# Behaviour at infinity for solutions of a mixed boundary value problem via
inversion

Jana Björn and Abubakar Mwasa  Jana Björn
Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden
<EMAIL_ADDRESS>ORCID: 0000-0002-1238-6751

Abubakar Mwasa
Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden;
Department of Mathematics, Busitema University, P.O.Box 236, Tororo, Uganda
<EMAIL_ADDRESS><EMAIL_ADDRESS>ORCID: 0000-0003-4077-3115

# Behaviour at infinity for solutions of a mixed nonlinear elliptic boundary
value problem via inversion

Jana Björn and Abubakar Mwasa  Jana Björn
Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden
<EMAIL_ADDRESS>ORCID: 0000-0002-1238-6751

Abubakar Mwasa
Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden;
Department of Mathematics, Busitema University, P.O.Box 236, Tororo, Uganda
<EMAIL_ADDRESS><EMAIL_ADDRESS>ORCID: 0000-0003-4077-3115

###### Abstract

We study a mixed boundary value problem for the quasilinear elliptic equation
$\operatorname{div}{\mathcal{A}}(x,\nabla u(x))=0$ in an open infinite
circular half-cylinder with prescribed continuous Dirichlet data on a part of
the boundary and zero conormal derivative on the rest. We prove the existence
and uniqueness of bounded weak solutions to the mixed problem and characterize
the regularity of the point at infinity in terms of
$p\mspace{1.0mu}$-capacities. For solutions with only Neumann data near the
point at infinity we show that they behave in exactly one of three possible
ways, similar to the alternatives in the Phragmén–Lindelöf principle.

_Key words and phrases_ : continuous Dirichlet data, existence and uniqueness
of solutions, mixed boundary value problem, Phragmén–Lindelöf trichotomy,
quasilinear elliptic equation, regularity at infinity, Wiener criterion.

Mathematics Subject Classification (2020): Primary: 35J25. Secondary: 35J62,
35B40.

##  1 Introduction

The _Dirichlet problem_ on a nonempty open set $\Omega\subset\mathbf{R}^{n}$
entails finding a function $u$ which solves a certain partial differential
equation in $\Omega$ with the prescribed boundary data $u=f$ on
$\partial\Omega$. When the boundary data $g:\partial\Omega\to\mathbf{R}$ are
taken as the normal derivative $\partial u/\partial\nu=g$, the problem is
called a _Neumann problem_. More general directional derivatives, such as the
conormal derivative $Nu$ considered below, are also possible. In this paper,
we study a _mixed boundary value problem_ for the quasilinear equation

$\operatorname{div}{\mathcal{A}}(x,\nabla u(x))=0,$ (1.1)

where a Dirichlet condition is prescribed on a part of the boundary, while the
rest carries the zero conormal derivative

$Nu(x):={\mathcal{A}}(x,\nabla u(x))\cdot\nu(x)=0,$

where $\nu(x)$ is the unit outer normal at $x$. Equation (1.1) is considered
in an infinite circular half-cylinder and the vector-valued function
${\mathcal{A}}$ satisfies the standard ellipticity conditions with a parameter
$p>1$. The $p\mspace{1.0mu}$-Laplace equation $\operatorname{div}(|\nabla
u|^{p-2}\nabla u)=0$ is included as a special case.

We prove the existence and uniqueness of bounded continuous weak solutions to
the above mixed boundary value problem with continuous Dirichlet boundary data
$f$ on a closed set $F\subset\partial G$, which can be bounded or unbounded.
More precisely, if $f\in C(F)$ is such that (for unbounded $F$)

$f(\infty):=\lim_{F\ni x\to\infty}f(x)\quad\text{exists and is finite,}$ (1.2)

then there exists a unique bounded continuous weak solution $u$ of (1.1) with
zero conormal derivative on $\partial G\setminus F$ and such that

$\lim_{x\to x_{0}}u(x)=f(x_{0})\quad\text{for all $x_{0}\in F$ outside a set
of $C_{p}$-capacity zero,}$

see Theorem 5.3 and Remark 5.4. Moreover, $u$ is Hölder continuous at all
points in the Neumann boundary $\partial G\setminus F$. For Dirichlet data of
Sobolev type, existence and uniqueness in the Sobolev sense are proved in
Theorem 5.6.

We also characterize the regularity of the point at infinity by a _Wiener type
criterion_. Roughly speaking, if the Dirichlet boundary $F$ is sufficiently
thick at $\infty$ in terms of a variational $p\mspace{1.0mu}$-capacity adapted
to the mixed problem, then for every $f\in C(F)$ satisfying (1.2), the unique
solution of the above mixed problem satisfies

$\lim_{x\to\infty}u(x)=f(\infty).$ (1.3)

Conversely, thickness of $F$ at $\infty$ is also necessary in order for (1.3)
to hold for all $f\in C(F)$, see Theorem 6.5. On the other hand, if $F$ is
bounded, i.e. when only the Neumann condition is used in a proximity of the
point at infinity, then we show in Theorem 7.4 that each solution $u$ of
equation (1.1) with zero conormal derivative near $\infty$ behaves in one of
the following three ways as $x\to\infty$:

  1. (i)

The solution has a finite limit $\displaystyle
u(\infty):=\lim_{x\to\infty}u(x)$.

  2. (ii)

The solution tends roughly linearly to either $\infty$ or $-\infty$.

  3. (iii)

The solution changes sign and approaches both $\infty$ and $-\infty$, i.e.

$\limsup_{x\to\infty}u(x)=\infty\quad\text{and}\quad\liminf_{x\to\infty}u(x)=-\infty.$

Similar trichotomy results at $\infty$ for solutions of the Neumann problem
for the linear uniformly elliptic equation $\operatorname{div}(A(x)\nabla
u)=0$ were obtained in Ibragimov–Landis [16], [17], Lakhturov [24] and
Landis–Panasenko [25]. For (sub/super)solutions of various PDEs in unbounded
domains with only Dirichlet data, similar alternative behaviour is often
referred to as Phragmén–Lindelöf principle and has been extensively studied.
See e.g. Gilbarg [9], Hopf [11], Horgan–Payne [12], [13], Jin–Lancaster [19],
[20], Lindqvist [26], Lundström [27], Quintanilla [30], Serrin [31] and Vitolo
[32].

Compared with pure Dirichlet and Neumann problems, the literature on mixed
boundary value problems is less extensive, especially in unbounded domains.
Mixed problems are sometimes called _Zaremba problems_ , mainly in the Russian
literature, since they were first considered for the Laplace equation $\Delta
u=0$ by Zaremba [35] in 1910. For linear uniformly elliptic equations of the
type $\operatorname{div}(A(x)\nabla u)=0$, they were studied by e.g. Ibragimov
[14], [15], Kerimov [21], [22] and Novruzov [29]. A mixed problem for linear
equations of nondivergence type was considered in Cao–Ibragimov–Nazarov [5]
and Ibragimov–Nazarov [18].

Existence of weak solutions for mixed and Neumann problems for linear
operators in very general unbounded domains was recently obtained using an
exhaustion with bounded domains by Chipot [6] and Chipot–Zube [7]. Wiśniewski
[33], [34] studied the decay at infinity of solutions to mixed problems with
coefficients approaching the Laplace operator in unbounded conical domains. On
the other hand, nonexistence Liouville type results for mixed problems of the
form $-\Delta u=f(u)$ in various unbounded domains were obtained in
Damascelli–Gladiali [8]. A priori estimates, unique solubility and traces for
linear elliptic systems of divergence type in Ahlfors regular cylindrical
domains with Dirichlet and Neumann data independent of the last variable were
obtained in Auscher–Egert [1] by the so-called DB-approach adapted from the
upper half-space. All the above papers deal with linear operators. We point
out that even for $p=2$, equation (1.1) considered here can be nonlinear. This
happens for example when ${\mathcal{A}}(x,q):=a(q/|q|)q$, where $a$ is a
sufficiently smooth strictly positive scalar function on the unit sphere; see
Example 2.3.

In Kerimov–Maz′ya–Novruzov [23], the regularity of the point at infinity for
the Zaremba problem for the Laplace equation $\Delta u=0$ in an infinite half-
cylinder was characterized by means of a Wiener type criterion. A similar
problem for certain weighted linear elliptic equations was studied in Björn
[3] and more recently for the $p\mspace{1.0mu}$-Laplace equation in
Björn–Mwasa [4]. The results in this paper partially extend the ones in [23],
[3] and [4] to general quasilinear elliptic equations of the form (1.1), but
we also address other properties of the solutions.

