $\mathcal{G}_{\delta}$ we connect
${\varphi}_{\delta}(t_{3})$ and ${\varphi}_{\delta}(t_{4})$ by a generalized
segment $[{\varphi}_{\delta}(t_{3}){\varphi}_{\delta}(t_{4})]$ with parameter
$0<\xi<\frac{\delta}{\\#F_{\delta}}$ in such a way that the generalized
segment lies in the same rectangle of the grid as the original curve
$\hat{\varphi}([t_{3},t_{4}])$)

In this way, the verification of whether the introduction of generalized
segments preserves injectivity reduces to one rectangle of the grid. As we
have only finitely many points in $F_{\delta}$, this can be assured by
choosing $0<\xi$ sufficiently small so that the newly introduced generalized
segments do not intersect each other or the previous segments.

Having chosen $\xi$ we define $\varphi_{\delta}$ on on the generalized
segments via the constant speed parametrization. It remains to extend
$\varphi_{\delta}$ onto the intervals $(0,t_{\text{min}}]$ and
$[t_{\text{max}},1)$, where $t_{\text{min}},t_{\text{max}}$ are the largest
and smallest $t\in F_{\delta}$ (so the fist and last crossing point with the
grid). We can do that by, for example, connecting $\varphi(0)$ and
$\varphi_{\delta}(t_{\text{min}})$ as well as
$\varphi_{\delta}(t_{\text{max}})$ and $\varphi(1)$ by a segment.

Let us notice that by construction, we have that for any
$K=K_{n,m}=[w_{n},w_{n+1}]\times[w_{m},w_{m+1}]$ (a closed rectangle formed by
the grid $\mathcal{G}_{\delta}$) we have that $\varphi_{\delta}(s)\in K$
exactly if $\varphi(s)\in K$.

Therefore

$\|\varphi_{\delta}-\varphi\|_{L^{\infty}((0,1);\mathbb{R}^{2})}\leq\operatorname{diam}K<4\delta\text{
and thus also
}\|\varphi_{\delta}-\varphi\|_{L^{p}((0,1);\mathbb{R}^{2})}<4\delta.$

Note that this uniform convergence works also for $p=1$.

Now we prove that $\varphi_{\delta}^{\prime}\to\varphi^{\prime}$ in
$L^{p}((0,1);\mathbb{R}^{2})$. Let us have a pair of neighbors $t_{1},t_{2}\in
F_{\delta}$. We know that for $t\in F_{\delta}$ we have
$\varphi_{\delta}(t)=\hat{\varphi}_{\delta}(t)$ and hence (5.7) and (5.5)
imply that

$\|\varphi_{\delta}(t_{2})-\varphi_{\delta}(t_{1})\|\leq\|\varphi(t_{2})-\varphi(t_{1})\|+4\nu<(1+\delta)\|\varphi(t_{2})-\varphi(t_{1})\|.$

Using Proposition 5.7 and the piecewise constant speed parametrization we
have, for each pair of neighbors $t_{1},t_{2}\in F_{\delta}$ with
$\varphi(t_{1})\neq\varphi(t_{2})$, that

(5.8) $\displaystyle\int_{t_{1}}^{t_{2}}|\varphi_{\delta}^{\prime}|^{p}$
$\displaystyle=\int_{t_{1}}^{t_{2}}\Big{(}-\hskip-13.0pt\int_{t_{1}}^{t_{2}}|\varphi_{\delta}^{\prime}(s)|ds\Big{)}^{p}dw$
$\displaystyle\leq\int_{t_{1}}^{t_{2}}\Big{(}\frac{1}{t_{2}-t_{1}}(1+\delta)\|\varphi_{\delta}(t_{2})-\varphi_{\delta}(t_{1})\|\Big{)}^{p}dw$
$\displaystyle\leq\int_{t_{1}}^{t_{2}}\Big{(}\frac{1}{t_{2}-t_{1}}(1+\delta)(1+\delta)\|\varphi(t_{2})-\varphi(t_{1})\|\Big{)}^{p}dw$
$\displaystyle\leq(1+\delta)^{2p}(t_{2}-t_{1})^{1-p}\Bigl{(}\int_{t_{1}}^{t_{2}}|\varphi^{\prime}(s)|ds\Bigr{)}^{p}$
$\displaystyle\leq(1+\delta)^{2p}\int_{t_{1}}^{t_{2}}|\varphi^{\prime}(s)|^{p}ds.$

