# Singular Sobolev spaces and applications to degenerate elliptic partial
differential equations

Djamel eddine Kebiche  Faculty of Mathematics, University of Vienna, Austria,
Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
<EMAIL_ADDRESS>
###### Abstract.

We present the notion of Singular Sobolev space $L_{w}^{p}(\Omega)$, which is
the space of all measurable functions $u$ on $\Omega$ such that $wu\in
L^{p}(\Omega)$, where $w$ is a Lebesgue measurable function, finite and non-
null a.e. in $\Omega$; the function $w$, is called elimination (singularities)
function. We highlight that the inclusion $L_{w}^{p}(\Omega)\subseteq
L_{\mathrm{loc}}^{1}(\Omega)$ is not required. In this setting, the notion of
distributional derivative is not well defined in general and hence it is
replaced with a new notion of weak derivative called weak derivative in the
sense of $L_{w,\text{loc}}^{1}(\Omega):=\\{u\mid wu\in
L_{\text{loc}}^{1}(\Omega)\\}$, which is defined using a suitable integration
by part formula. For a family $\\{w_{\alpha}\\}$ of elimination functions, we
then consider the subspace $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$ of
$L_{w}^{p}(\Omega)$ of functions whose $\alpha$-weak derivative in the sense
of $L_{w,\text{loc}}^{1}(\Omega)$ exists and belongs to
$L_{w_{\alpha}}^{p}(\Omega)$ for all multi-index $\alpha$ with $|\alpha|\leq
m$, and give a sufficient condition on the family $\\{w_{\alpha}\\}$ that
guarantees the Cauchy completeness of the space
$W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$. We also define the trace operator Tr on
$W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$. The main aim of this construction is
the study of the following Cauchy problem

$\begin{cases}-\sum_{i,j=1}^{d}D_{j}(a_{ij}D_{i}u)+(b\cdot D)u+cu=f&\text{in
}\Omega\\\ \text{Tr}(u)=0&\text{on }\partial\Omega\end{cases}$

where in particular the bilinear form $\sum_{i,j=1}^{d}a_{ij}\xi_{i}\xi_{j}$
is positive almost everywhere in $\Omega$, and for all
$\xi:=(\xi_{1},...,\xi_{d})\in\mathbb{R}^{d}$, but it degenerates on some null
set $N$ of $\Omega$. We highlight that, in general, the solution is not
locally integrable, and sufficient conditions that guarantee the non-
integrability of the solution are given. Additional results such as Poincaré
inequality (adapted to our framework) are proved. We also show that the
solution satisfies the differential equation in the sense of
$\mathcal{D}^{\prime}(\Omega\backslash N)$ as well.

D. Kebiche has been supported by grant P33538 of the Austrian Science Fund FWF

##  1\. Introduction

In the present paper, we want to deal with Cauchy problems of the form

$\begin{cases}-\sum_{i,j=1}^{d}D_{j}(a_{ij}D_{i}u)+(b\cdot D)u+cu=f&\text{in
}\Omega\\\ \text{Tr}(u)=0&\text{on }\partial\Omega,\end{cases}$ (1.1)

where the bilinear form $\sum_{i,j=1}^{d}a_{ij}\xi_{i}\xi_{j}$ is assumed to
be positive almost everywhere in $\Omega$, and for all
$\xi:=(\xi_{1},...,\xi_{d})\in\mathbb{R}^{d}$. It is well-known that if this
form degenerates on some null set $N$ of $\Omega$, this causes major
difficulties in adapting the theory of weak solutions of the boundary-value
problem (1.1). Indeed, weak solutions may still exist, but they do not belong
any longer to Sobolev spaces. They are indeed generally sought in weighted
Sobolev spaces (see [4, 7] and reference therein) or in degenerate Sobolev
spaces (see [5]).

In our paper, we will assume that the bilinear form
$\sum_{i,j=1}^{d}a_{ij}(x)\xi_{i}\xi_{j}$ satisfies

$\exists\mu,C>0\,\forall\xi\in\mathbb{R}^{d}:\,\mu
w^{4}(x)|\xi|^{2}\leq\sum_{i,j=1}^{d}a_{ij}(x)\xi_{i}\xi_{j}\leq
Cw^{4}(x)|\xi|^{2}\,\,\,\mathrm{a.e.}\,\text{in }\,\Omega$

for some $w\in\mathcal{C}^{1}(\Omega)$ where $N:=\\{x\in\Omega\mid w(x)=0\\}$
is a Lebesgue negligible set. We highlight that no assumption on the
integrability of $w^{-1}$ is considered. We will seek the weak solution in a
new Sobolev-like spaces called singular Sobolev spaces in which the functions
are not necessary locally integrable.

For some given elimination function $w$ (see Def. 1), the singular Sobolev
space $L_{w}^{p}(\Omega)$ is defined as the space of measurable functions $u$
such that $uw\in L^{p}(\Omega)$. The inclusion $L_{w}^{p}(\Omega)\subseteq
L_{\mathrm{loc}}^{1}(\Omega)$ is not required, and a new notion of weak
derivative called weak derivative in the sense of
$L_{w,\mathrm{loc}}^{1}(\Omega)$ is introduced (see Def. 7). This new notion
of weak derivative is weaker that the notion of weak derivative in the sense
of $L_{\mathrm{loc}}^{1}(\Omega)$, and is adapted to smooth elements on
$\Omega\backslash N$ of the space $L_{w}^{p}(\Omega)$.

For a family $\\{w_{\alpha}\\}$ of elimination functions, we introduce in
Subsection 2.3, the subspace $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$ of
$L_{w}^{p}(\Omega)$ of those $\alpha$-weak derivative in the sense of
$L_{w,\text{loc}}^{1}(\Omega)$ exists and belongs to
$L_{w_{\alpha}}^{p}(\Omega)$ for all $\alpha\in\mathbb{N}^{d}$ with
$|\alpha|\leq m$. In Thm. 14, we give a sufficient condition on the family of
elimination functions $\\{w_{\alpha}\\}$ that guarantees the Cauchy
completeness of the space $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$. We will also
prove that $W_{\\{w^{|\alpha|+1}\\}}^{m,p}(\Omega)$ is continuously embedded
into $W_{\text{loc}}^{m,p}(\Omega\backslash N)$.

In Subsection 2.4, we will see that it is possible to define a trace operator
Tr on the space $W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$, in case
$|w|>\delta>0$ on $\Omega\backslash K$ where $K\Subset\Omega$, or in case
$\nabla w\in L^{\infty}(\Omega)$.

Subsection 2.7 is devoted to the solvability of the boundary-value problem
(1.1) where in Thm. 28 and in Thm. 29 sufficient conditions that guarantee the
existence and uniqueness of a solution in the Hilbert space
$H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$ (which is a subspace of
$W_{\\{w^{|\alpha|+1}\\}}^{1,2}(\Omega)$ whose elements have a null trace) are
given. Moreover, we will show that the solution of (1.1) satisfies the
differential equation of (1.1) in the sense of
$\mathcal{D}^{\prime}(\Omega\backslash N)$.

Finally, in Prop. 31 we give assumptions under which the solution is not
locally integrable.

##  2\. Singular Sobolev spaces and weak derivative in the sense of
$L_{w,\mathrm{loc}}^{1}(\Omega)$

###  2.1. The space $L_{w}^{p}(\Omega)$

We start by giving the definition of elimination functions. Such functions
play a role similar to the role of weight functions in weighted Sobolev
spaces, but they are not necessary positive.

######  Definition 1.

Let $\Omega$ be an open set. We denote by $W(\Omega)$ the set of all
measurable, a.e. in $\Omega$ non-null and finite functions $w=w(x)$,
$x\in\Omega$. Elements of $W(\Omega)$ will be called elimination
(singularities) functions on $\Omega$.

Throughout this paper, we will denote by $N$ the set of real zeros of $w$.

In the sequel, we write $\Omega_{N}$ instead of $\Omega\backslash N$.

We define now the notion of singular Sobolev spaces.

######  Definition 2.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $w\in W(\Omega)$ and
$p\in[1,\infty]$. We define $L_{w}^{p}(\Omega)$ as the space of measurable
functions $u$ on $\Omega$ such that $uw\in L^{p}(\Omega)$. This space will be
equipped with the norm $||\cdot||_{L_{w}^{p}}$ defined by

$||u||_{L_{w}^{p}}:=||uw||_{L^{p}}\,\,\,\forall u\in L_{w}^{p}(\Omega).$

The space $L_{w}^{2}(\Omega)$ is equipped with the scalar product
$(\cdot,\cdot)_{L_{w}^{2}}$ defined by

$(u,v)_{L_{w}^{2}}:=(wu,wv)_{L^{2}}\,\,\,\forall u,v\in L_{w}^{2}(\Omega)$
(2.1)

where $(\cdot,\cdot)_{L^{2}}$ is the usual scalar product of the space
$L^{2}(\Omega)$.

Moreover, we write $u\in L_{w,\mathrm{loc}}^{p}(\Omega)$ if $u\in
L_{w}^{p}(K)$ for all $K\Subset\Omega$.

One can notice (when $p$ is finite) that
$L_{w}^{p}(\Omega)=L^{p}(\Omega,|w|^{p})$, i.e. $L_{w}^{p}(\Omega)$ is the
weighted Sobolev space with the weight function $|w|^{p}$.

######  Remark 3.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $w\in W(\Omega)$ and
$p\in[1,\infty]$. Clearly, the space $L_{w}^{p}(\Omega)$ is continuously
embedded into $L^{p}(M_{n}^{c})$ for all $n$, where
$M_{n}:=\\{x\in\Omega\mid|w(x)|\leq 1/n\\}$ and $M_{n}^{c}$ is the
complementary (within $\Omega$) of $M_{n}$. If $w^{-1}\in
L_{\text{loc}}^{\infty}(\Omega_{N})$, e.g. $w\in\mathcal{C}(\Omega)$, then the
space $L_{w}^{p}(\Omega)$ is continuously embedded into
$L_{\text{loc}}^{p}(\Omega_{N})$.