In order to achieve our results, we make use of the change of variables
introduced in Björn [3], and later adopted by Björn–Mwasa [4], to transform
the infinite half-cylinder $G$ and the quasilinear elliptic equation (1.1)
into a unit half-ball and a degenerate elliptic equation

$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u}(\xi))=0,$ (1.4)

with the $A_{p}$-weight $w(\xi)=|\xi|^{p-n}$, see Section 3.

After the above transformation, we are able to eliminate the Neumann data by
reflecting the unit half-ball and equation (1.4) to the whole unit ball,
leaving only the Dirichlet data. This is done in Section 4. In Section 5, we
then use tools for Dirichlet problems from Heinonen–Kilpeläinen–Martio [10]
and Björn–Björn–Mwasa [2] to prove the existence and uniqueness of continuous
weak solutions to the mixed boundary value problem for (1.1) in the cylinder,
with zero conormal derivative and continuous Dirichlet data.

In Section 6, we show that regularity at infinity for the mixed problem for
(1.1) is equivalent to the regularity of the origin for the Dirichlet problem
for (1.4). This can in turn be characterized in terms of weighted
$p\mspace{1.0mu}$-capacities using a Wiener criterion, provided by [10,
Theorem 21.30] and Mikkonen [28]. This Wiener criterion for (1.4) is then
transferred back to the cylinder to characterize the regularity at infinity
for (1.1) by means of a variational $p\mspace{1.0mu}$-capacity adapted to the
cylinder.

Finally, in Section 7, we use estimates from [10, Sections 6 and 7] for
capacitary potentials and singular solutions of (1.4) to prove the above
Phragmén–Lindelöf type trichotomy (i)–(iii).

Acknowledgement. J. B. was partially supported by the Swedish Research Council
grant 2018-04106. A. M. was supported by the SIDA (Swedish International
Development Cooperation Agency) project 316-2014 “Capacity building in
Mathematics and its applications” under the SIDA bilateral program with the
Makerere University 2015–2020, contribution No. 51180060.

##  2 Notation and preliminaries

Let $G=B^{\prime}\times(0,\infty)\subset\mathbf{R}^{n}$ be an open infinite
circular half-cylinder, where

$B^{\prime}=\\{x^{\prime}\in\mathbf{R}^{n-1}:|x^{\prime}|<1\\}$

is the unit ball in $\mathbf{R}^{n-1}$. Points in
$\mathbf{R}^{n}=\mathbf{R}^{n-1}\times\mathbf{R}$, $n\geq 2$, are denoted by
$x=(x^{\prime},x_{n})=(x_{1},\mathinner{.}\nonscript\\!.\nonscript\\!\mathinner{.},x_{n-1},x_{n})$.
Let $F$ be a closed subset of the closure ${\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}$ of $G$.
Assume that $F$ contains the base $B^{\prime}\times\\{0\\}$ of $G$. We
consider the following mixed boundary value problem

$\begin{cases}\operatorname{div}{\mathcal{A}}(x,\nabla u)=0,&\text{in
}G\setminus F,\\\ u=f,&\text{on }F_{0}:=F\cap\partial(G\setminus F),\text{
(Dirichlet data)},\\\ Nu:={\mathcal{A}}(x,\nabla u)\cdot\nu=0&\text{on
}\partial G\setminus F,\text{ (generalized Neumann data)},\end{cases}$ (2.1)

where $Nu$ is the conormal derivative, $\nu$ is the unit outer normal of $G$
and $\partial G$ denotes the boundary of $G$.

Let $1<p<\infty$ be fixed. The mapping ${\mathcal{A}}:{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\times\mathbf{R}^{n}\to\mathbf{R}^{n}$ in (2.1) is assumed to
satisfy the standard ellipticity and boundedness conditions:

  * •

${\mathcal{A}}(\cdot,q)$ is measurable for all $q\in\mathbf{R}^{n}$,

  * •

${\mathcal{A}}(x,\cdot)$ is continuous for a.e. $x\in{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}$,

  * •

There are constants $0<\alpha_{1}\leq\alpha_{2}<\infty$ such that for all
$q,q_{1},q_{2}\in\mathbf{R}^{n}$, $0\neq\lambda\in\mathbf{R}$ and a.e.
$x\in{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}$,

$\displaystyle{\mathcal{A}}(x,q)\cdot q$ $\displaystyle\geq\alpha_{1}|q|^{p},$
(2.2) $\displaystyle|{\mathcal{A}}(x,q)|$
$\displaystyle\leq\alpha_{2}|q|^{p-1},$ (2.3)
$\displaystyle{\mathcal{A}}(x,\lambda q)$
$\displaystyle=\lambda|\lambda|^{p-2}{\mathcal{A}}(x,q),$ (2.4)

and

$({\mathcal{A}}(x,q_{1})-{\mathcal{A}}(x,q_{2}))\cdot(q_{1}-q_{2})>0\quad\text{when
}q_{1}\neq q_{2}.$ (2.5)

The quasilinear elliptic equation $\operatorname{div}{\mathcal{A}}(x,\nabla
u)=0$ and the conormal derivative in (2.1) will be considered in the weak
sense as follows.

######  Definition 2.1.

A function

$u\in W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F):=\\{u|_{{\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}\setminus
F}:u\in W^{1,p}_{\rm loc}(\mathbf{R}^{n}\setminus F)\\},$

is a _weak solution_ of the equation $\operatorname{div}{\mathcal{A}}(x,\nabla
u)=0$ in $G\setminus F$ with _zero conormal derivative_ on $\partial
G\setminus F$ if the integral identity

$\int_{G\setminus F}{\mathcal{A}}(x,\nabla u)\cdot\nabla\varphi\,dx=0$ (2.6)

holds for all $\varphi\in C_{0}^{\infty}(\mathbf{R}^{n}\setminus F)$, where
$\cdot$ denotes the scalar product in $\mathbf{R}^{n}$ and
$C_{0}^{\infty}(\Omega)$ is the space of all infinitely many times
continuously differentiable functions with compact support in
$\Omega\subset\mathbf{R}^{n}$.

######  Remark 2.2.

If the whole boundary $\partial G$ is contained in $F$, then there is no
Neumann condition and the mixed boundary value problem reduces to a purely
Dirichlet problem.

As usual, the local space $W^{1,p}_{\rm loc}(\Omega)$ (and later $H^{1,p}_{\rm
loc}(\Omega,w)$) consists of those functions $u$ which belong to
$W^{1,p}(\Omega^{\prime})$ (resp. $H^{1,p}(\Omega^{\prime},w)$) for all
$\Omega^{\prime}\Subset\Omega$, where $\Omega^{\prime}\Subset\Omega$ means
that $\overline{\Omega^{\prime}}$ is a compact subset of $\Omega$.

As mentioned in the introduction, equation (1.1) can be nonlinear even for
$p=2$, as illustrated by the following example.

######  Example 2.3.

For $p=2$ and $x,q\in\mathbf{R}^{n}$, let

${\mathcal{A}}(x,q)=\begin{cases}0&\text{if }q=0,\\\
a\bigl{(}\frac{q}{|q|}\bigr{)}q&\text{if }q\neq 0,\end{cases}$ (2.7)

where the scalar function $a$ is strictly positive and continuous on the unit
sphere $\partial B(0,1)$ in $\mathbf{R}^{n}$ and such that

$\frac{a(\theta^{\prime})}{a(\theta)}>\frac{1-\sin\alpha}{1+\sin\alpha}\quad\text{for
any $\theta,\theta^{\prime}\in\partial B(0,1)$ with
$\theta\cdot\theta^{\prime}=\cos\alpha>0$,}$ (2.8)

where $\alpha=\alpha(\theta,\theta^{\prime})$ is the acute angle between
$\theta$ and $\theta^{\prime}$. Then ${\mathcal{A}}$ satisfies the ellipticity
conditions (2.2)–(2.5).