In case $\varphi(t_{1})=\varphi(t_{2})$ we can estimate similarly and use
(5.7) to obtain

(5.9) $\displaystyle\int_{t_{1}}^{t_{2}}|\varphi_{\delta}^{\prime}|^{p}$
$\displaystyle\leq\int_{t_{1}}^{t_{2}}\Big{(}\frac{1}{t_{2}-t_{1}}(1+\delta)\|\varphi_{\delta}(t_{2})-\varphi_{\delta}(t_{1})\|\Big{)}^{p}dw$
$\displaystyle\leq\int_{t_{1}}^{t_{2}}\Big{(}\frac{1}{t_{2}-t_{1}}(1+\delta)4\eta\Big{)}^{p}dw$
$\displaystyle\leq C(p)(t_{2}-t_{1})^{1-p}\eta^{p}.$

Summing over neighbors in $F_{\delta}$ we have using (5.6) that

$\displaystyle\int_{0}^{1}|\varphi_{\delta}^{\prime}(s)|^{p}ds\leq$
$\displaystyle(1+\delta)^{2p}\int_{0}^{1}|\varphi^{\prime}(s)|^{p}ds+C\delta.$

For $p\in(1,\infty)$ the fact that $\|\varphi_{\delta}^{\prime}\|_{p}$ is
bounded means that for any infinitesimal sequence $\delta_{k}\to 0$ we can
find a subsequence such that
$\varphi_{\delta_{k_{j}}}\mathop{\rule[3.0pt]{16.0pt}{0.4pt}\rule[3.05pt]{0.05pt}{0.4pt}\rule[3.1pt]{0.05pt}{0.4pt}\rule[3.15pt]{0.05pt}{0.4pt}\rule[3.2pt]{0.05pt}{0.4pt}\rule[3.25pt]{0.05pt}{0.4pt}\rule[3.3pt]{0.05pt}{0.4pt}\rule[3.35pt]{0.05pt}{0.4pt}\rule[3.4pt]{0.05pt}{0.4pt}\rule[3.45pt]{0.05pt}{0.4pt}\rule[3.5pt]{0.05pt}{0.4pt}\rule[3.55pt]{0.05pt}{0.4pt}\rule[3.6pt]{0.05pt}{0.4pt}\rule[3.65pt]{0.05pt}{0.4pt}\rule[3.7pt]{0.05pt}{0.4pt}\rule[3.75pt]{0.05pt}{0.4pt}\rule[3.8pt]{0.05pt}{0.4pt}\rule[3.85pt]{0.05pt}{0.4pt}\rule[3.9pt]{0.05pt}{0.4pt}\rule[3.95pt]{0.05pt}{0.4pt}\rule[4.0pt]{0.05pt}{0.4pt}\rule[4.05pt]{0.05pt}{0.4pt}\rule[4.1pt]{0.05pt}{0.4pt}\rule[4.15pt]{0.05pt}{0.4pt}\rule[4.2pt]{0.05pt}{0.4pt}\rule[4.25pt]{0.05pt}{0.4pt}\rule[4.3pt]{0.05pt}{0.4pt}\rule[4.35pt]{0.05pt}{0.4pt}\rule[4.4pt]{0.05pt}{0.4pt}\rule[4.45pt]{0.05pt}{0.4pt}\rule[4.5pt]{0.05pt}{0.4pt}\rule[4.55pt]{0.05pt}{0.4pt}\rule[4.6pt]{0.05pt}{0.4pt}\rule[4.65pt]{0.05pt}{0.4pt}\rule[4.7pt]{0.05pt}{0.4pt}\rule[4.75pt]{0.05pt}{0.4pt}\rule[4.8pt]{0.05pt}{0.4pt}\rule[4.85pt]{0.05pt}{0.4pt}\rule[4.9pt]{0.05pt}{0.4pt}\rule[4.95pt]{0.05pt}{0.4pt}\rule[5.0pt]{0.05pt}{0.4pt}\rule[5.05pt]{0.05pt}{0.4pt}\rule[5.1pt]{0.05pt}{0.4pt}\rule[5.15pt]{0.05pt}{0.4pt}\rule[5.2pt]{0.05pt}{0.4pt}\rule[5.25pt]{0.05pt}{0.4pt}\rule[5.3pt]{0.05pt}{0.4pt}\rule[5.35pt]{0.05pt}{0.4pt}\rule[5.4pt]{0.05pt}{0.4pt}\rule[5.45pt]{0.05pt}{0.4pt}\rule[5.5pt]{0.05pt}{0.4pt}\rule[5.55pt]{0.05pt}{0.4pt}\rule[5.6pt]{0.05pt}{0.4pt}\rule[5.65pt]{0.05pt}{0.4pt}\rule[5.7pt]{0.05pt}{0.4pt}\rule[5.75pt]{0.05pt}{0.4pt}\rule[5.8pt]{0.05pt}{0.4pt}\rule[5.85pt]{0.05pt}{0.4pt}\rule[5.9pt]{0.05pt}{0.4pt}\rule[5.95pt]{0.05pt}{0.4pt}\rule[6.0pt]{0.05pt}{0.4pt}~{}}\limits\varphi$
in $W^{1,p}((0,1);\mathbb{R}^{2})$. The inequality above implies that
$\|\varphi^{\prime}_{\delta_{k_{j}}}\|_{L^{p}((0,1);\mathbb{R}^{2})}$ is
converging to $\|\varphi^{\prime}\|_{L^{p}((0,1);\mathbb{R}^{2})}$ and hence
strict convexity of $L^{p}$, $1<p<\infty$, implies using Clarkson’s
inequalities in $L^{p}$ that we have also strong convergence in $W^{1,p}$.