Let $\mathcal{S}$ be the Schwartz space. For a fixed non-null polynomial $P$
we set

$\mathcal{S}_{P}=\\{\psi\in\mathcal{S}:\,\psi=P\varphi,\varphi\in\mathcal{S}\\}$

which is a linear subspace of $\mathcal{S}$ equipped with the sequence of semi
norms of $\mathcal{S}$. Having denoted by $\mathcal{S}^{\prime}_{P}$ the dual
space of $\mathcal{S}{}_{P}$, we have the following proposition.

######  Proposition 4.

Let $P$ be a non-null polynomial function and let $p\in[1,\infty]$. Then, any
element $f\in L_{P}^{p}(\mathbb{R}^{d})$ defines an element of
$\mathcal{S}^{\prime}_{P}$ in following way

$\forall\psi\in\mathcal{S}_{P}:\,\langle
f,\psi\rangle_{\mathcal{S}^{\prime}_{P},\mathcal{S}{}_{P}}:=\int_{\mathbb{R}^{d}}f(x)\psi(x)\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}.$

###### Proof.

Let $f\in L_{P}^{p}(\mathbb{R}^{d})$, $\psi\in\mathcal{S}_{P}$ and pick
$\varphi$ so that $\psi=P\varphi$. By Hölder’s inequality we have

$\left|\langle
f,\psi\rangle_{\mathcal{S}^{\prime}_{P},\mathcal{S}_{P}}\right|\leq\int_{\mathbb{R}^{d}}\left|f(x)P(x)\varphi(x)\right|\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}\leq||f||_{L_{P}^{p}}||\varphi||_{L^{p^{\prime}}}\leq
C||f||_{L_{P}^{p}}||\varphi||_{k}$

for some $k\in\mathbb{N}$ and $C>0$ that depend only on $d$ and $p$, where
$p^{\prime}$ is the conjugate exponent of $p$. By Thm. 1 of [1], the linear
continuous multiplication map
$M_{P}:\mathcal{S}\longrightarrow\mathcal{S}_{P}$ defined by
$M_{P}\varphi=P\varphi$ has a continuous inverse
$M_{P}^{-1}:\mathcal{S}_{P}\longrightarrow\mathcal{S}$. It follows that

$\exists C^{\prime}>0\,\exists
k^{\prime}\in\mathbb{N}\,\forall\psi\in\mathcal{S}_{P}:\,\left|\langle
f,\psi\rangle_{\mathcal{S}^{\prime}_{P},\mathcal{S}_{P}}\right|\leq
C^{\prime}||f||_{L_{P}^{p}}||\psi||_{k^{\prime}}$

which complete the proof. ∎

We shall now prove that the space $L_{w}^{p}(\Omega)$ is Cauchy complete.

######  Theorem 5.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $w\in W(\Omega)$, and
$p\in[1,\infty]$. Then, the space $L_{w}^{p}(\Omega)$ is Cauchy complete.

###### Proof.

This follows from Thm. 3.11 of [2] by noting that
$L_{w}^{p}(\Omega)=L^{p}(\mu)$ where $\mu$ is the measure defined for any
measurable set $E\subset\Omega$ by

$\mu(E):=\int_{E}|w|^{p}\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$

when $p$ is finite, and

$\mu(E):=||w||_{L^{\infty}(E)}$

when $p=\infty$. ∎

We prove now that the space $\mathcal{D}(\Omega)$ is dense in
$L_{w}^{p}(\Omega)$ when $p\in[1,\infty)$ and $w$ is smooth using the density
of $\mathcal{D}(\Omega)$ in $L^{p}(\Omega)$.

######  Theorem 6.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $p\in[1,\infty)$ and let
$w\in\mathcal{C}^{\infty}(\Omega)\cap W(\Omega)$. Then, $\mathcal{D}(\Omega)$
is dense in $L_{w}^{p}(\Omega)$.

###### Proof.

Let $f\in L_{w}^{p}(\Omega)$ and let $(\varphi_{n})$ be a sequence of
$\mathcal{D}(\Omega)$ that converges to $wf$ in $L^{p}(\Omega)$. Let
$\eta\in\mathcal{C}^{\infty}(\mathbb{R}^{d})$ be such that

$\eta\equiv 1\,\,\,\text{on }\,\left[-1,1\right]^{c},\,\eta\equiv 0\text{ on
}\left[-\frac{1}{2},\frac{1}{2}\right],\,0\leq\eta\leq 1,$

and for all $n\in\mathbb{N}$, set $\chi_{n}:=\eta_{n}\circ w$, where
$\eta_{n}(\cdot):=\eta(n\cdot)$. Clearly,
$\chi_{n}\in\mathcal{C}^{\infty}(\Omega)$ and

$\forall n\in\mathbb{N}:\,\chi_{n}\equiv 1\,\,\,\text{on
}M_{n}^{c},\,\chi_{n}\equiv 0\text{ on }M_{2n},\,\text{and }0\leq\chi_{n}\leq
1$ (2.2)

where the sequence $(M_{n})$ is as defined in Rem. 3. It follows that

$||\varphi_{n}\chi_{n}-fw||_{L^{p}}\leq||\varphi_{n}-fw||_{L^{p}}+||(1-\chi_{n})fw||_{L^{p}}.$
(2.3)

Since the set $N$ is Lebesgue negligible, the sequence $(\chi_{n})$ converges
a.e. in $\Omega$ to $1$. Indeed, for all $x\in\Omega_{N}$ there exists
$n_{0}\in\mathbb{N}$ such that $x\in M_{n_{0}}^{c}$, which implies that
$\chi_{n}(x)=1$ for all $n\geq n_{0}$ because the sequence $(M_{n})$ is
decreasing. It follows from (2.3) and the dominated convergence theorem that
the sequence $(\chi_{n}\varphi_{n})$ converges to $fw$ in $L^{p}(\Omega)$. For
all $n\in\mathbb{N}$, set $\psi_{n}:=w^{-1}\varphi_{n}\chi_{n}$ which belongs
to $\mathcal{D}(\Omega)$ since
$w^{-1}\chi_{n}\in\mathcal{C}^{\infty}(\Omega)$. Hence, by construction, we
have that $\psi_{n}\to f$ in $L_{w}^{p}(\Omega)$. ∎

In our next paper [6], we will prove that the density of $\mathcal{D}(\Omega)$
in $L_{w}^{p}(\Omega)$ holds provided that $w\in W(\Omega)\cap
L^{\infty}(\Omega)$.

###  2.2. The weak derivative in the sense of $L_{w,\mathrm{loc}}^{1}(\Omega)$

As discussed above, in general, elements of $L_{w}^{p}(\Omega)$ are not
locally integrable over $\Omega$, and hence the notion of weak derivative in
the sense of $L_{\text{loc}}^{1}(\Omega)$ to elements of $L_{w}^{p}(\Omega)$
is not well defined. However, since $L_{w}^{p}(\Omega)\subseteq
L_{\text{loc}}^{1}(\Omega_{N})$ when $w\in W(\Omega)$ (see Rem. 3), one can
use the weak derivative in the sense of $L_{\text{loc}}^{1}(\Omega_{N})$ to
define the derivative in $L_{w}^{p}(\Omega)$. Indeed, this is possible, but
the problem with this definition is that it does not take into account the
behavior of elements of $L_{w}^{p}(\Omega)$ near the set $N$, since an element
of $\mathcal{D}(\Omega_{N})$ vanish on some neighborhood of $N$.

####  2.2.1. Definition and proprieties of the weak derivative in the sense of
$L_{w,\mathrm{loc}}^{1}(\Omega)$

######  Definition 7.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $\alpha\in\mathbb{N}^{d}$
with $|\alpha|\neq 0$, $p\in[1,\infty]$, and $w\in
W(\Omega)\cap\mathcal{C}^{|\alpha|}(\Omega)$. We say that $f\in
L_{w,\text{loc}}^{1}(\Omega)$ is $\alpha$-weakly differentiable in the sense
of $L_{w,\text{loc}}^{1}(\Omega)$ if there exists an element of
$L_{w^{|\alpha|+1},\text{loc}}^{1}(\Omega)$ denoted by $D_{w}^{\alpha}f$
satisfying

$\forall\varphi\in\mathcal{D}(\Omega):\,\int_{\Omega}f\partial^{\alpha}(w^{|\alpha|+1}\varphi)\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=(-1)^{|\alpha|}\int_{\Omega}w^{|\alpha|+1}\varphi
D_{w}^{\alpha}f\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}.$
(2.4)

In the sequel, for some given $f\in L_{w,\text{loc}}^{1}(\Omega)$, we will
frequently write $\exists D_{w}^{\alpha}f$ to say that the $\alpha$-weak
derivative of $f$ in the sense of $L_{w,\text{loc}}^{1}(\Omega)$ exists (and
hence belongs to $L_{w^{|\alpha|+1},\text{loc}}^{1}(\Omega)$).

######  Remark 8.

Let $\Omega$, $\alpha$, $p$, $w$ be as in the previous definition.

  1. (i)

The weak derivative in the sense of $L_{w,\text{loc}}^{1}(\Omega)$ (when it
exists) is unique. The proof is identical to the proof of the uniqueness of
the weak derivative in the sense of $L_{\text{loc}}^{1}(\Omega)$.