Indeed, the only nontrivial verification is that of (2.5). For this, we can
clearly assume that $|q|=1$ and

$0<a(\theta^{\prime})\leq a(\theta),\quad\text{where }\theta=q\text{ and
}\theta^{\prime}=\frac{q^{\prime}}{|q^{\prime}|}.$

Condition (2.5) then means that the angle between the vectors $q^{\prime}-q$
and $\frac{a(\theta^{\prime})}{a(\theta)}q^{\prime}-q$ is strictly less than
$\frac{\pi}{2}$. Since $a(\theta^{\prime})/a(\theta)\leq 1$, a geometric
consideration shows that this is clearly satisfied if
$\alpha\geq\frac{\pi}{2}$ or if

$\alpha<\frac{\pi}{2}\text{ and }|q^{\prime}|\leq\frac{1}{\cos\alpha}.$

So assume that $\alpha<\frac{\pi}{2}$ and $|q^{\prime}|>1/\cos\alpha$. Aplying
the cosine theorem to the triangles spanned by the vectors $q$ and
$q^{\prime}$, or by

$q\text{ and
}q^{\prime\prime}:=\frac{a(\theta^{\prime})}{a(\theta)}q^{\prime},$

respectively, as well as to the triangle spanned by $q^{\prime}-q$ and
$q^{\prime\prime}-q$, we see that the angle between $q^{\prime}-q$ and
$q^{\prime\prime}-q$ is $<\frac{\pi}{2}$ if and only if

$|q|^{2}+|q^{\prime}|^{2}-2|q|\,|q^{\prime}|\cos\alpha+|q|^{2}+|q^{\prime\prime}|^{2}-2|q|\,|q^{\prime\prime}|\cos\alpha>|q^{\prime}-q^{\prime\prime}|^{2}.$

Since $|q|=1$, this is equivalent to

$1-\biggl{(}1+\frac{a(\theta^{\prime})}{a(\theta)}\biggr{)}|q^{\prime}|\cos\alpha+\frac{a(\theta^{\prime})}{a(\theta)}|q^{\prime}|^{2}>0,$

i.e.

$\frac{a(\theta^{\prime})}{a(\theta)}>\frac{|q^{\prime}|\cos\alpha-1}{|q^{\prime}|(|q^{\prime}|-\cos\alpha)}.$

Maximizing the right-hand side over $|q^{\prime}|>1/\cos\alpha$, we find that
its maximum is attained when

$|q^{\prime}|=\frac{1+\sin\alpha}{\cos\alpha}\quad\text{and equals
}\frac{1-\sin\alpha}{1+\sin\alpha}.$

This justifies the requirement (2.8).

A concrete example of a function satisfying (2.8) is $a(\theta)=e^{\theta\cdot
q_{0}}$, for any fixed vector $q_{0}\in\mathbf{R}^{n}$ with
$|q_{0}|<1/\sqrt{2}$. Indeed, with this choice,

$\frac{a(\theta^{\prime})}{a(\theta)}=e^{(\theta^{\prime}-\theta)\cdot
q_{0}}>1-\frac{|\theta^{\prime}-\theta|}{\sqrt{2}},$

while

$|\theta^{\prime}-\theta|^{2}=2-2\theta\cdot\theta^{\prime}=2-2\cos\alpha\leq
2\sin^{2}\alpha,$

from which (2.8) readily follows.

Clearly, a modification of (2.7) and (2.8) can be made so that
${\mathcal{A}}(x,q)$ and $a(x,\theta)$ also depend on $x\in\mathbf{R}^{n}$.

Throughout the paper, unless otherwise stated, $C$ will denote any positive
constant whose real value is not important and need not be the same at each
point of use. It can even vary within a line. By $a\lesssim b$, we mean that
there exists a positive constant $C$, independent of $a$ and $b$, such that
$a\leq Cb$. Similarly, $a\gtrsim b$ means $b\lesssim a$, while $a\simeq b$
stands for $a\lesssim b\lesssim a$.

##  3 Transformation of the half-cylinder

In this section, the quasilinear elliptic operator
$\operatorname{div}{\mathcal{A}}(x,\nabla u)$ in $G$ is shown to correspond to
a weighted quasilinear elliptic operator on the unit half-ball. We will use
the following change of variables introduced in Björn [3, Section 3].

Let $\kappa>0$ be a fixed constant. Define the mapping

$T:\mathbf{R}^{n}\longrightarrow
T(\mathbf{R}^{n})=\mathbf{R}^{n}\setminus\\{(\xi^{\prime},\xi_{n})\in\mathbf{R}^{n}:\xi^{\prime}=0\text{
and }\xi_{n}\leq 0\\}$

by $T(x^{\prime},x_{n})=(\xi^{\prime},\xi_{n})$, where

$\xi^{\prime}=\frac{2e^{-\kappa
x_{n}}x^{\prime}}{1+|x^{\prime}|^{2}}\quad\text{and}\quad\xi_{n}=\frac{e^{-\kappa
x_{n}}(1-|x^{\prime}|^{2})}{1+|x^{\prime}|^{2}}.$ (3.1)

We will use $x=(x^{\prime},x_{n})$ for points in ${\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}$ and
$\xi=(\xi^{\prime},\xi_{n})=(\xi_{1},\mathinner{.}\nonscript\\!.\nonscript\\!\mathinner{.},\xi_{n-1},\xi_{n})\in\mathbf{R}^{n}$
for points in the space transformed by $T$. Note that

$\displaystyle T(G)$ $\displaystyle=\\{\xi\in\mathbf{R}^{n}:|\xi|<1\text{ and
}\xi_{n}>0\\},$ $\displaystyle T({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}})$ $\displaystyle=\\{\xi\in\mathbf{R}^{n}:0<|\xi|\leq 1\text{
and }\xi_{n}\geq 0\\}$

are the open and the closed upper unit half-ball, respectively, with the
origin $\xi=0$ removed. From (3.1) it is easy to see that

$|\xi|=|T(x)|=e^{-\kappa x_{n}}\to 0\quad\text{as }x_{n}\to\infty,$

that is, the point at infinity for the half-cylinder $G$ corresponds to the
origin $\xi=0$ in $T(G)$.

The mapping $T$ is a smooth diffeomorphism between $\mathbf{R}^{n}$ and
$T(\mathbf{R}^{n})$, see Björn–Mwasa [4, Lemma 3.1]. A direct calculation
shows that the inverse mapping $T^{-1}$ of $T$ is given by

$x^{\prime}=\frac{\xi^{\prime}}{|\xi|+\xi_{n}}\quad\text{and}\quad
x_{n}=-\frac{1}{\kappa}\log|\xi|.$

In the following lemma we show how the operator
$\operatorname{div}{\mathcal{A}}(x,\nabla u)$ on the half-cylinder is
transformed under $T$ to the unit half-ball, cf. [4, Section 3].

######  Lemma 3.1.

Let $u,v\in W^{1,p}(\Omega)$ for some open set $\Omega\subset G$ and let
$\tilde{u}=u\circ T^{-1}$ and $\tilde{v}=v\circ T^{-1}$. Then for any
measurable set $A\subset\Omega$,

$\int_{A}{\mathcal{A}}(x,\nabla u)\cdot\nabla
v\,dx=\int_{T(A)}{\mathcal{B}}(\xi,\nabla\tilde{u})\cdot\nabla\tilde{v}\,d\xi,$
(3.2)

where ${\mathcal{B}}$ is for $\xi=Tx\in T({\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}})$ and
$q\in\mathbf{R}^{n}$ defined by

${\mathcal{B}}(\xi,q)=|J_{T}(x)|^{-1}dT(x){\mathcal{A}}(x,dT^{*}(x)q).$ (3.3)

Here, $J_{T}(x)=\det(dT(x))$ denotes the Jacobian of $T$ at $x$ and
$dT^{*}(x)$ is the transpose of the differential $dT(x)$ of $T$ at $x$, seen
as $(n\times n)$-matrices. In (3.3), both $q$ and
${\mathcal{A}}(x,dT^{*}(x)q)$ are regarded as column vectors for the matrix
multiplication to make sense.

  * Proof.

First, we rewrite the scalar product on the left-hand side of (3.2) using
matrix multiplication as

${\mathcal{A}}(x,\nabla u)\cdot\nabla v=(\nabla v)^{*}{\mathcal{A}}(x,\nabla
u),$

where both ${\mathcal{A}}(x,\nabla u)$ and $\nabla v$ are seen as column
vectors. Using the change of variables $\xi=T(x)$, together with the chain
rule

$\nabla u(x)=dT^{*}(x)\nabla\tilde{u}(\xi),\quad\text{where }\xi=T(x),$

we get

$\displaystyle\int_{A}(\nabla v)^{*}{\mathcal{A}}(x,\nabla u)\,dx$
$\displaystyle\quad\quad\quad=\int_{T(A)}(dT^{*}(x)\nabla\tilde{v})^{*}{\mathcal{A}}(x,dT^{*}(x)\nabla\tilde{u})|J_{T}(x)|^{-1}\,d\xi$
$\displaystyle\quad\quad\quad=\int_{T(A)}|J_{T}(x)|^{-1}dT(x){\mathcal{A}}(x,dT^{*}(x)\nabla\tilde{u})\cdot\nabla\tilde{v}\,d\xi$
$\displaystyle\quad\quad\quad=\int_{T(A)}{\mathcal{B}}(\xi,\nabla\tilde{u})\cdot\nabla\tilde{v}\,d\xi.\qed$

In view of the integral identity (2.6), Lemma 3.1 shows that the quasilinear
equation (1.1) on $G\setminus F$ will be transformed by $T$ into the equation

$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u})=0\quad\text{on
}T(G\setminus F),$ (3.4)

with a proper interpretation of the function spaces and the zero Neumann
condition.