Step 3 - Approximation in $W^{1,1}((0,1);\mathbb{R}^{2})$:

Now we prove that $\varphi_{\delta}^{\prime}\to\varphi^{\prime}$ in
$L^{1}((0,1);\mathbb{R}^{2})$. We will need to do this differently to above as
we cannot use uniform convexity of the space. Choose
$\alpha\in(0,\tfrac{1}{4})$ and for the choice of $\alpha$ find $\beta>0$ such
that

$\text{ for every }E\subset(0,1)\text{ with }\mathcal{L}^{1}(E)<\beta\text{ we
have }\int_{E}|\varphi^{\prime}|<\alpha.$

Because, for almost every $t\in(0,1)$ it holds that

$\lim_{r\to
0^{+}}-\hskip-13.0pt\int_{t-r}^{t+r}|\varphi^{\prime}(s)-\varphi^{\prime}(t)|ds=0$

we can find small parameters $\lambda,r\in(0,\alpha)$ such that
$\mathcal{L}^{1}((0,1)\setminus H_{\lambda,r})<\beta$ where

$H_{\lambda,r}=\Big{\\{}t\in(0,1);\ \exists\varphi^{\prime}(t),\
|\varphi^{\prime}(t)|\in\\{0\\}\cup(\lambda,\infty),\sup_{\rho\in(0,r)}-\hskip-13.0pt\int_{t-\rho}^{t+\rho}|\varphi^{\prime}(s)-\varphi^{\prime}(t)|ds\leq\tfrac{1}{8}\lambda\Big{\\}}.$

We also denote
$H_{\lambda,r}^{0}=H_{\lambda,r}\cap\\{t;\varphi^{\prime}(t)=0\\}$ and
$H_{\lambda,r}^{+}=H_{\lambda,r}\setminus H_{\lambda,r}^{0}$. We show that for
any

(5.10) $0<\delta<\frac{1}{8}\lambda r$

we have a good estimate of $\int|\varphi^{\prime}-\varphi^{\prime}_{\delta}|$.