  2. (ii)

Take $\alpha$ with $|\alpha|=1$ and let $f\in L_{w}^{p}(\Omega)$. Assume that
$\exists D_{w}^{\alpha}f\in L_{w^{2}}^{p}(\Omega)$, and that
$f\partial^{\alpha}(w^{2})$, $w^{2}f\in L^{p}(\Omega)$, e.g. $\Omega$ is
bounded and $\nabla w\in L^{\infty}(\Omega)$, then (2.4) yields

$\forall\varphi\in\mathcal{D}(\Omega):\,\int_{\Omega}\left(f\partial^{\alpha}(w^{2})+w^{2}D_{w}^{\alpha}f\right)\varphi\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=-\int_{\Omega}w^{2}f\partial^{\alpha}\varphi\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$

which implies that $w^{2}f$ is $\alpha$-weakly differentiable in the sense of
$L_{\text{loc}}^{1}(\Omega)$ (or simply $\exists D^{\alpha}(w^{2}f)$) and
$D^{\alpha}(w^{2}f)=f\partial^{\alpha}(w^{2})+w^{2}D_{w}^{\alpha}f\in
L^{p}(\Omega)$. Consequently, if we assume that the above conditions hold for
all $\alpha\in\mathbb{N}^{d}$ with $|\alpha|=1$ we get that $w^{2}f\in
W^{1,p}(\Omega)$.

  3. (iii)

The reason for defining the weak derivative in the sense of
$L_{w,\text{loc}}^{1}(\Omega)$ in such a way can be explained as follows: One
can show that for any $\varphi\in\mathcal{D}(\Omega)$, there exists a
compactly supported continuous function $h_{\alpha}$ on $\Omega$ such that
$\partial^{\alpha}(w^{|\alpha|+1}\varphi)=wh_{\alpha}$, which implies that

$\forall\varphi\in\mathcal{D}(\Omega):\,\left|\int_{\Omega}f\partial^{\alpha}(w^{|\alpha|+1}\varphi)\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}\right|\leq||f||_{L_{w}^{1}(K)}||h_{\alpha}||_{L^{\infty}(K)},$

where $K$ is the support of $h_{\alpha}$. Hence, the left hand side of (2.4)
is always finite. Moreover, for any test function $\varphi$, the functional
$L_{w}^{p}(\Omega)\longrightarrow\mathbb{R}$,
$f\longrightarrow\int_{\Omega}f\partial^{\alpha}(w^{|\alpha|+1}\varphi)\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$ is continuous. This property will be
used to prove the completeness of the spaces
$W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$ defined below.

  4. (iv)

By density of $\mathcal{D}(\Omega)$ in $\mathcal{C}_{c}^{|\alpha|}(\Omega)$,
we can equally well use the space $\mathcal{C}_{c}^{|\alpha|}(\Omega)$ instead
of the space $\mathcal{D}(\Omega)$ in (2.4). This is because

$\left|\int_{\Omega}f\partial^{\alpha}(w^{|\alpha|+1}\varphi)\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}\right|\leq||fw||_{L^{1}(K)}\sum_{0\leq\beta\leq\alpha}C_{\beta}||g_{\beta}||_{L^{\infty}(K)}||\partial^{\beta}\varphi||_{\infty},$

where $K$ is the support of $\varphi$, $g_{\beta}$ is a continuous function
that depends on $w$, $\alpha$ and $\beta$, and $||\cdot||_{\infty}$ is the
uniform norm.

######  Example 9.

Let $f(x)=1/x^{2}\in L_{x^{\text{2}}}^{2}(-1,1)$. Then, $\exists
D_{x^{2}}f=-2x^{-3}\in L_{x^{3}}^{2}(-1,1)$.

####  2.2.2. Comparison with the weak derivative in the sense of
$L_{\mathrm{loc}}^{1}(\Omega)$

Since $L_{\text{loc}}^{1}(\Omega)\subseteq L_{w,\text{loc}}^{1}(\Omega)$ when
$w$ is locally bounded, one can ask if the weak derivative in the sense of
$L_{\text{loc}}^{1}(\Omega)$ implies the weak derivative in the sense of
$L_{w,\text{loc}}^{1}(\Omega)$. The answer is given in the following
proposition

######  Proposition 10.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $\alpha\in\mathbb{N}^{d}$,
$w\in W(\Omega)\cap\mathcal{C}^{|\alpha|}(\Omega)$ and $f\in
L_{\mathrm{loc}}^{1}(\Omega)$. If $\exists D^{\alpha}f\in
L_{\mathrm{loc}}^{1}(\Omega)$ then $\exists D_{w}^{\alpha}f$ and
$D^{\alpha}f=D_{w}^{\alpha}f$.

###### Proof.

Recall that, by density of $\mathcal{D}(\Omega)$ in the space
$\mathcal{C}_{c}^{|\alpha|}(\Omega)$, we can replace the space
$\mathcal{D}(\Omega)$ in the definition of the weak derivative in the sense of
$L_{\text{loc}}^{1}(\Omega)$ with the space
$\mathcal{C}_{c}^{|\alpha|}(\Omega)$. Hence, it suffices to use the definition
of the weak derivative in the sense of $L_{\text{loc}}^{1}(\Omega)$ with the
function $\varphi w^{|\alpha|+1}\in\mathcal{C}_{c}^{|\alpha|}(\Omega)$ instead
of $\varphi\in\mathcal{D}(\Omega)$. ∎

####  2.2.3. Comparison with the weak derivative in the sense of
$L_{\mathrm{loc}}^{1}(\Omega_{N})$

In the next proposition, we will see that the weak derivative in the sense of
$L_{w,\mathrm{loc}}^{1}(\Omega)$ implies the weak derivative in the sense of
$L_{\mathrm{loc}}^{1}(\Omega_{N})$.

######  Proposition 11.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $\alpha\in\mathbb{N}^{d}$
with $|\alpha|\neq 0$, $w\in W(\Omega)\cap\mathcal{C}^{|\alpha|}(\Omega)$ and
$f\in L_{w,\mathrm{loc}}^{1}(\Omega)$. If $f$ is $\alpha$-weakly
differentiable in the sense of $L_{w,\mathrm{loc}}^{1}(\Omega)$, then the
$\alpha$-weak derivative of $f$ in the sense of
$L_{\mathrm{loc}}^{1}(\Omega_{N})$, denoted by $D^{\alpha}f$, exists (and
hence belongs to $L_{\mathrm{loc}}^{1}(\Omega_{N})$) and coincides with
$D_{w}^{\alpha}f$.

###### Proof.

Recall from (iv) of Rem. 8 that we can equivalently replace the space of test
functions $\mathcal{D}(\Omega)$ with the space
$\mathcal{C}_{c}^{|\alpha|}(\Omega)$ in (2.4). Hence, for any test function
$\varphi\in\mathcal{D}(\Omega_{N})$, it suffices to use (2.4) with $\varphi
w^{-|\alpha|-1}\in\mathcal{C}_{c}^{|\alpha|}(\Omega)$. ∎

###  2.3. The space $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$

In this section, we introduce the space $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$
and prove its completeness.

In the rest of the paper, for any $m\in\mathbb{N}$, we will denote by
$\pi_{m}$ the set of multi-indices $\alpha\in\mathbb{N}^{d}$ with
$|\alpha|\leq m$. We will also use the notation $\pi_{m}^{k}$ ($k<m$) for the
set $\pi_{m}\backslash\pi_{k}$.

If $\\{w_{\alpha}\\}_{\alpha\in\pi_{m}}$ is a collection of elimination
functions then the function $w_{0}$ ($0\in\mathbb{N}^{d}$) will be simply
denoted by $w$.

######  Definition 12.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $p\in[1,\infty]$,
$m\in\mathbb{N}$, and $\\{w_{\alpha}\\}_{\alpha\in\pi_{m}}$ be a collection of
elimination functions on $\Omega$ such that $w\in\mathcal{C}^{m}(\Omega)$. We
set

$W_{\\{w_{\alpha}\\}}^{m,p}(\Omega):=\\{u\in
L_{w}^{p}(\Omega)\mid\forall\alpha\in\pi_{m}^{0}:\,\exists D_{w}^{\alpha}u\in
L_{w_{\alpha}}^{p}(\Omega)\\}.$

This space is equipped with the norm

$\forall u\in
W_{\\{w_{\alpha}\\}}^{m,p}(\Omega):\,||u||_{W_{\\{w_{\alpha}\\}}^{m,p}}:=\left(\sum_{|\alpha|\leq
m}||D_{w}^{\alpha}u||_{L_{w_{\alpha}}^{p}}^{p}\right)^{1/p}$

when $p<\infty$, and

$\forall u\in
W_{\\{w_{\alpha}\\}}^{m,\infty}(\Omega):\,||u||_{W_{\\{w_{\alpha}\\}}^{m,\infty}}:=\sum_{|\alpha|\leq
m}||D_{w}^{\alpha}u||_{L_{w_{\alpha}}^{\infty}}.$

When $p=2$, the space $W_{\\{w_{\alpha}\\}}^{m,2}(\Omega)$ will be denoted by
$H_{\\{w_{\alpha}\\}}^{m}(\Omega)$ and its norm is generated by the scalar
product

$\forall u,v\in
H_{\\{w_{\alpha}\\}}^{m}(\Omega):\,(u,v)_{H_{\\{w_{\alpha}\\}}^{m}}:=\sum_{|\alpha|\leq
m}(D_{w}^{\alpha}u,D_{w}^{\alpha}v)_{L_{w_{\alpha}}^{2}}.$

In the sequel, the space $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$ will be denoted
by $W_{\\{w^{|\alpha|+1}\\}}^{m,p}(\Omega)$ when $w_{\alpha}=w^{|\alpha|+1}$
for all $\alpha\in\pi_{m}^{0}$.

Now, we use Prop. 11 to show that the space
$W_{\\{w^{|\alpha|+1}\\}}^{m,p}(\Omega)$ is continuously embedded into the
space $W_{\text{loc}}^{m,p}(\Omega_{N})$.

######  Corollary 13.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $p\in[1,\infty]$,
$m\in\mathbb{N}$, and $w\in W(\Omega)\cap\mathcal{C}^{m}(\Omega)$. Then the
space $W_{\\{w^{|\alpha|+1}\\}}^{m,p}(\Omega)$ is continuously embedded into
$W_{\mathrm{loc}}^{m,p}(\Omega_{N})$.