To prove the fundamental properties of the transformed operator
${\mathcal{B}}(\xi,\nabla\tilde{u})$, we will need the following estimates
from Björn–Mwasa [4, Lemma 3.3].

######  Lemma 3.2.

There exist constants $C_{1},C_{2}>0$ such that if $x,y\in
B^{\prime}\times\mathbf{R}$ and $x_{n}\leq y_{n}$, then

$C_{1}e^{-\kappa y_{n}}|x-y|\leq|T(x)-T(y)|\leq C_{2}e^{-\kappa x_{n}}|x-y|.$

In particular, if $x\in B^{\prime}\times\mathbf{R}$ and $q\in\mathbf{R}^{n}$
then

$|dT^{*}(x)q|\simeq|dT(x)q|\simeq e^{-\kappa
x_{n}}|q|\quad\text{and}\quad|J_{T}(x)|\simeq e^{-\kappa nx_{n}},$

where the comparison constants in $\simeq$ depend on $\kappa$, but are
independent of $x$ and $q$.

##  4 Properties of $\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u})$ and
removing the Neumann data

In order to apply the theory of degenerate elliptic equations developed in
Heinonen–Kilpeläinen–Martio [10], we need to first show that the mapping
${\mathcal{B}}$ in (3.4) satisfies ellipticity assumptions similar to
(2.2)–(2.5).

######  Theorem 4.1.

The mapping ${\mathcal{B}}:T({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}})\times\mathbf{R}^{n}\to\mathbf{R}^{n}$, defined by (3.3),
satisfies for all $q\in\mathbf{R}^{n}$ and a.e. $\xi\in T({\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}})$ the
following ellipticity and boundedness conditions

${\mathcal{B}}(\xi,q)\cdot
q\gtrsim\widetilde{w}(\xi)|q|^{p}\quad\text{and}\quad|{\mathcal{B}}(\xi,q)|\lesssim\widetilde{w}(\xi)|q|^{p-1},$

where $\widetilde{w}(\xi)=|\xi|^{p-n}$ is a weight function and the comparison
constants in $\gtrsim$ and $\lesssim$ are independent of $\xi$ and $q$.

  * Proof.

From (3.3), we have that for all $q\in\mathbf{R}^{n}$ and a.e. $\xi=Tx\in
T({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}})$,

$\displaystyle{\mathcal{B}}(\xi,q)\cdot q$
$\displaystyle=|J_{T}(x)|^{-1}dT(x){\mathcal{A}}(x,dT^{*}(x)q)\cdot q$
$\displaystyle=|J_{T}(x)|^{-1}{\mathcal{A}}(x,dT^{*}(x)q)\cdot(dT^{*}(x)q).$

Now applying (2.2) together with Lemma 3.2, we get that

${\mathcal{B}}(\xi,q)\cdot q\gtrsim e^{\kappa nx_{n}}|dT^{*}(x)q|^{p}\simeq
e^{-\kappa(p-n)x_{n}}|q|^{p}=\widetilde{w}(\xi)|q|^{p},$

which completes the proof of the first part. For the second part, we have
using (3.3), (2.3) and Lemma 3.2 that

$\displaystyle|{\mathcal{B}}(\xi,q)|$
$\displaystyle=|J_{T}(x)|^{-1}|dT(x){\mathcal{A}}(x,dT^{*}(x)q)|\simeq|J_{T}(x)|^{-1}e^{-\kappa
x_{n}}|{\mathcal{A}}(x,dT^{*}(x)q)|$ $\displaystyle\lesssim e^{(n-1)\kappa
x_{n}}|dT^{*}(x)q|^{p-1}\simeq
e^{-\kappa(p-n)x_{n}}|q|^{p-1}=\widetilde{w}(\xi)|q|^{p-1}.\qed$

######  Theorem 4.2.

The mapping ${\mathcal{B}}:T({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}})\times\mathbf{R}^{n}\to\mathbf{R}^{n}$, defined by (3.3),
satisfies for a.e. $\xi\in T({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}})$ and all $q_{1},q_{2}\in\mathbf{R}^{n}$ the monotonicity
condition

$({\mathcal{B}}(\xi,q_{1})-{\mathcal{B}}(\xi,q_{2}))\cdot(q_{1}-q_{2})>0\quad\text{when
}q_{1}\neq q_{2}.$

  * Proof.

Using (3.3), we have for all $q_{1},q_{2}\in\mathbf{R}^{n}$,

$\displaystyle({\mathcal{B}}(\xi,q_{1})-{\mathcal{B}}(\xi,q_{2}))\cdot(q_{1}-q_{2})$
$\displaystyle\qquad=|J_{T}(x)|^{-1}((dT(x){\mathcal{A}}(x,dT^{*}(x)q_{1})-dT(x){\mathcal{A}}(x,dT^{*}(x)q_{2}))\cdot(q_{1}-q_{2}))$
$\displaystyle\qquad=|J_{T}(x)|^{-1}(({\mathcal{A}}(x,dT^{*}(x)q_{1})-{\mathcal{A}}(x,dT^{*}(x)q_{2}))\cdot(dT^{*}(x)q_{1}-dT^{*}(x)q_{2})).$

Since $|J_{T}(x)|^{-1}>0$, we have by (2.5) that the last expression is always
nonnegative. Moreover, it is zero if and only if
$dT^{*}(x)q_{1}=dT^{*}(x)q_{2}$, implying that $q_{1}=q_{2}$ as $dT^{*}(x)$ is
invertible. ∎

It is clear from Theorems 4.1 and 4.2, together with the homogeneous condition
(2.4), that the assumptions (3.4)–(3.7) in Heinonen–Kilpeläinen–Martio [10]
are satisfied for ${\mathcal{B}}$ with the weight function
$\widetilde{w}(\xi)=|\xi|^{p-n}$, $\xi\in\mathbf{R}^{n}\setminus\\{0\\}$.
Moreover, ${\mathcal{B}}(\xi,q)$ is measurable in $\xi$ and continuous in $q$.

The weight $\widetilde{w}(\xi)$ belongs to the Muckenhoupt $A_{p}$ class and
the associated measure $d\mu(\xi)=\widetilde{w}(\xi)\,d\xi$ is doubling and
supports a $p\mspace{1.0mu}$-Poincaré inequality on $\mathbf{R}^{n}$. Such
weights are suitable for the study of partial differential equations and
Sobolev spaces, see [10, Chapters 15 and 20] for a detailed exposition. We
follow [10, Chapters 1 and 3] giving the following definitions.

######  Definition 4.3.

For an open set $\Omega\subset\mathbf{R}^{n}$, the weighted Sobolev space
$H^{1,p}_{0}(\Omega,\widetilde{w})$ is the completion of
$C_{0}^{\infty}(\Omega)$ with respect to the norm

$\|u\|_{H^{1,p}(\Omega,\widetilde{w})}:=\biggl{(}\int_{\Omega}\bigl{(}|u(\xi)|^{p}+|\nabla
u(\xi)|^{p}\bigr{)}\widetilde{w}(\xi)\,d\xi\biggr{)}^{1/p}.$

Similarly, $H^{1,p}(\Omega,\widetilde{w})$ is the completion of the set

$\\{\varphi\in
C^{\infty}(\Omega):\|\varphi\|_{H^{1,p}(\Omega,\widetilde{w})}<\infty\\}$

in the $H^{1,p}(\Omega,\widetilde{w})$-norm.

If the weight $\widetilde{w}\equiv 1$, then the symbol $\widetilde{w}$ is
dropped and in this case we have the usual Sobolev space $W^{1,p}(\Omega)$.

######  Definition 4.4.

A function $u\in H^{1,p}_{\rm loc}(\Omega,\widetilde{w})$ in an open set
$\Omega\subset\mathbf{R}^{n}$ is said to be a _weak solution_ of the equation
$\operatorname{div}{\mathcal{B}}(\xi,\nabla u)=0$ if for all functions
$\varphi\in C_{0}^{\infty}(\Omega)$, the following integral identity holds

$\int_{\Omega}{\mathcal{B}}(\xi,\nabla u)\cdot\nabla\varphi\,d\xi=0.$ (4.1)

To use the tools developed in Heinonen–Kilpeläinen–Martio [10] for Dirichlet
problems, the part of the boundary, $T(\partial G\setminus F)$, where the zero
conormal derivative is prescribed, will be removed. This will be done by a
reflection in the hyperplane

$\\{\xi\in\mathbf{R}^{n}:\xi_{n}=0\\}.$

By a reflection, we mean the mapping $P:\mathbf{R}^{n}\to\mathbf{R}^{n}$
defined by

$P\xi=P(\xi^{\prime},\xi_{n})=(\xi^{\prime},-\xi_{n}).$

Let $D$ be the open set consisting of $T(G\setminus F)$ together with
$PT(G\setminus F)$ and $T(\partial G\setminus F)$, i.e.