We sum (5.8) for $p=1$ over all intervals $(t_{1},t_{2})\subset(0,1)\setminus
H_{\lambda,r}^{+}$ and we obtain using $\varphi^{\prime}=0$ on
$H_{\lambda,r}\setminus H_{\lambda,r}^{+}$ and $\mathcal{L}^{1}((0,1)\setminus
H_{\lambda,r})<\beta$ that

(5.11) $\displaystyle\sum_{(t_{1},t_{2})\subset(0,1)\setminus
H_{\lambda,r}^{+}}$
$\displaystyle\int_{t_{1}}^{t_{2}}|\varphi^{\prime}(s)-\varphi_{\delta}^{\prime}(s)|ds\leq\sum_{(t_{1},t_{2})\subset(0,1)\setminus
H_{\lambda,r}^{+}}\int_{t_{1}}^{t_{2}}\bigl{(}|\varphi^{\prime}(s)|+|\varphi_{\delta}^{\prime}(s)|\bigr{)}ds$
$\displaystyle\leq(1+(1+\delta)^{2})\sum_{(t_{1},t_{2})\subset(0,1)\setminus
H_{\lambda,r}^{+}}\int_{t_{1}}^{t_{2}}|\varphi^{\prime}(s)|ds$
$\displaystyle\leq(1+(1+\delta)^{2})\sum_{(t_{1},t_{2})\subset(0,1)\setminus
H_{\lambda,r}}\int_{t_{1}}^{t_{2}}|\varphi^{\prime}(s)|ds\leq C\alpha.$

It remains to consider intervals $[t_{1},t_{2}]$, $t_{1},t_{2}\in F_{\delta}$,
such that there is a $t\in(t_{1},t_{2})\cap H_{\lambda,r}^{+}$. The first step
is to prove using (5.10) that $[t_{1},t_{2}]$ is entirely contained in
$[t-r,t+r]$. Using $t\in H_{\lambda,r}^{+}$ and definition of
$H_{\lambda,r}^{+}$ we have that

$|\varphi(t+r)-\varphi(t)-\varphi^{\prime}(t)r|=\Big{|}\int_{t}^{t+r}\bigl{(}\varphi^{\prime}(s)-\varphi^{\prime}(t)\bigr{)}ds\Big{|}\leq\int_{t-r}^{t+r}|\varphi^{\prime}(s)-\varphi^{\prime}(t)|ds\leq\tfrac{1}{8}\lambda
r$

and since $|\varphi^{\prime}(t)r|\geq\lambda r$ we can use (5.10) to obtain

$|\varphi(t+r)-\varphi(t)|\geq\frac{7}{8}\lambda r\geq
7\delta>\operatorname{diam}K$

and so $\varphi(t+r)$ lies in a different rectangle $K$ of the grid
$\mathcal{G}_{\delta}$ than does $\varphi(t)$. The same holds for
$\varphi(t-r)$. In other words $[t_{1},t_{2}]$ is entirely contained in
$[t-r,t+r]$.

Call $\rho:=\max\\{|t-t_{1}|,|t-t_{2}|\\}<|t_{1}-t_{2}|$. Using $t\in
H_{\lambda,r}^{+}$ and definition of $H_{\lambda,r}^{+}$ we estimate
$\varphi^{\prime}$ as follows

$\int_{t_{1}}^{t_{2}}|\varphi^{\prime}(s)-\varphi^{\prime}(t)|\;ds\leq\int_{t-\rho}^{t+\rho}|\varphi^{\prime}(s)-\varphi^{\prime}(t)|ds\leq
2\rho\frac{1}{8}\lambda=2|t_{2}-t_{1}|\frac{1}{8}\lambda$

and summing over these intervals $t_{1},t_{2}\in F_{\delta}$ containing some
$t\in H_{\lambda,r}^{+}$ we get

$\sum_{(t_{1},t_{2})\cap
H_{\lambda,r}^{+}\neq\emptyset}\int_{t_{1}}^{t_{2}}|\varphi^{\prime}(s)-\varphi^{\prime}(t)|\;ds\leq\frac{\lambda}{4}\leq\frac{\alpha}{4}.$

Concerning $\varphi_{\delta}^{\prime}$ we have firstly that

(5.12)
$\Big{|}\frac{\varphi(t_{2})-\varphi(t_{1})}{t_{2}-t_{1}}-\varphi^{\prime}(t)\Big{|}\leq-\hskip-13.0pt\int_{t-\rho}^{t+\rho}|\varphi^{\prime}(s)-\varphi^{\prime}(t)|ds\leq\frac{\lambda}{4}\leq\frac{\alpha}{4}.$