###### Proof.

Let $u\in W_{\\{w^{|\alpha|+1}\\}}^{m,p}(\Omega)$. By definition, we have that
$u\in L_{w}^{p}(\Omega)$ and $\exists D_{w}^{\alpha}u\in
L_{w^{|\alpha|+1}}^{p}(\Omega)$ for all $\alpha\in\pi_{m}^{0}$. By Prop. 11,
we have that the weak derivative $D^{\alpha}u$ of $u$ in the sense of
$L_{\text{loc}}^{1}(\Omega_{N})$ exists and coincides with $D_{w}^{\alpha}f$.
Moreover, we have

$\forall K\Subset\Omega_{N}:\,||u||_{W^{m,p}(K)}\leq\max_{|\alpha|\leq
m}\sup_{x\in
K}|w(x)^{-1}|^{|\alpha|+1}||u||_{W_{\\{w^{|\alpha|+1}\\}}^{m,p}(\Omega)}$

which complete the proof. ∎

We now prove that the spaces $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$ are Cauchy
complete.

######  Theorem 14.

Let $\Omega\subseteq\mathbb{R}^{d}$ be an open set, $p\in[1,\infty]$,
$m\in\mathbb{N}$, and let $\\{w_{\alpha}\\}$ be a collection of elimination
functions on $\Omega$ such that $w\in\mathcal{C}^{m}(\Omega)$ and

$\forall\alpha\in\pi_{m}^{0}\,\forall K\Subset\Omega\,\exists C_{K}\,\forall
x\in K:\,|w(x)|^{|\alpha|+1}\leq C_{K}|w_{\alpha}(x)|.$ (2.5)

Then, the space $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$ is Cauchy complete.

###### Proof.

Let $(u_{n})$ be a Cauchy sequence of $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$. By
definition, for all $\alpha\in\pi_{m}$, the sequence $(D_{w}^{\alpha}u_{n})$
is a Cauchy sequence of $L_{w_{\alpha}}^{p}(\Omega)$. By completeness of
$L_{w_{\alpha}}^{p}(\Omega)$ (see Thm. 5) there exists a unique $u_{\alpha}\in
L_{w_{\alpha}}^{p}(\Omega)$ limit of the sequence $(D_{w}^{\alpha}u_{n})$ in
$L_{w_{\alpha}}^{p}(\Omega)$. Set $u:=u_{0}$. We shall now prove that $u$ is
weakly differentiable in the sense of $L_{w,\text{loc}}^{1}(\Omega)$ with
$D_{w}^{\alpha}u=u_{\alpha}$ for all $\alpha\in\pi_{m}^{0}$. Indeed, by
assumption we have

$\forall\varphi\in\mathcal{D}(\Omega):\,\int_{\Omega}u_{n}\partial^{\alpha}\left(w^{|\alpha|+1}\varphi\right)\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}=(-1)^{|\alpha|}\int_{\Omega}\varphi
w^{|\alpha|+1}D_{w}^{\alpha}u_{n}\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}.$
(2.6)

We have that
$\lim_{n\to\infty}\int_{\Omega}u_{n}\partial^{\alpha}\left(w^{|\alpha|+1}\varphi\right)\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=\int_{\Omega}u\partial^{\alpha}\left(w^{|\alpha|+1}\varphi\right)\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$ by (iii) of Rem. 8. On the other
hand, (2.5) implies that

$\left|\int_{\Omega}(D_{w}^{\alpha}u_{n})w^{|\alpha|+1}\varphi\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}\right|\leq
C_{\text{supp}(\varphi)}\int_{\Omega}\left|(D_{w}^{\alpha}u_{n})w_{\alpha}\varphi\right|\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}\leq
C_{\text{supp}(\varphi)}||D_{w}^{\alpha}u_{n}||_{L_{w_{\alpha}}^{p}}||\varphi||_{L^{p^{\prime}}}$

which implies that the right hand side of (2.6) converges to
$(-1)^{|\alpha|}\int_{\Omega}\varphi
w^{|\alpha|+1}u_{\alpha}\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$ when
$n\to\infty$. It follows that

$\forall\varphi\in\mathcal{D}(\Omega):\,\int_{\Omega}u\partial^{\alpha}\left(w^{|\alpha|+1}\varphi\right)\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}=(-1)^{|\alpha|}\int_{\Omega}\varphi
w^{|\alpha|+1}u_{\alpha}\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$

which proves that $\exists D_{w}^{\alpha}u=u_{\alpha}\in
L_{w_{\alpha}}^{p}(\Omega)\subseteq L_{w^{|\alpha|+1},\text{loc}}^{1}(\Omega)$
(where the last inclusion follows from (2.5)), which implies that $u\in
W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$. Therefore, the sequence $(u_{n})$
converges to $u$ in $W_{\\{w_{\alpha}\\}}^{m,p}(\Omega)$, and hence the proof
is completed. ∎

######  Remark 15.

  1. (i)

If $w_{\alpha}=|w|^{\sigma}$ for some $\sigma\in(0,|\alpha|+1)$ then (2.5)
holds. This is because $|w(x)|^{|\alpha|+1}\leq|w(x)|^{\sigma}$ on
$\\{x\in\Omega\mid|w(x)|\leq 1\\}$, and $|w|^{|\alpha|+1-\sigma}$ is bounded
on any compact subset of $\Omega$.

  2. (ii)

Clearly, the space $W_{\\{w^{|\alpha|+1}\\}}^{m,p}(\Omega)$ is Cauchy complete
since (2.5) trivially holds.

###  2.4. The trace operator

Now, we discuss the different possibilities to define a trace operator on
$W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$.

######  Remark 16.

Let $\Omega$ be a bounded open set with a sufficiently smooth boundary111So
that the trace operator on $W^{1,p}(\Omega)$ would be well defined.,
$p\in[1,\infty)$, $w\in W(\Omega)\cap\mathcal{C}^{1}(\Omega)$ and $f\in
W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$.

  1. (i)

If we assume that

$\exists K\Subset\Omega\,\exists\delta>0\,\forall x\in\Omega\backslash
K:\,|w(x)|\geq\delta$ (2.7)

(or equivalently $M_{n_{0}}\Subset\Omega$ for some $n_{0}\in\mathbb{N}$) then
one can show that $\partial\Omega\subseteq\partial M_{n}^{c}$ for all $n\geq
n_{0}$. Choose $n\geq n_{0}$. We claim that

$\chi_{n}f\in
W^{1,p}(\Omega)\,\,\,\text{and}\,\,\,\forall\alpha\in\pi_{1}^{0}:\,D^{\alpha}(\chi_{n}f)=f\partial^{\alpha}\chi_{n}+\chi_{n}D_{w}^{\alpha}f,$

where $(\chi_{n})$ is the sequence of $\mathcal{C}^{1}(\Omega)$ defined in the
proof of Thm. 6. Indeed, by construction, the function $\chi_{n}$ is bounded
and supported in $M_{2n}^{c}$ which implies that

$||\chi_{n}D_{w}^{\alpha}f||_{L^{p}(\Omega)}\leq||w^{-|\alpha|-1}\chi_{n}w^{|\alpha|+1}D_{w}^{\alpha}f||_{L^{p}(\Omega)}\leq(2n)^{|\alpha|+1}||w^{|\alpha|+1}D_{w}^{\alpha}f||_{L^{p}(\Omega)}$
(2.8)

Moreover, since $\chi_{n}\equiv 1$ on $M_{n}^{c}$, and (by (2.7))
$M_{n}\Subset\Omega$, we get

$\exists C>0\,\forall\alpha\in\pi_{1}^{0}\,\forall
x\in\Omega:\,|\partial^{\alpha}\chi_{n}|=|n(\partial^{\alpha}w)\eta^{\prime}(nw(x))|\leq
Cn,$

where the constant $C$ does not depend on $n$, which implies that

$||f\partial^{\alpha}\chi_{n}||_{L^{p}(\Omega)}\leq
Cn^{2}||wf||_{L^{p}(\Omega)}$ (2.9)

and hence $f\chi_{n}$, $f\partial^{\alpha}\chi_{n}+\chi_{n}D_{w}^{\alpha}f\in
L^{p}(\Omega)$. It remains to prove that $\chi_{n}f$ is weakly differentiable
in the sense of $L_{\text{loc}}^{1}(\Omega)$ and that
$D^{\alpha}(\chi_{n}f)=f\partial^{\alpha}\chi_{n}+\chi_{n}D_{w}^{\alpha}f$. To
do so, take $\alpha\in\pi_{1}^{0}$, $\varphi\in\mathcal{D}(\Omega)$, and
recall from Prop. 11 that $D_{w}^{\alpha}f$ is also the $\alpha$-weak
derivative of $f$ in the sense of $L_{\text{loc}}^{1}(\Omega_{N})$, which
implies that

$\int_{\Omega}f\partial^{\alpha}(\varphi\chi_{n})\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=-\int_{\Omega}\varphi\chi_{n}D_{w}^{\alpha}f\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$

because $\varphi\chi_{n}\in\mathcal{C}_{c}^{1}(\Omega_{N})$. Hence

$\int_{\Omega}f\chi_{n}\partial^{\alpha}\varphi\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=\int_{\Omega}f(\partial^{\alpha}(\chi_{n}\varphi)-\varphi\partial^{\alpha}\chi_{n})\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=-\int_{\Omega}\varphi(\chi_{n}D_{w}^{\alpha}f+f\partial^{\alpha}\chi_{n})\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}.$

Therefore, our claim is proved. If we denote by
$|_{\partial\Omega}:W^{1,p}(\Omega)\longrightarrow L^{p}(\partial\Omega)$ the
trace operator, then $(f\chi_{n})|_{\partial\Omega}$ is well defined. Since
$\chi_{n}\equiv 1$ on $\Omega\backslash K\subseteq M_{n}^{c}$, we have that
$(f\chi_{n})|_{\partial\Omega}=f|_{\partial\Omega}$ which implies that
$|_{\partial\Omega}$ can be extended to the space
$W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$. Furthermore, from (2.8), (2.9) and
the trace theorem in $W^{1,p}(\Omega)$ we get

$||f|_{\partial\Omega}||_{L^{p}(\partial\Omega)}\leq
C||\chi_{n}f||_{W^{1,p}(\Omega)}\leq
Cn^{2}||f||_{W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)}$ (2.10)

which proves that $|_{\partial\Omega}$ is continuous on
$W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$.