$D=B(0,1)\setminus\widetilde{F},\quad\text{where }\widetilde{F}=T(F)\cup
PT(F)\cup\\{0\\}.$

The point at infinity in $G$ corresponds to the origin $\xi=0$ in $B(0,1)$.
Note that $\widetilde{F}$ is closed and that the base
$B^{\prime}\times\\{0\\}$ of $G$ is mapped onto the upper unit half-sphere
$\\{\xi\in\partial B(0,1):\xi_{n}>0\\}$. By assumption, the base
$B^{\prime}\times\\{0\\}\subset F$ and so the whole boundary $\partial
D\subset\widetilde{F}$ carries the Dirichlet condition.

Extend ${\mathcal{B}}(\xi,q)$ from $T({\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}})$ to the
reflected half-ball $PT({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}})$ by

${\mathcal{B}}(\xi,q)=\begin{cases}P{\mathcal{B}}(P\xi,Pq)&\text{if
}\xi_{n}<0,\\\ 0&\text{if }\xi=0.\end{cases}$ (4.2)

The standard ellipticity and monotonicity assumptions for ${\mathcal{B}}$
still hold after the extension from $T({\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}})$ to
$PT({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}})$.

With this reflection, we will now be able to eliminate the Neumann boundary
data on $T(\partial G\setminus F)$ so that only the Dirichlet data on
$\partial D$ remain. In Theorem 4.7, we will show that $u\in W^{1,p}_{\rm
loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$ is a weak solution of the equation (1.1) in
$G\setminus F$ with zero conormal derivative on $\partial G\setminus F$ if and
only if the symmetric reflection of $u\circ T^{-1}$ is a weak solution of
(3.4) in $D$. We shall use the function spaces identified in Björn–Mwasa [4].

######  Lemma 4.5.

([4, Lemma 4.6]) Assume that $u\in L^{1}_{\rm loc}(U)$ with the distributional
gradient $\nabla u\in L^{1}_{\rm loc}(U)$ for some open set $U\subset
B^{\prime}(0,R)\times\mathbf{R}$ and let $\tilde{u}=u\circ T^{-1}$. Then for
any measurable set $A\subset U$,

$\displaystyle\int_{A}|\nabla u|^{p}\,dx$
$\displaystyle\simeq\int_{T(A)}|\nabla\tilde{u}|^{p}\widetilde{w}\,d\xi,$
$\displaystyle\int_{A}|u|^{p}e^{-p\kappa x_{n}}\,dx$
$\displaystyle\simeq\int_{T(A)}|\tilde{u}|^{p}\widetilde{w}\,d\xi,$

with comparison constants depending on $R$ but independent of $A$ and $u$.

######  Proposition 4.6.

Let $u\in W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$. Then the function

$\tilde{u}(\xi^{\prime},\xi_{n})=\begin{cases}(u\circ
T^{-1})(\xi^{\prime},\xi_{n})&\text{if }\xi\in T({\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}\setminus
F),\\\ (u\circ T^{-1})(\xi^{\prime},-\xi_{n})&\text{if }\xi\in PT(G\setminus
F),\end{cases}$ (4.3)

belongs to $H^{1,p}_{\rm loc}(D,\widetilde{w})$. Conversely, if $\tilde{v}\in
H^{1,p}_{\rm loc}(D,\widetilde{w})$, then $\tilde{v}\circ T\in W^{1,p}_{\rm
loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$.

  * Proof.

By the definition of $W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}\setminus
F)$, we can assume that $u\in W^{1,p}_{\rm loc}(\mathbf{R}^{n}\setminus F)$.
Then $u\in W^{1,p}(U)$ for every $U\Subset\mathbf{R}^{n}\setminus F$. To show
that $\tilde{u}\in H^{1,p}_{\rm loc}(D,\widetilde{w})$, let $B\Subset D$ be a
ball. Assume without loss of generality that $B$ is centred in
$T({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$ and set $U:=T^{-1}(B)$. Choose a sequence
$u_{j}\in C^{\infty}(\mathbf{R}^{n}\setminus F)$ such that $u_{j}\to u$ in
$W^{1,p}(U)$. In particular, $u_{j}$ are Lipschitz on $U$. Define
$\tilde{u}_{j}:=u_{j}\circ T^{-1}$ restricted to $B\cap T({\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}})$.

By Lemma 3.2, the functions $\tilde{u}_{j}$ are Lipschitz. Their extensions
across the hyperplane $\xi_{n}=0$ to the whole $B$, by the reflection
$\tilde{u}_{j}(\xi)=\tilde{u}_{j}(P\xi)$, are still Lipschitz. Lemma 4.5
implies that $\tilde{u}$ can be approximated in the
$H^{1,p}(B,\widetilde{w})$-norm by these Lipschitz extensions, that is

$\|\tilde{u}_{j}-\tilde{u}\|_{H^{1,p}(B,\widetilde{w})}\lesssim\|u_{j}-u\|_{W^{1,p}(U)}\to
0\quad\text{as }j\to\infty.$

Hence, $\tilde{u}\in H^{1,p}(B,\widetilde{w})$. Since $B$ was arbitrary, we
have that $\tilde{u}\in H^{1,p}_{\rm loc}(D,\widetilde{w})$.

Conversely, let $V\Subset\mathbf{R}^{n}\setminus F$. Then $T(V)\Subset D$ and
hence $\tilde{v}\in H^{1,p}(T(V),\widetilde{w})$. By Lemma 4.5 and the fact
that $e^{-p\kappa x_{n}}\simeq 1$ on $V$, we have that $\tilde{v}\circ T$
belongs to $W^{1,p}(V)$. Since $V$ was arbitrary, we get that $\tilde{v}\circ
T\in W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$. ∎

######  Theorem 4.7.

Let $u\in W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$ be a weak solution of the equation

$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0\quad\text{in $G\setminus F$}$
(4.4)

with zero conormal derivative on $\partial G\setminus F$, that is, the
integral identity (2.6) holds for all $\varphi\in
C_{0}^{\infty}(\mathbf{R}^{n}\setminus F)$. Let $\tilde{u}$ be as in (4.3).
Then $\tilde{u}\in H^{1,p}_{\rm loc}(D,\widetilde{w})$ is a weak solution of
the equation

$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u})=0\quad\text{in }D.$
(4.5)

Conversely, if $\bar{u}\in H^{1,p}_{\rm loc}(D,\widetilde{w})$ is a weak
solution of (4.5) such that $\bar{u}=\bar{u}\circ P$, then $\bar{u}\circ T\in
W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$, with $\bar{u}$ restricted to
$T({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$, is a weak solution of the equation (4.4) with
zero conormal derivative on $\partial G\setminus F$.

  * Proof.

The fact that $\tilde{u}\in H^{1,p}_{\rm loc}(D,\widetilde{w})$ follows from
Proposition 4.6. To prove the first implication, let ${{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}\in C_{0}^{\infty}(D)$ be an arbitrary test function.
Clearly, ${{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}\circ T|_{{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F}$ is a restriction of a function from
$C_{0}^{\infty}(\mathbf{R}^{n}\setminus F)$. In Lemma 3.1, replace
$v,\tilde{v}$ and $A$ with ${{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}\circ T,{{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}$ and $G\setminus F$, respectively. Lemma 3.1 and the
integral identity (2.6) then give

$\int_{T(G\setminus
F)}{\mathcal{B}}(\xi,\nabla\tilde{u})\cdot\nabla{{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}\,d\xi=\int_{G\setminus F}{\mathcal{A}}(x,\nabla
u)\cdot\nabla({{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}\circ T)\,dx=0.$ (4.6)

Now the change of variables $\zeta=P\xi$, together with (4.2) and the fact
that $\tilde{u}=\tilde{u}\circ P$, yields

$\int_{PT(G\setminus
F)}{\mathcal{B}}(\xi,\nabla\tilde{u})\cdot\nabla{{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}\,d\xi=\int_{T(G\setminus
F)}{\mathcal{B}}(\xi,\nabla\tilde{u})\cdot\nabla({{\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}\circ P)\,d\xi,$ (4.7)

cf. [4, Lemma 6.1]. Since ${{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}\circ P\in C_{0}^{\infty}(D)$, we see as in (4.6) that
the last integral is zero. Adding the left-hand sides of (4.6) and (4.7) shows
that $\tilde{u}$ is a weak solution of (4.5).