Our $\varphi_{\delta}$ has constant speed parametrization and hence either for
every $s\in(t_{1},t_{2})$ we have

$\frac{\varphi_{\delta}(t_{2})-\varphi_{\delta}(t_{1})}{t_{2}-t_{1}}=\varphi_{\delta}^{\prime}(s)$

or we use generalized segment between $\varphi_{\delta}(t_{1})$ and
$\varphi_{\delta}(t_{2})$ and then using Proposition 5.7 we get

$\Big{|}\frac{\varphi_{\delta}(t_{2})-\varphi_{\delta}(t_{1})}{t_{2}-t_{1}}-\varphi_{\delta}^{\prime}(s)\Big{|}\leq\xi\Big{|}\frac{\varphi_{\delta}(t_{2})-\varphi_{\delta}(t_{1})}{t_{2}-t_{1}}\Big{|}.$

Using (5.7) we thus obtain in both cases

(5.13)
$\Big{|}\frac{\varphi(t_{2})-\varphi(t_{1})}{t_{2}-t_{1}}-\varphi_{\delta}^{\prime}(s)\Big{|}\leq\frac{8\varepsilon}{t_{2}-t_{1}}+\xi\Big{|}\frac{\varphi(t_{2})-\varphi(t_{1})}{t_{2}-t_{1}}\Big{|}.$

Together with $t\in H_{\lambda,r}^{+}$, the definition of $H_{\lambda,r}^{+}$,
(5.12), (5.13), $\varepsilon<\frac{\delta}{\\#F_{\delta}}$ and
$\xi<\frac{\delta}{\\#F_{\delta}}$ this gives

$\displaystyle\sum_{(t_{1},t_{2})\cap
H_{\lambda,r}^{+}\neq\emptyset}\int_{t_{1}}^{t_{2}}|\varphi^{\prime}_{\delta}(s)-\varphi^{\prime}(s)|\;ds\leq$
$\displaystyle\sum_{(t_{1},t_{2})\cap
H_{\lambda,r}^{+}\neq\emptyset}\int_{t_{1}}^{t_{2}}|\varphi^{\prime}(t)-\varphi^{\prime}(s)|\;ds+$
$\displaystyle+\sum_{(t_{1},t_{2})\cap
H_{\lambda,r}^{+}\neq\emptyset}\int_{t_{1}}^{t_{2}}\Bigl{|}\varphi^{\prime}(t)-\frac{\varphi(t_{2})-\varphi(t_{1})}{t_{2}-t_{1}}\Bigr{|}\;ds$
$\displaystyle+\sum_{(t_{1},t_{2})\cap
H_{\lambda,r}^{+}\neq\emptyset}\int_{t_{1}}^{t_{2}}\Bigl{|}\varphi_{\delta}^{\prime}(s)-\frac{\varphi(t_{2})-\varphi(t_{1})}{t_{2}-t_{1}}\Bigr{|}\;ds$
$\displaystyle\leq$ $\displaystyle\sum_{(t_{1},t_{2})\cap
H_{\lambda,r}^{+}\neq\emptyset}\frac{\lambda}{8}(t_{2}-t_{1})+\sum_{(t_{1},t_{2})\cap
H_{\lambda,r}^{+}\neq\emptyset}\frac{\alpha}{4}(t_{2}-t_{1})$
$\displaystyle+\sum_{(t_{1},t_{2})\cap
H_{\lambda,r}^{+}\neq\emptyset}\bigl{(}8\varepsilon+\xi
2\|\varphi\|_{L^{\infty}((0,1);\mathbb{R}^{2})}\bigr{)}$ $\displaystyle\leq$
$\displaystyle\frac{\alpha}{8}+\frac{\alpha}{4}+8\delta+\delta
2\|\varphi\|_{L^{\infty}((0,1);\mathbb{R}^{2})}.$

In combination with (5.11) this implies that $\varphi_{\delta}\to\varphi$ in
$W^{1,p}((0,1);\mathbb{R}^{2})$ as $\delta\to 0_{+}$. ∎

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