  2. (ii)

If we assume that $\nabla w\in L^{\infty}(\Omega)$, then $w$ is also bounded
and uniformly continuous, because $\Omega$ is bounded. Hence, we can assume
that $w\in\mathcal{C}^{0}(\overline{\Omega})$. It follows in particular that
$f\partial^{\alpha}(w^{2})$, $fw^{2}\in L^{p}(\Omega)$ for all
$\alpha\in\pi_{1}^{0}$, which implies, as we have seen in (ii) of Rem. 8, that
$w^{2}f\in W^{1,p}(\Omega)$ with
$D^{\alpha}(fw^{2})=w^{2}D_{w}^{\alpha}f+f\partial^{\alpha}w^{2}$ for all
$\alpha\in\pi_{1}^{0}$. It follows that $(w^{2}f)|_{\partial\Omega}$ is well
defined and belongs to $L^{p}(\partial\Omega)$, or equivalently
$f|_{\partial\Omega}\in L_{w^{2}|_{\partial\Omega}}^{p}(\partial\Omega)$.
Moreover, we have that the map
$W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)\longrightarrow W^{1,p}(\Omega)$,
$f\longrightarrow w^{2}f$ is continuous. Indeed,

$\displaystyle||w^{2}f||_{W^{1,p}}^{p}$
$\displaystyle\leq||w^{2}f||_{L^{p}}^{p}+2^{p-1}\sum_{|\alpha|=1}||f\partial^{\alpha}w^{2}||_{L^{p}}^{p}+2^{p-1}\sum_{|\alpha|=1}||w^{2}D_{w}^{\alpha}f||_{L^{p}}^{p}$
$\displaystyle\leq(||w||_{L^{\infty}}^{p}+2^{p}\sum_{|\alpha|=1}||\partial^{\alpha}w||_{L^{\infty}}^{p})||wf||_{L^{p}}^{p}+2^{p-1}\sum_{|\alpha|=1}||w^{2}D_{w}^{\alpha}f||_{L^{p}}^{p}$
$\displaystyle\leq C||f||_{W_{\\{w^{|\alpha|+1}\\}}^{1,p}}^{p}.$

Using this inequality and the trace theorem in $W^{1,p}(\Omega)$ we get

$||(w^{2}f)|_{\partial\Omega}||_{L^{p}(\partial\Omega)}\leq
C||w^{2}f||_{W^{1,p}(\Omega)}\leq
C||f||_{W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)}$ (2.11)

which implies that the operator
$W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)\longrightarrow L^{p}(\partial\Omega)$,
$f\longrightarrow(w^{2}f)|_{\partial\Omega}$ is continuous.

Therefore, we can define the trace operator on the space
$W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$ either by setting
$\text{Tr}(f)=f|_{\partial\Omega}\in L^{p}(\partial\Omega)$, or by setting
$\text{Tr}(f)=(w^{2}f)|_{\partial\Omega}\in L^{p}(\partial\Omega)$ depending
on the assumptions satisfied by the elimination function $w$.

######  Remark 17.

In the favorite situation in which assumption (2.7) holds and $\nabla w\in
L^{\infty}(\Omega)$, both $f|_{\partial\Omega}$ and
$(w^{2}f)|_{\partial\Omega}$ are well defined and belong to
$L^{p}(\partial\Omega)$. We have also that
$w\in\mathcal{C}^{0}(\overline{\Omega})$ and
$w^{-1}\in\mathcal{C}^{0}(\overline{M_{n}^{c}})$ for all $n$, which implies
that

$\sup_{x\in\partial\Omega}|w(x)|<\infty,\,\,\,\sup_{x\in\partial\Omega}|w^{-1}(x)|\leq\sup_{x\in\partial
M_{n}^{c}}|w^{-1}(x)|<\infty$

where $n$ is chosen sufficiently large so that
$\partial\Omega\subseteq\partial M_{n}^{c}$. It follows that

$\sup_{x\in\partial\Omega}|w^{-2}(x)|^{-1}||f|_{\partial\Omega}||_{L^{p}(\partial\Omega)}\leq||(w^{2}f)|_{\partial\Omega}||_{L^{p}(\partial\Omega)}\leq\sup_{x\in\partial\Omega}|w(x)|^{2}||f|_{\partial\Omega}||_{L^{p}(\partial\Omega)}$
(2.12)

for all $f\in W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$, which means that the
operators $f\longrightarrow f|_{\partial\Omega}$,
$f\longrightarrow(fw^{2})|_{\partial\Omega}$ are equivalent, and hence
$f|_{\partial\Omega}=0$ if and only if $(w^{2}f)|_{\partial\Omega}=0$.

These different possibilities motivates the following definition

######  Definition 18.

Let $\Omega$ be a bounded open set with sufficiently smooth boundary,
$p\in[1,\infty)$, and $w\in W(\Omega)\cap\mathcal{C}^{1}(\Omega)$. We say that
the trace operator
$\text{Tr}:W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)\longrightarrow
L^{p}(\partial\Omega)$ is well defined

  1. (i)

if assumption (2.7) holds, and we write

$\forall f\in
W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega):\,\text{Tr}(f):=f|_{\partial\Omega}\in
L^{p}(\partial\Omega)$ (2.13)

  2. (ii)

if assumption (2.7) does not holds, $\nabla w\in L^{\infty}(\Omega)$, and we
write

$\forall f\in
W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega):\,\text{Tr}(f):=(w^{2}f)|_{\partial\Omega}\in
L^{p}(\partial\Omega).$ (2.14)

As we have seen in the beginning of this subsection, under the assumptions
given in the latter definition, the trace operator is continuous (see (2.10)
and (2.11)).

###  2.5. The space $W_{\\{w^{|\alpha|+1}\\},0}^{1,p}$

In this subsection we define the subspace $W_{\\{w^{|\alpha|+1}\\},0}^{1,p}$
of $W_{\\{w^{|\alpha|+1}\\}}^{1,p}$and we prove that is Cauchy complete.

######  Definition 19.

Let $\Omega$ be an open set with sufficiently smooth boundary,
$p\in[1,\infty)$, and $w\in W(\Omega)\cap\mathcal{C}^{1}(\Omega)$. Assume that
the trace operator is well defined. Then we set

$W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega):=\\{f\in
W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)\mid\text{Tr}(f)=0\\}.$

######  Remark 20.

  1. (i)

Recall that we have $w^{2}f\in W^{1,p}(\Omega)$ for all $f\in
W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$ in case $\nabla w\in
L^{\infty}(\Omega)$ and $\Omega$ is bounded (see (ii) of Rem. 17). Moreover,
we claim that $w^{2}f\in W_{0}^{1,p}(\Omega)$ for all $f\in
W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega)$ regardless the way the trace
operator is defined. Indeed, this is trivial if the trace operator is defined
as in (2.14). However, if the trace operator is defined as in (2.13), then
$\text{Tr}(f)=0$ implies that $w^{2}f|_{\partial\Omega}=0$ (by Rem. 17) and
hence the claim is proved.

  2. (ii)

In the favorite situation in which assumption (2.7) holds and $\nabla w\in
L^{\infty}(\Omega)$, we have

$W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega):=\\{f\in
W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)\mid f|_{\partial\Omega}=0\\}$

and $w^{2}f\in W_{0}^{1,p}(\Omega)$ for all $f\in
W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega)$.

  3. (iii)

The completion of $\mathcal{D}(\Omega)$ with respect to the norm of
$W_{\\{w^{|\alpha|+1}\\}}^{1,p}(\Omega)$ is included in
$W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega)$.

######  Example 21.

We have that $x^{-2}-1\in H_{\\{w^{|\alpha|+1}\\},0}^{1}(0,1)$ with $w=x^{2}$.

######  Proposition 22.

Let $\Omega$ be a bounded open set with sufficiently smooth boundary,
$p\in[1,\infty)$, and $w\in W(\Omega)\cap\mathcal{C}^{1}(\Omega)$. Assume that
the trace operator is well defined. Then the space
$W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega)$ is Cauchy complete.

###### Proof.

It is sufficient to prove that the space
$W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega)$ is closed since it is a subspace of
a Banach space (See (ii) of Rem. 15). However, the closedness follows from the
fact that $W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega)$ is the kernel of the
trace operator Tr. ∎

###  2.6. Poincaré inequality

We first derive an interesting inequality in which we prove that the term
$||f\nabla(w^{2})||_{L^{p}}$ is bounded by $||w^{2}D_{w}f||_{L^{p}}$. This
inequality will be used to prove an interesting Poincaré inequality under the
assumption $\nabla w\in L^{\infty}(\Omega)$.