Conversely, first we recall from [10, Lemma 3.11] that if $\bar{u}$ is a weak
solution of (4.5), then the integral identity (4.1) holds for all test
functions in $H^{1,p}_{0}(D,\widetilde{w})$ with compact support in $D$. Let
$\varphi\in C_{0}^{\infty}(\mathbf{R}^{n}\setminus F)$ be an arbitrary test
function. Define

${\widetilde{\varphi}}(\xi^{\prime},\xi_{n}):=\begin{cases}(\varphi\circ
T^{-1})(\xi^{\prime},\xi_{n})&\text{if }\xi\in T({\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}),\\\
(\varphi\circ T^{-1})(\xi^{\prime},-\xi_{n})&\text{if }\xi\in PT(G),\\\
0&\text{otherwise.}\end{cases}$

Note that ${\widetilde{\varphi}}={\widetilde{\varphi}}\circ P$ and that
${\widetilde{\varphi}}$ has compact support in $D$. Proposition 4.6 therefore
implies that ${\widetilde{\varphi}}\in H^{1,p}_{0}(D,\widetilde{w})$. Since
$\bar{u}=\bar{u}\circ P$, the integral identity (4.1), tested with
${\widetilde{\varphi}}$, gives

$\int_{T(G\setminus
F)}{\mathcal{B}}(\xi,\nabla\bar{u})\cdot\nabla{\widetilde{\varphi}}\,d\xi+\int_{PT(G\setminus
F)}{\mathcal{B}}(\xi,\nabla\bar{u})\cdot\nabla{\widetilde{\varphi}}\,d\xi=\int_{D}{\mathcal{B}}(\xi,\nabla\bar{u})\cdot\nabla{\widetilde{\varphi}}\,d\xi=0.$

Observe that the integrals on the left-hand side are the same as in (4.7) with
$\tilde{u},{{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle\varphi}\kern 0.0pt}$\hss}{\varphi}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle\varphi}\kern
0.0pt}$\hss}{\varphi}}}}$ replaced by $\bar{u},{\widetilde{\varphi}}$ and are
thus equal. It follows that

$\int_{T(G\setminus
F)}{\mathcal{B}}(\xi,\nabla\bar{u})\cdot\nabla{\widetilde{\varphi}}\,d\xi=0.$

Replacing $u,v,\tilde{v}$ and $A$ in Lemma 3.1 by $\bar{u}\circ
T,\varphi,{\widetilde{\varphi}}$ and $G\setminus F$, respectively, we then get

$\int_{G\setminus F}{\mathcal{A}}(x,\nabla(\bar{u}\circ
T))\cdot\nabla\varphi\,dx=\int_{T(G\setminus
F)}{\mathcal{B}}(\xi,\nabla\bar{u})\cdot\nabla{\widetilde{\varphi}}\,d\xi=0.$

Since $\varphi$ was chosen arbitrarily, we conclude that $\bar{u}\circ T$ is a
weak solution of (4.4) with zero conormal derivative, as in Definition 2.1.
Lastly, Proposition 4.6 shows that $\bar{u}\circ T\in W^{1,p}_{\rm
loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$. ∎

######  Remark 4.8.

The weak solution $\tilde{u}$ can be modified on a set of measure zero, so
that it becomes Hölder continuous in $D$, see Heinonen–Kilpeläinen–Martio [10,
Theorems 3.70 and 6.6]. Hence, for the corresponding continuous representative
of $u$, the limit

$\lim_{G\setminus F\in x\to x_{0}}u(x)$

exists and is finite for every $x_{0}$ on the Neumann boundary $\partial
G\setminus F$. Moreover, $u$ is Hölder continuous at $x_{0}$, by Lemma 3.2.

##  5 Existence and uniqueness of the solutions

In this section, we shall prove the existence and uniqueness of weak solutions
to the equation $\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus
F$ with zero conormal derivative on $\partial G\setminus F$ and continuous
Dirichlet data. We shall also show that the solution attains its continuous
Dirichlet data except possibly on a set of Sobolev $C_{p}$-capacity zero.
Recall that $F$ is a closed subset of ${\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}$, which
contains the base $B^{\prime}\times\\{0\\}$, and that the Dirichlet boundary
data $f$ are prescribed on $F_{0}:=F\cap\partial(G\setminus F)$.

######  Definition 5.1.

Let $K\subset\mathbf{R}^{n}$ be a compact set. The Sobolev
$(p,\widetilde{w})$-capacity of $K$ is

$C_{p,\widetilde{w}}(K)=\inf_{v}\int_{\mathbf{R}^{n}}(|v|^{p}+|\nabla
v|^{p})\widetilde{w}\,dx,$

where the infimum is taken over all $v\in C_{0}^{\infty}(\mathbf{R}^{n})$ (or
equivalently, all continuous $v\in H^{1,p}_{0}(\mathbf{R}^{n},\widetilde{w})$)
such that $v\geq 1$ on $K$, see Heinonen–Kilpeläinen–Martio [10, Section 2.35
and Lemma 2.36].

The capacity $C_{p,\widetilde{w}}$ can be extended to general sets as a
Choquet capacity, see [10, Chapter 2]. In particular, for all Borel sets
$E\subset\mathbf{R}^{n}$,

$C_{p,\widetilde{w}}(E)=\sup\\{C_{p,\widetilde{w}}(K):K\subset E\text{
compact}\\}.$ (5.1)

We also say that a property holds _$C_{p,\widetilde{w}}$ -quasieverywhere_ if
the set where it fails has zero $C_{p,\widetilde{w}}$-capacity. If
$\widetilde{w}\equiv 1$, then we have the usual Sobolev $C_{p}$-capacity. The
following lemma shows that $T$ preserves sets of zero capacity.

######  Lemma 5.2.

Let $E\subset{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}$. Then $C_{p}(E)=0$ if and only if
$C_{p,\widetilde{w}}(T(E))=0$.

  * Proof.

The fact that if $C_{p,\widetilde{w}}(T(E))=0$ then $C_{p}(E)=0$ follows from
Lemma 6.6 in Björn–Mwasa [4].

Conversely, assume that $C_{p}(E)=0$. Replacing $E$ by a Borel set
$E_{0}\supset E$ with zero capacity, we can assume that $E$ is a Borel set.
Let $K\subset T(E)$ be compact. Then $T^{-1}(K)\subset E$ is compact and hence
$C_{p}(T^{-1}(K))=0$. For $\varepsilon>0$, choose $\varphi\in
C^{\infty}_{0}(\mathbf{R}^{n})$ such that $\varphi\geq 1$ on $T^{-1}(K)$ and
$\|\varphi\|_{W^{1,p}(\mathbf{R}^{n})}<\varepsilon$. Multiplying $\varphi$ by
a suitable cut-off function, we can assume that $\varphi(x)=0$ when
$x_{n}\leq-1$. Define

${\widetilde{\varphi}}(\xi^{\prime},\xi_{n}):=\begin{cases}(\varphi\circ
T^{-1})(\xi^{\prime},\xi_{n})&\text{if }\xi\in
T({\,\overline{\\!B^{\prime}}}\times\mathbf{R}),\\\ (\varphi\circ
T^{-1})(\xi^{\prime},-\xi_{n})&\text{if }\xi\in
PT(B^{\prime}\times\mathbf{R}),\\\ 0&\text{if }\xi=0.\end{cases}$

Then ${\widetilde{\varphi}}$ is Lipschitz continuous, by Lemma 3.2, belongs to
$H^{1,p}_{0}(\mathbf{R}^{n},\widetilde{w})$ and is such that
${\widetilde{\varphi}}\geq 1$ on $K$. Using Lemma 4.5 and the fact that
$\varphi(x)=0$ when $x_{n}\leq-1$, we have

$\|{\widetilde{\varphi}}\|_{H^{1,p}(\mathbf{R}^{n},\widetilde{w})}\simeq\|{\widetilde{\varphi}}\|_{H^{1,p}(T(B^{\prime}\times\mathbf{R}),\widetilde{w})}\lesssim\|\varphi\|_{W^{1,p}(B^{\prime}\times\mathbf{R})}\leq\|\varphi\|_{W^{1,p}(\mathbf{R}^{n})}<\varepsilon,$

with comparison constants independent of $\varphi$ and $\varepsilon$. Letting
$\varepsilon\to 0$, gives that $C_{p,\widetilde{w}}(K)=0$ and hence (5.1)
concludes the proof. ∎

The following existence and uniqueness theorem is the main result of this
section. Note that there may also exist unbounded solutions.