######  Theorem 23.

Let $\Omega\subseteq\mathbb{R}^{d}$ ($d\geq 2$) be an open set with
sufficiently smooth boundary, $p\in[1,d)$, and $w\in
W(\Omega)\cap\mathcal{C}^{1}(\Omega)$, with $\nabla w\in L^{\infty}(\Omega)$.
Assume that

$\exists\sigma>0\,\forall x\neq 0:\,|\nabla
w(x)|_{p}:=\left(\sum_{i=1}^{d}|\partial_{i}w(x)|^{p}\right)^{1/p}\leq\sigma\frac{|w(x)|}{|x|}$
(2.15)

where $|x|$ is the euclidean norm of $\mathbb{R}^{d}$, and that

$0<\frac{2\sigma p}{d-p}<1.$ (2.16)

Then

$\forall f\in
W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega):\,||f\nabla(w^{2})||_{L^{p}}\leq\frac{2\sigma
p}{d-p-2\sigma p}||w^{2}D_{w}f||_{L^{p}}.$ (2.17)

###### Proof.

Let $f\in W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega)$. We recall from (i) of
Rem. 20 that $fw^{2}\in W_{0}^{1,p}(\Omega)$ when $\nabla w\in
L^{\infty}(\Omega)$, and from (ii) of Rem. 17 that
$D^{\alpha}(fw^{2})=w^{2}D_{w}^{\alpha}f+f\partial^{\alpha}w^{2}$ for all
$\alpha\in\pi_{1}^{0}$. Thus, using (2.15) and Hardy inequality we get

$\displaystyle||f\nabla(w^{2})||_{L^{p}}:=2\left(\sum_{i=1}^{d}||fw\partial_{i}w||_{L^{p}}^{p}\right)^{\frac{1}{p}}$
$\displaystyle\leq 2\sigma\left\|\frac{fw^{2}}{x}\right\|_{L^{p}}$
$\displaystyle\leq 2\sigma\frac{p}{d-p}\left\|D(fw^{2})\right\|_{L^{p}}$
$\displaystyle\leq
2\sigma\frac{p}{d-p}\left\|w^{2}D_{w}f\right\|_{L^{p}}+2\sigma\frac{p}{d-p}\left\|f\nabla(w^{2})\right\|_{L^{p}}$

This inequality together with (2.16) gives (2.17). ∎

######  Example 24.

The typical example of a function satisfying (2.15), when $p=2$ is the
function $w(x)=|x|^{\sigma}$ with $\sigma\geq 2$.

Similarly, we have

######  Theorem 25.

Let $\Omega\subseteq\mathbb{R}$ be an open set, $p\in(1,\infty)$, and $w\in
W(\Omega)\cap\mathcal{C}^{1}(\Omega)$ with $w^{\prime}\in L^{\infty}(\Omega)$.
Assume that

$\exists\sigma>0\,\forall x\neq
0:\,|w^{\prime}(x)|\leq\sigma\frac{|w(x)|}{|x|},\,\,\,0<\frac{2\sigma
p}{p-1}<1.$

Then

$\forall f\in
W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega):\,||f(w^{2})^{\prime}||_{L^{p}}\leq\frac{2\sigma
p}{p-1-2\sigma p}||w^{2}D_{w}f||_{L^{p}}.$

The proof is identical to the proof of Thm. 23.

Using Poincaré inequality and Thm. 23 we get

######  Corollary 26.

Let $\Omega\subseteq\mathbb{R}^{d}$ ($d\geq 2$) be a bounded open set with
sufficiently smooth boundary, $p\in[1,d)$, and $w\in
W(\Omega)\cap\mathcal{C}^{1}(\Omega)$ with $\nabla w\in L^{\infty}(\Omega)$.
Assume that (2.15) and (2.16) hold. Then

$\exists C_{\Omega}>0\,\forall f\in
W_{\\{w^{|\alpha|+1}\\},0}^{1,p}(\Omega):\,||w^{2}f||_{L^{p}}\leq
C_{\Omega}||w^{2}D_{w}f||_{L^{p}}.$

###### Proof.

This is a direct consequence of Poincaré inequality in $W_{0}^{1,p}(\Omega)$
and Thm. 23. ∎

###  2.7. Degenerate elliptic PDE

In the sequel, if $\alpha\in\pi_{1}^{0}$, we write $D_{w}^{i}$ instead of
$D_{w}^{\alpha}$ where $i$ is the index of the non-null component of $\alpha$.
We also write $D_{w}$ for the operator $(D_{w}^{i})$.

Let consider the boundary-value problem

$\begin{cases}-\sum_{i,j=1}^{d}D_{j}(a_{ij}D_{i}u)+(b\cdot D)u+cu=f&\text{in
}\Omega\\\ \text{Tr}(u)=0&\text{on }\partial\Omega\end{cases}$ (2.18)

where $(a_{ij})$, $f$, $b=(b_{i})$, $c$ are some given functions. We will
specify later what regularity conditions we have to assume on these functions.
We say that $u\in H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$ is a weak solution
of the boundary-value problem (2.18) if

$\forall v\in
H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega):\,B_{w}(u,v)=(f,v)_{L^{2}},$ (2.19)

where $B_{w}(\cdot,\cdot):H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)\times
H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)\longrightarrow\mathbb{R}$ is the
bilinear form given by

$B_{w}(u,v):=\sum_{i,j=1}^{d}\int_{\Omega}a_{ij}D_{w}^{i}uD_{w}^{j}v\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}+\int_{\Omega}(b\cdot
D_{w})uv\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}+\int_{\Omega}cuv\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}.$
(2.20)

We will see in the following proposition, that if $u\in
H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$ is a weak solution of (2.18), then the
differential equation of (2.18) holds in the sense of
$\mathcal{D}^{\prime}(\Omega_{N})$ provided the coefficient and the external
term satisfy suitable assumptions.

######  Proposition 27.

Let $\Omega$ be an open set, $w\in W(\Omega)\cap\mathcal{C}^{1}(\Omega)$, and
let $(a_{ij})$, $b=(b_{i})$, $c$, $f$ be such that

$\forall i,j=1,...,d:\,a_{ij},\,b_{i},\,c\in
L_{\mathrm{loc}}^{\infty}(\Omega_{N})\,\text{ and }\,f\in
L_{\mathrm{loc}}^{2}(\Omega_{N}).$ (2.21)

Assume that $u\in H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$ is a weak solution
of (2.18). Then, the differential equation of (2.18) holds in the sense of
$\mathcal{D}^{\prime}(\Omega_{N})$.

###### Proof.

Since $u\in H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$, Cor. 13 implies that
$u\in H_{\text{loc}}^{1}(\Omega_{N})$. Moreover, (2.21) implies that

$\forall i,j=1,...,d:\,a_{ij}D_{w}^{i}u,\,b_{i}D_{w}^{i}u,\,cu,\,f\in
L_{\text{loc}}^{2}(\Omega_{N})\subseteq\mathcal{D}^{\prime}(\Omega_{N}).$

For notational simplicity, we write $\langle\cdot,\cdot\rangle$ instead of
$\langle\cdot,\cdot\rangle_{\mathcal{D}^{\prime}(\Omega_{N}),\mathcal{D}(\Omega_{N})}$.
Hence, for any $\varphi\in\mathcal{D}(\Omega_{N})$ and for any $i$,
$j=1,...,d$ we have

$\int_{\Omega}a_{ij}D_{w}^{i}u\partial_{j}\varphi\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}=\langle
a_{ij}D_{w}^{i}u,\partial_{j}\varphi\rangle=-\langle
D_{j}(a_{ij}D_{i}u),\varphi\rangle$

where we used the fact that $D_{w}^{i}u$ coincides with the weak derivative in
the sense of $L_{\text{loc}}^{1}(\Omega_{N})$. Moreover, we have
$(f,\varphi)_{L^{2}}=\langle f,\varphi\rangle$. It follows that

$B_{w}(u,\varphi)=\left\langle-\sum_{i,j=1}^{d}D_{j}(a_{ij}D_{i}u)+\left(b\cdot
D\right)u+cu,\varphi\right\rangle=\langle f,\varphi\rangle$

which implies that

$-\sum_{i,j=1}^{d}D_{j}(a_{ij}D_{i}u)+\left(b\cdot
D\right)u+cu=f,\,\,\,\text{in }\mathcal{D}^{\prime}(\Omega_{N})$

and hence the proof is completed. ∎

In the next two theorems, we will give sufficient assumptions that guarantee
the existence of a unique solution to the boundary-value problem (2.18).

######  Theorem 28.

Let $\Omega\subseteq\mathbb{R}^{d}$ be a bounded open set with a sufficiently
smooth boundary, and $w$, $(a_{ij})$, $b=(b_{i})$, $c$, $f$ be such that

  1. (i)

$w\in\mathcal{C}^{1}(\Omega)\cap W(\Omega)$ and (2.7) holds

  2. (ii)

$f\in L_{w^{-1}}^{2}(\Omega)$, $a_{ij}\in L_{w^{-4}}^{\infty}(\Omega)$,
$b_{i}\in L_{w^{-3}}^{\infty}(\Omega)$, $c\in L_{w^{-2}}^{\infty}(\Omega)$ for
all $i,j=1,...,d$

  3. (iii)

$\exists\mu>0\,\forall\xi:=(\xi_{1},...,\xi_{d})\in\mathbb{R}^{d}:\sum_{i,j=1}^{d}a_{ij}(x)\xi_{i}\xi_{j}\geq\mu
w^{4}(x)|\xi|^{2}$ a.e. in $\Omega$

  4. (iv)

$\exists\sigma>0:\,c(x)\geq\sigma w^{2}(x)$ a.e. in $\Omega$ and
$d^{\frac{1}{2}}||b||_{L_{w^{-3}}^{\infty}}<2\sqrt{\mu\sigma}$, where
$||b||_{L_{w^{-3}}^{\infty}}:=\max_{i}||b_{i}||_{L_{w^{-3}}^{\infty}}$.

Then, the boundary-value problem (2.18) has a unique weak solution $u\in
H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$. In addition, we have the estimate

$\exists\gamma>0:\,||u||_{H_{\\{w^{|\alpha|+1}\\}}^{1}}\leq\frac{1}{\gamma}||f||_{L_{w^{-1}}^{2}}.$
(2.22)

Moreover, the solution satisfies the differential equation of (2.18) in the
sense of $\mathcal{D}^{\prime}(\Omega_{N})$.

Note that under the assumption (i), the trace operator is defined as in
(2.13).

###### Proof.

We check that the bilinear form $B_{w}$ defined in (2.20) satisfies the
assumptions of the Lax-Milgram theorem in the Hilbert space
$H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$.

By (ii), for any $u$, $v\in H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$ we have
that

$\left|B_{w}(u,v)\right|\leq
C\sum_{i,j=1}^{d}||w^{2}D_{w}^{i}u||_{L^{2}}||w^{2}D_{w}^{j}v||_{L^{2}}+C\sum_{i=1}^{d}||w^{2}D_{w}^{i}u||_{L^{2}}||wv||_{L^{2}(\Omega)}\\\
+C||wu||_{L^{2}}||wv||_{L^{2}(\Omega)}\leq
C||u||_{H_{\\{w^{|\alpha|+1}\\}}^{1}}||v||_{H_{\\{w^{|\alpha|+1}\\}}^{1}}$

for some non-negative constant $C$, which proves the continuity of the
bilinear form $B_{w}$.

For the coercivity, using (iii), (iv) and Hölder’s inequality we get

$\mu||w^{2}D_{w}u||_{L^{2}}^{2}+\sigma||wu||_{L^{2}}^{2}-d^{\frac{1}{2}}||b||_{L_{w^{-3}}^{\infty}}||w^{2}D_{w}u||_{L^{2}}||wu||_{L^{2}}\leq
B_{w}(u,u)$ (2.23)

Thanks to (iv), we can choose $\gamma$ sufficiently small so that
$0<\gamma\leq\min(\mu,\sigma)$ and
$d^{\frac{1}{2}}||b||_{L_{w^{-3}}^{\infty}}<2\sqrt{(\mu-\gamma)(\sigma-\gamma)}$
and hence the left hand side of (2.23) is bounded from bellow by
$\gamma(||w^{2}D_{w}u||_{L^{2}}^{2}+||wu||_{L^{2}}^{2})$ which proves the
coercivity of the bilinear form $B_{w}$. On the other hand, (ii) yields

$|(f,v)_{L^{2}}|=|(w^{-1}f,wv)_{L^{2}}|\leq||w^{-1}f||_{L^{2}}||wv||_{L^{2}}\leq||f||_{L_{w^{-1}}^{2}}||v||_{H_{\\{w^{|\alpha|+1}\\}}^{1}}$
(2.24)

for any $v\in H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$. Therefore, we can
conclude by the Lax-Milgram theorem.

For the estimate (2.22), it follows easily from (2.19), (2.24) and the
coercivity of the bilinear form $B_{w}$. Indeed, if we denote by $u$ the
unique weak solution of (2.18), we get

$\gamma||u||_{H_{\\{w^{|\alpha|+1}\\}}^{1}}^{2}\leq
B_{w}(u,u)=(f,u)\leq||f||_{L_{w^{-1}}^{2}}||u||_{H_{\\{w^{|\alpha|+1}\\}}^{1}}.$

The last part of the theorem follows from Prop. 27. It suffices to see that
(ii) implies (2.21) because $w\in\mathcal{C}^{1}(\Omega)$. ∎

######  Theorem 29.

Let $\Omega\subseteq\mathbb{R}^{d}$ ($d\geq 3$) be a bounded open set with a
sufficiently smooth boundary, $w$, $(a_{ij})$, $b=(b_{i})$, $c$, $f$ be such
that

  1. (i)

$w\in\mathcal{C}^{1}(\Omega)\cap W(\Omega)$ with $\nabla w\in
L^{\infty}(\Omega)$ and (2.15), (2.16) hold with $p=2$

  2. (ii)

$f\in L_{w^{-1}}^{2}(\Omega)$, $a_{ij}\in L_{w^{-4}}^{\infty}(\Omega)$,
$b_{i}\in L_{w^{-4}}^{\infty}(\Omega)$, $c\in L_{w^{-2}}^{\infty}(\Omega)$ for
all $i,j=1,...,d$

  3. (iii)

$\exists\mu>0\,\forall\xi:=(\xi_{1},...,\xi_{d})\in\mathbb{R}^{d}:\sum_{i,j=1}^{d}a_{ij}(x)\xi_{i}\xi_{j}\geq\mu
w^{4}(x)|\xi|^{2}$ a.e. in $\Omega$

  4. (iv)

$\exists\sigma>0:\,c(x)\geq w^{2}(x)\sigma$ a.e. in $\Omega$, and

$0<\mu-C_{\Omega}d^{\frac{1}{2}}||b||_{L_{w^{-4}}^{\infty}}$ (2.25)

Then, the boundary-value problem (2.18) has a unique weak solution $u\in
H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$.

Moreover, the solution satisfies (2.22) and the differential equation of
(2.18) in the sense of $\mathcal{D}^{\prime}(\Omega_{N})$.

###### Proof.

The boundedness of the bilinear form $B_{w}$ can be proved as in the proof of
the previous theorem, and we use the fact that $w$ is bounded (because
$\Omega$ is bounded and $\nabla w\in L^{\infty}(\Omega)$).

For the coercivity, using (iii), (iv) and Hölder’s inequality we get

$\mu||w^{2}D_{w}u||_{L^{2}}^{2}+\sigma||wu||_{L^{2}}^{2}-d^{\frac{1}{2}}||b||_{L_{w^{-4}}^{\infty}}||w^{2}D_{w}u||_{L^{2}}||w^{2}u||_{L^{2}}\leq
B_{w}(u,u)$

Now we use Cor. 26 in the previous inequality to find that

$(\mu-
C_{\Omega}d^{\frac{1}{2}}||b||_{L_{w^{-4}}^{\infty}})||w^{2}D_{w}u||_{L^{2}}^{2}+\sigma||wu||_{L^{2}}^{2}\leq
B_{w}(u,u)$

and then choose $\gamma:=\min((\mu-
C_{\Omega}d^{\frac{1}{2}}||b||_{L_{w^{-4}}^{\infty}}),\sigma)$ which is non-
negative thanks to (iv). Thus, the bilinear form $B_{w}$ is coercive. On the
other hand, (2.24) holds. Therefore, by the Lax-Milgram theorem, there exists
a unique weak solution $u\in H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$ of the
boundary-value problem (2.18).

The remaining parts were already proved in the previous theorem. ∎

######  Example 30.

Let $(\tilde{a}_{ij})$, $(\tilde{b})=(\tilde{b}_{i})$, $\tilde{c}$ and
$\tilde{f}$ be such that

  1. (i)

$\forall i,j=1,...,d:\,\tilde{a}_{ij},\,\tilde{b}_{i},\,\tilde{c}\in
L^{\infty}(\Omega)$ and $\tilde{f}\in L^{2}(\Omega)$

  2. (ii)

$\exists\mu>0$ such that
$\sum_{i,j}\tilde{a}_{ij}\xi_{i}\xi_{j}\geq\mu|\xi|^{2}$ a.e. in $\Omega$ and
for all $\xi\in\mathbb{R}^{d}$

  3. (iii)

$\exists\sigma>0$ such that $\tilde{c}\geq\sigma$ a.e. in $\Omega$, and
$d^{\frac{1}{2}}||\tilde{b}||_{L^{\infty}(\Omega)}<2\sqrt{\mu\sigma}$.

Then, for a fixed $w$ satisfying assumption (i) of Thm. 28, the functions
$(a_{ij})$, $(b)=(b_{i})$, $c$ and $f$ defined by

$\forall
i,j=1,...,d:\,a_{ij}=w^{4}\tilde{a}_{ij},\,b_{i}=w^{\text{3}}\tilde{b}_{i},\,c=w^{\text{2}}\tilde{c},\,f=w\tilde{f}$

satisfy the assumptions (ii)-(iv) of Thm. 28.

Using the same argument, we can give examples of functions $(a_{ij})$,
$(b)=(b_{i})$, $c$ and $f$ satisfying the assumptions (ii)-(iv) of Thm. 29.

In the next proposition we give sufficient conditions under which the solution
is not locally integrable.

######  Proposition 31.

Let $\Omega\subseteq\mathbb{R}^{d}$ be a bounded open set with sufficiently
smooth boundary, and let $w\in W(\Omega)\cap\mathcal{C}^{2}(\Omega)$ be such
that (2.7) holds. Let $u\in H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$ be such
that (2.19) holds where $(a_{ij})$, $(b_{i})$, $c$ and $f$ are as in (ii) of
Thm. 28. Assume that the following assertions hold

  1. (i)

$\forall i,j=1,...,d:\,a_{ij}\in\mathcal{C}^{1}(\Omega)$ and
$\forall\alpha\in\pi_{1}^{0}:\,\partial^{\alpha}a_{ij}\in
L_{w^{-3},\mathrm{loc}}^{\infty}(\Omega)$

  2. (ii)

$\forall i=1,...,d:\,b_{i}\in\mathcal{C}^{1}(\Omega)$ and
$\forall\alpha\in\pi_{1}^{0}:\,\partial^{\alpha}b_{i}\in
L_{w^{-2},\mathrm{loc}}^{\infty}(\Omega)$

  3. (iii)

$f\geq 0$ $\mathrm{a.e.}$ in $\Omega$ and $fw^{-2}\not\in
L_{\mathrm{loc}}^{1}(\Omega)$

Then, $u\not\in L_{\mathrm{loc}}^{1}(\Omega)$.