######  Theorem 5.3.

Assume that $F_{0}$ is unbounded and let $f\in C(F_{0})$ be such that

$f(\infty):=\lim_{F_{0}\ni x\to\infty}f(x)\quad\text{exists and is finite}.$

Then there exists a unique bounded continuous weak solution $u\in W^{1,p}_{\rm
loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$ of the mixed problem for the equation
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus F$ with zero
conormal derivative on $\partial G\setminus F$ and such that

$\lim_{G\setminus F\ni x\to x_{0}}u(x)=f(x_{0})\quad\text{for
$C_{p}$-quasievery $x_{0}\in F_{0}$.}$ (5.2)

######  Remark 5.4.

The proof below shows that the conclusion of Theorem 5.3 holds also when
$F_{0}$ is bounded. Moreover, since the origin has
$(p,\widetilde{w})$-capacity zero, the solution $u$ is then independent of the
value $f(\infty)$ assigned to $\xi=0$,

  * Proof.

Define

$\tilde{f}(\xi):=\begin{cases}(f\circ T^{-1})(\xi)&\text{if }\xi\in
T(F_{0}),\\\ (f\circ(PT)^{-1})(\xi)&\text{if }\xi\in PT(F_{0}\cap G),\\\
f(\infty)&\text{if }\xi=0.\end{cases}$ (5.3)

Then $\tilde{f}\in C(\partial D)$. By Björn–Björn–Mwasa [2, Theorem 3.12],
there exists a unique bounded continuous weak solution $\bar{u}\in
H^{1,p}_{\rm loc}(D,\widetilde{w})$ of the equation (4.5) such that

$\lim_{D\ni\xi\to\xi_{0}}\bar{u}(\xi)=\tilde{f}(\xi_{0})\quad\text{for
$C_{p,\widetilde{w}}$-quasievery $\xi_{0}\in\partial D$,}$ (5.4)

i.e. for all $\xi_{0}\in\partial D\setminus Z$ for some set $Z\subset\partial
D$ with $C_{p,\widetilde{w}}(Z)=0$. Note that (4.2) holds in $D$ and
$\tilde{f}=\tilde{f}\circ P$. So (5.4) gives

$\lim_{D\ni\xi\to\xi_{0}}\bar{u}(P\xi)=\lim_{D\ni\xi\to
P\xi_{0}}\bar{u}(\xi)=\tilde{f}(P\xi_{0})=\tilde{f}(\xi_{0})$

for all $\xi_{0}\in\partial D\setminus P(Z)$. That is, $\bar{u}\circ P$ also
satisfies (5.4) and $\bar{u}\circ P\in H^{1,p}_{\rm loc}(D,\widetilde{w})$.
Since $\varphi\circ P\in C_{0}^{\infty}(D)$ if and only if $\varphi\in
C_{0}^{\infty}(D)$, the change of variables $\zeta=P\xi$ together with (4.2)
shows that the integral identity (4.1) holds for $\bar{u}\circ P$ as well.
Thus $\bar{u}\circ P$ is also a bounded continuous weak solution of (4.5) in
$D$, satisfying (5.4). By the uniqueness in [2, Theorem 3.12], we conclude
that $\bar{u}=\bar{u}\circ P$.

Define $u:=\bar{u}\circ T$, with $\bar{u}$ restricted to $T({\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}\setminus
F)$. Theorem 4.7 shows that $u$ is a continuous weak solution of
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus F$ with zero
conormal derivative, as in Definition 2.1. Since $\bar{u}$ satisfies (5.4), it
then follows that $u$ satisfies (5.2) for every $x_{0}\in F_{0}\setminus
Z_{0}$, where $Z_{0}:=T^{-1}(Z\cap T({\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}))$ with
$C_{p}(Z_{0})=0$ by Lemma 5.2.

To prove the uniqueness, suppose that $v\in W^{1,p}_{\rm
loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$ is a bounded continuous weak solution of the
equation $\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ satisfying (5.2) for
all $x_{0}\in F_{0}\setminus Z^{\prime}_{0}$ with $C_{p}(Z^{\prime}_{0})=0$.
Let $\tilde{v}$ be as in (4.3) with $u$ replaced by $v$. Then by Theorem 4.7,
$\tilde{v}$ is a continuous weak solution of (4.5).

Since $v$ satisfies (5.2), it follows that $\tilde{v}$ satisfies (5.4) for
each $\xi_{0}\in\partial D\setminus Z^{\prime}$, where
$Z^{\prime}:=T(Z^{\prime}_{0})\cup PT(Z^{\prime}_{0})\cup\\{0\\}$. Now we have
that $C_{p,\widetilde{w}}(T(Z^{\prime}_{0}))=0$ by Lemma 5.2, and so
$C_{p,\widetilde{w}}(PT(Z^{\prime}_{0}))=0$, by reflection. The origin $0$ has
zero $(p,\widetilde{w})$-capacity by [4, Lemma 7.6], and it follows by
subadditivity that $C_{p,\widetilde{w}}(Z^{\prime})=0$.

By [2, Theorem 3.12], the solution of (4.5) satisfying (5.4) is unique, in
other words $\tilde{v}=\bar{u}$ and so $v=u$. ∎

The following definition is adopted from Björn–Mwasa [4].

######  Definition 5.5.

The space $L^{1,p}_{\kappa}(G\setminus F)$ consists of all measurable
functions $v$ on $G\setminus F$ such that the norm

$\|v\|_{L^{1,p}_{\kappa}(G\setminus F)}=\biggl{(}\int_{G\setminus
F}\bigl{(}|v(x)|^{p}e^{-p\kappa x_{n}}+|\nabla
v(x)|^{p}\bigr{)}\,dx\biggr{)}^{1/p}<\infty,$

where $\nabla
v=(\partial_{1}v,\mathinner{.}\nonscript\\!.\nonscript\\!\mathinner{.},\partial_{n}v)$
is the distributional gradient of $v$. The space
$L^{1,p}_{\kappa,0}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$ is the completion of
$C_{0}^{\infty}(\mathbf{R}^{n}\setminus F)$ in the above
$L^{1,p}_{\kappa}(G\setminus F)$-norm.

Note that the space $L^{1,p}_{\kappa}(G\setminus F)$ is contained in
$W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$. The following result generalizes [4, Theorem
6.3] to elliptic divergence type equations.

######  Theorem 5.6.

Let $f\in L^{1,p}_{\kappa}(G\setminus F)$. Then there exists a unique
continuous weak solution $u\in L^{1,p}_{\kappa}(G\setminus F)$ of the equation
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus F$ with zero
conormal derivative on $\partial G\setminus F$ and such that $u-f\in
L^{1,p}_{\kappa,0}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$.

  * Proof.

Let $\tilde{f}$ be defined as in (4.3), with $u$ replaced by $f$. Then
$\tilde{f}\in H^{1,p}(D,\widetilde{w})$, by [4, Proposition 5.3]. By [10,
Theorems 3.17 and 3.70], there exists a unique continuous weak solution
$\tilde{u}\in H^{1,p}(D,\widetilde{w})$ of the degenerate equation (4.5) such
that $\tilde{u}-\tilde{f}\in H^{1,p}_{0}(D,\widetilde{w})$. Since
${\mathcal{B}}(\xi,Pq)=P{\mathcal{B}}(P\xi,q)$ by (4.2), we infer from [4,
Corollary 6.2] (with ${\mathcal{A}}(\xi,q)$ in [4] replaced by
${\mathcal{B}}(\xi,q)$) that $\tilde{u}=\tilde{u}\circ P$. By Theorem 4.7,
$u:=\tilde{u}\circ T$ (with $\tilde{u}$ restricted to $T(G\setminus F)$) is a
weak solution of $\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in
$G\setminus F$ with zero conormal derivative on $\partial G\setminus F$.
Moreover, $u\in L^{1,p}_{\kappa}(G\setminus F)$ by [4, Proposition 5.3] and
$u-f\in L^{1,p}_{\kappa,0}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$ by [4, Proposition 5.5].

To prove the uniqueness, suppose that $v\in L^{1,p}_{\kappa}(G\setminus F)$ is
a continuous weak solution of the equation
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus F$ with zero
conormal derivative on $\partial G\setminus F$ and $v-f\in
L^{1,p}_{\kappa,0}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$. Let $\tilde{v}$ be as in (4.3), with $u$
replaced by $v$. Theorem 4.7 then implies that $\tilde{v}$ satisfies the
equation (4.5). Moreover, [4, Proposition 5.5] shows that
$\tilde{v}-\tilde{f}\in H^{1,p}_{0}(D,\widetilde{w})$. From the uniqueness of
solutions to (4.5) we thus get that $\tilde{v}=\tilde{u}$, and so $v=u$. ∎

##  6 Boundary regularity at infinity

We saw in Remark 4.8 that weak solutions of the equation
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ with zero conormal derivative
are continuous in $G\setminus F$ and at the Neumann boundary $\partial
G\setminus F$. Moreover, if the Dirichlet boundary data $f$ are continuous on
$F_{0}$, then the solution is continuous at $F_{0}$, except possibly for a set
of Sobolev $C_{p}$-capacity zero. We now study continuity at the point at
infinity.