###### Proof.

Use (2.19) with $v=\varphi\chi_{n}w^{-2}\in\mathcal{C}_{c}^{2}(\Omega)$
(because supp$(\chi_{n})\subseteq M_{2n}^{c}$) where
$\varphi\in\mathcal{D}(\Omega)$ and $(\chi_{n})$ is the sequence of
$\mathcal{C}^{2}(\Omega)$ given in the proof of Thm. 6, and recall that
$D_{w}^{j}(\varphi\chi_{n}w^{-2})$ coincides with
$\partial_{j}(\varphi\chi_{n}w^{-2})$, we get

$\sum_{i,j=1}^{d}\int_{\Omega}a_{ij}D_{w}^{i}u\partial_{j}(\varphi\chi_{n}w^{-2})\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}+\sum_{i=1}^{d}\int_{\Omega}b_{i}\varphi\chi_{n}w^{-2}D_{w}^{i}u\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}\\\
+\int_{\Omega}cu\varphi\chi_{n}w^{-2}\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=\int_{\Omega}f\varphi\chi_{n}w^{-2}\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$ (2.26)

Using the weak derivative in the sense of $L_{w,\mathrm{loc}}^{1}(\Omega)$
with
$a_{ij}w^{-2}\partial_{j}(\varphi\chi_{n}w^{-2})\in\mathcal{C}_{c}^{1}(\Omega)$
we get

$\int_{\Omega}a_{ij}D_{w}^{i}u\partial_{j}(\varphi\chi_{n}w^{-2})\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=-\int_{\Omega}u\partial_{i}(a_{ij}\partial_{j}(\varphi\chi_{n}w^{-2}))\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$ (2.27)

Similarly, using the weak derivative in the sense of
$L_{w,\mathrm{loc}}^{1}(\Omega)$ with the function
$b_{i}\varphi\chi_{n}w^{-4}\in\mathcal{C}_{c}^{1}(\Omega)$ we get

$\int_{\Omega}b_{i}\varphi\chi_{n}w^{-2}D_{w}^{i}u\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=-\int_{\Omega}u\partial_{i}(b_{i}\varphi\chi_{n}w^{-2})\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}$ (2.28)

Using (2.27) and (2.28) in (2.26) we get

$-\sum_{i,j=1}^{d}\int_{\Omega}u(\partial_{i}(a_{ij}\partial_{j}(\varphi\chi_{n}w^{-2})\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}-\sum_{i=1}^{d}\int_{\Omega}u\partial_{i}(b_{i}\varphi\chi_{n}w^{-2})\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}\\\
+\int_{\Omega}cu\varphi\chi_{n}w^{-2}\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=\int_{\Omega}f\varphi\chi_{n}w^{-2}\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}.$ (2.29)

We now show that if $u\in L_{\mathrm{loc}}^{1}(\Omega)$ then the left hand
side of the latter equation is bounded whereas the right hand side is
unbounded, which leads to a contradiction. Indeed, assume by contradiction
that $u\in L_{\mathrm{loc}}^{1}(\Omega)$. We have

$\partial_{i}(a_{ij}\partial_{j}(\varphi\chi_{n}w^{-2})=\chi_{n}(\partial_{i}a_{ij}\partial_{j}(w^{-2}\varphi)+a_{ij}\partial_{ij}(w^{-2}\varphi))\\\
+\partial_{j}\chi_{n}(w^{-2}\varphi\partial_{i}a_{ij}+a_{ij}\partial_{i}(w^{-2}\varphi))+\partial_{i}\chi_{n}(a_{ij}\partial_{j}(w^{-2}\varphi)+a_{ij}w^{-2}\varphi\partial_{ij}\chi_{n}$
(2.30)

Developing the expression
$\partial_{i}a_{ij}\partial_{j}(w^{-2}\varphi)+a_{ij}\partial_{ij}(w^{-2}\varphi)$
we get

$\partial_{i}a_{ij}(-2(\partial_{j}w)w^{-3}\varphi+w^{-2}\partial_{j}\varphi)+\\\
a_{ij}(w^{-2}\partial_{ij}\varphi-2(\partial_{j}w)w^{-3}\partial_{i}\varphi-2(\partial_{i}w)w^{-3}\partial_{j}\varphi-2(\partial_{ij}w)w^{-3}\varphi+6(\partial_{i}w)(\partial_{j}w)w^{-4}\varphi$

By assumption (i), and assumption (ii) of Thm. 28 we have

$\exists C>0\,\forall
n\in\mathbb{N}:\,|\chi_{n}(\partial_{i}a_{ij}\partial_{j}(w^{-2}\varphi)+a_{ij}\partial_{ij}(w^{-2}\varphi))|\leq
C1_{V}$ (2.31)

where $1_{V}$ is the characteristic function of the set
$V:=\mathrm{supp}(\varphi)$. Note now that, for all $\alpha\in\pi_{2}^{0}$ we
have $\partial^{\alpha}\chi_{n}\equiv 0$ on $M_{n}^{c}$ (because by
$\chi_{n}\equiv 1$ on $M_{n}^{c}$), and that $|\partial^{\alpha}\chi_{n}|\leq
Cn^{|\alpha|}$ on $M_{n}$ for all $n$ sufficiently large, where $C>0$ is
independent of $n$. This is because $M_{n}\Subset\Omega$ (by assumption (2.7))
(and hence $\partial^{\alpha}w$ is bounded on $M_{n}$), and because
$|\eta_{n}^{(k)}|\leq Cn^{|k|}$ where the constant $C>0$ depends on $k$ but
not on $n$. It follows that

$\exists C>0\,\exists n_{0}\in\mathbb{N}\,\forall n\geq
n_{0}\,\forall\alpha\in\pi_{2}^{0}:\,|w^{|\alpha|}\partial^{\alpha}\chi_{n}|\leq
C$ (2.32)

since $|w|\leq 1/n$ on $M_{n}$. Thus, one can easily prove (as above), using
assumption (i), assumption (ii) of Thm. 28, and (2.32) that

$\left|\partial_{j}\chi_{n}(w^{-2}\varphi\partial_{i}a_{ij}+a_{ij}\partial_{i}(w^{-2}\varphi))+\partial_{i}\chi_{n}(a_{ij}\partial_{j}(w^{-2}\varphi)+a_{ij}w^{-2}\varphi\partial_{ij}\chi_{n}\leq\right|C1_{V}.$
(2.33)

for all $n\geq n_{0}$. It follows from (2.30), (2.31) and (2.33) that

$\forall n\geq
n_{0}:\,\left|\sum_{i,j=1}^{d}\int_{\Omega}u(\partial_{i}(a_{ij}\partial_{j}(\varphi\chi_{n}w^{-2})))\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}\right|\leq C||u||_{L^{1}(V)}.$ (2.34)

Using similar arguments one can prove that

$\forall n\geq
n_{0}:\,\left|\sum_{i=1}^{d}\int_{\Omega}u\partial_{i}(b_{i}\varphi\chi_{n}w^{-2})+uc\varphi\chi_{n}w^{-2}\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}\right|\leq C||u||_{L^{1}(V)}$ (2.35)

We deduce from (2.29), (2.34), (2.35) that

$\exists C>0\,\forall n\geq
n_{0}:\,\int_{\Omega}f\varphi\chi_{n}w^{-2}\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}\leq C$

Let now $K\Subset\Omega$ be such that $fw^{-2}\not\in L^{1}(K)$ and choose
$\varphi$ so that $\varphi\geq 0$ on $\Omega$ and $\varphi\equiv 1$ on $K$. It
follows that $\int_{K}f\chi_{n}w^{-2}\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm d}x$}}\leq
C$ for all $n\geq n_{0}$, which leads to a contradiction when choosing $n$
sufficiently large since

$\lim_{n\to\infty}\int_{K}f\chi_{n}w^{-2}\,\mathchoice{\hbox{\rm
d}x}{\hbox{\rm d}x}{\scalebox{0.75}{$\hbox{\rm
d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=\int_{K}fw^{-2}\,\mathchoice{\hbox{\rm d}x}{\hbox{\rm
d}x}{\scalebox{0.75}{$\hbox{\rm d}x$}}{\scalebox{0.35}{$\hbox{\rm
d}x$}}=\infty$

by the monotone convergence theorem. ∎

######  Remark 32.

If

$\forall i,j=1,...,d:\,a_{ij}=\tilde{a}_{ij}w^{4},\,b_{i}=\tilde{b}_{i}w^{3}$

where

$\forall
i,j=1,...,d:\,\tilde{a}_{ij}\in\mathcal{C}^{1}(\Omega),\,\tilde{b}_{i}\in\mathcal{C}^{1}(\Omega),$

then assumptions (i), (ii) of the latter proposition hold.

######  Example 33.

For any integer $m\geq 1$, and for any $\beta\in(d/2-2m,d/2)$, the boundary-
value problem

$\begin{cases}-\sum_{i=1}^{d}D_{i}(|x|^{8m}D_{i}u)+|x|^{4m}u=|x|^{-\beta+2m}&\text{in
}\\{x\in\mathbb{R}^{d}\mid|x|<1\\}\\\ u=0&\text{on
}\\{x\in\mathbb{R}^{d}\mid|x|=1\\}\end{cases}$

has a unique non-locally integrable solution $u\in
H_{\\{|x|^{2m(|\alpha|+1)}\\},0}^{1}$ thanks to Thm. 28 and to Prop. 31.

##  3\. Conclusion

The key idea of the paper is the notion of weak derivative in the sense of
$L_{w,\text{loc}}^{1}(\Omega)$ which extends the notion of the weak derivative
in the sense of $L_{\text{loc}}^{1}(\Omega)$. Using this notion, we have been
able to construct Sobolev-like spaces with non-locally integrable functions,
and solve a large class of degenerate elliptic PDE.

The next step in the direction of this paper would be to find sufficient
conditions that ensure the density of the space $\mathcal{D}(\Omega)$ in
$H_{\\{w^{|\alpha|+1}\\},0}^{1}(\Omega)$. This can be used in particular to
provide more sufficient conditions (when $b_{i}\in\mathcal{C}^{1}(\Omega)$ for
all $i$) that ensure the existence and uniqueness of a weak solution of the
boundary-value problem (2.18).

We will also investigate whether, for some given functions $(a_{ij})$,
$(b_{i})$, $c$, $f$, there exists an elimination function $w$ that satisfies
the assumptions of Thm. 28 or the assumptions of Thm. 29.

Moreover, we also plan to investigate the interior regularity of the solution,
that is, the belonging of the solution to $H_{\text{loc}}^{m}(\Omega_{N})$,
and the global regularity, that is, the existence of a second order weak
derivative in the sense of $L_{w,\text{loc}}^{1}(\Omega)$ of the solution, and
the belonging of the latter to some singular Sobolev space.

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