We follow Björn–Mwasa [4, Section 8] giving the following definition.

######  Definition 6.1.

Assume that $F$ is unbounded. We say that the point at $\infty$ is _regular_
for the mixed problem (2.1) for the equation
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus F$ with zero
conormal derivative on $\partial G\setminus F$ if for all Dirichlet boundary
data $f\in C(F_{0})$ with a finite limit

$\lim_{F_{0}\ni x\to\infty}f(x)=:f(\infty),$ (6.1)

the unique bounded continuous weak solution $u$ of (2.1), provided by Theorem
5.3, satisfies

$\lim_{G\setminus F\ni x\to\infty}u(x)=f(\infty).$ (6.2)

Remark 5.4 shows that the point at infinity is always irregular when $F_{0}$
is bounded. Note that due to the conormal derivative condition, the regularity
at $\infty$ in Definition 6.1 differs from the usual notion of boundary
regularity for the Dirichlet problem in unbounded domains, as in e.g.
Heinonen–Kilpeläinen–Martio [10, Section 9.5]. The following is our first step
in characterizing the regularity of the point at infinity for the mixed
problem (2.1).

######  Proposition 6.2.

The point at $\infty$ is regular for the mixed problem (2.1) for the equation
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus F$ if and only
if the origin $0\in\partial D$ is regular with respect to the equation

$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u}(\xi))=0\quad\text{in }D,$
(6.3)

where ${\mathcal{B}}$ is as in (3.3).

Before the proof of Proposition 6.2, we give the definition below, see [10,
Section 9.5].

######  Definition 6.3.

A point $\xi_{0}\in\partial D$ is regular for the equation (6.3) if

$\lim_{D\ni\xi\to\xi_{0}}\bar{u}(\xi)=\bar{f}(\xi_{0})\quad\text{for all
}\bar{f}\in C(\partial D),$

where $\bar{u}$ is the Perron solution of (6.3) with the boundary data
$\bar{f}$, as in [10, Section 9.1].

Note that $\bar{f}$, being continuous, is resolutive, i.e. the upper and the
lower Perron solutions coincide and are equal to the Perron solution
$\bar{u}$, see [10, Theorem 9.25]. Moreover, by Björn–Björn–Mwasa [2, Theorem
3.12], the Perron solution $\bar{u}$ is the only bounded continuous weak
solution of (6.3) that attains the boundary values $\bar{f}$
$C_{p,\widetilde{w}}$-quasieverywhere on $\partial D$ in the sense of (5.4).

  * Proof of Proposition 6.2.

Assume that $0\in\partial D$ is a regular point with respect to (6.3). Let
$f\in C(F_{0})$ be such that the limit in (6.1) exists and is finite. Let $u$
be the unique bounded continuous weak solution of
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus F$ with zero
conormal derivative, provided for $f$ by Theorem 5.3. Define $\tilde{u}$ and
$\tilde{f}$ as in (4.3) and (5.3), respectively. Then by Theorem
4.7$,\tilde{u}$ is a bounded continuous weak solution of
$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u})=0$ in $D$ and attains
the boundary values $\tilde{f}$ $C_{p,\widetilde{w}}$-quasieverywhere on
$\partial D$. Note that $\tilde{f}\in C(\partial D)$. Thus, by [2, Theorem
3.12], $\tilde{u}$ is the Perron solution of (6.3) with the boundary data
$\tilde{f}$.

Since $0\in\partial D$ is regular, we have that

$\lim_{G\setminus F\ni x\to\infty}u(x)=\lim_{D\ni\xi\to
0}\tilde{u}(\xi)=\tilde{f}(0)=f(\infty).$

Thus, $u$ satisfies (6.2) and so $\infty$ is regular for the mixed problem for
the equation $\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus
F$.

Conversely, assume that $\infty$ is regular for the mixed problem (2.1) and
let $\bar{f}\in C(\partial D)$. The function $\bar{f}$ is not necessarily
symmetric, so we consider

$\bar{f}_{1}=\min\\{\bar{f},\bar{f}\circ
P\\}\quad\text{and}\quad\bar{f}_{2}=\max\\{\bar{f},\bar{f}\circ P\\},$

which are symmetric, i.e. $\bar{f}_{j}=\bar{f}_{j}\circ P$, $j=1,2$. Let
$\bar{u}$ and $\bar{u}_{j}$ be the Perron solutions of (6.3) with boundary
data $\bar{f}$ and $\bar{f}_{j}$, respectively. Define
$u_{j}:=\bar{u}_{j}\circ T$, $j=1,2$, with $\bar{u}_{j}$ restricted to
$T({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$. Then by Theorem 4.7, $u_{j}$ are bounded
continuous weak solutions of the mixed problem (2.1) satisfying (5.2) with
boundary data $f_{j}:=\bar{f}_{j}\circ T|_{F_{0}}$. Note that $f_{j}$ are
continuous on $F_{0}$ and that the limits

$\lim_{F_{0}\ni x\to\infty}f_{j}(x)=f_{j}(\infty)\quad\text{exist and are
finite}.$

Since $\infty$ is regular for (2.1), the solutions $u_{j}$ satisfy

$\lim_{G\setminus F\ni x\to\infty}u_{j}(x)=f_{j}(\infty)=\bar{f}(0).$

It now follows that

$\lim_{D\ni\xi\to 0}\bar{u}_{j}(\xi)=\bar{f}(0).$

But $\bar{f}_{1}\leq\bar{f}\leq\bar{f}_{2}$ and thus
$\bar{u}_{1}\leq\bar{u}\leq\bar{u}_{2}$ by the definition of Perron solutions.
We conclude that

$\lim_{D\ni\xi\to 0}\bar{u}(\xi)=\bar{f}(0),$

and so $0\in\partial D$ is regular for the equation (6.3). ∎

Regular points with respect to
$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u}(\xi))=0$ in $D$ are
characterized by the _Wiener criterion_ , see Heinonen–Kilpeläinen–Martio [10,
Theorem 21.30
(i)$\mathchoice{\quad\Longleftrightarrow\quad}{\Leftrightarrow}{\Leftrightarrow}{\Leftrightarrow}$(v)]
and Mikkonen [28]. Note that the definitions for regularity of a point
$\xi\in\partial D$ with respect to (6.3) in terms of both Sobolev and Perron
solutions are equivalent, see [10, Theorem 9.20].

Recall that $D=B_{1}\setminus\widetilde{F}$, where $\widetilde{F}=T(F)\cup
PT(F)\cup\\{0\\}$. Note that the ball
$B_{r}:=\\{\xi\in\mathbf{R}^{n}:|\xi|<r\\}$ in $B_{1}$ corresponds to the
truncated cylinder

$G_{t}:=\\{x\in{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}:x_{n}>t\\}={\,\overline{\\!B^{\prime}}}\times(t,\infty)\quad\text{with
}t=-\frac{1}{\kappa}\log r\geq 0$

in ${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}$ and that $G_{t}$ contains the lateral boundary but not its
base $B^{\prime}\times\\{t\\}$. With the above notation, we see that $G_{2t}$
and $G_{t-1}$ correspond to $B_{r^{2}}$ and $B_{2r}$, respectively. We follow
[4, Section 7] giving the following definition.

######  Definition 6.4.

Let $K\subset G_{t-1}$ be a compact set, where $t\geq 1$. The _( Neumann)
variational $p\mspace{1.0mu}$-capacity_ of $K$ with respect to $G_{t-1}$ is

$\operatorname{cap}_{p,G_{t-1}}(K)=\inf_{v}\int_{G_{t-1}}|\nabla v|^{p}\,dx,$

where the infimum is taken over all functions $v\in
C_{0}^{\infty}(\mathbf{R}^{n})$ satisfying $v\geq 1$ on $K$ and $v=0$ on
${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus G_{t-1}$.

Just as $C_{p,\widetilde{w}}$, the capacity $\operatorname{cap}_{p,G_{t-1}}$
is also a Choquet capacity. This was proved in [4, Section 7], even though we
only need $\operatorname{cap}_{p,G_{t-1}}$ for compact sets here. In [4,
Section 8], the Wiener criterion from [10, Section 6.16] is rewritten in terms
of $\operatorname{cap}_{p,G_{t-1}}$ and we thus get the following criterion
for the boundary regularity at $\infty$ for the mixed boundary value problem
(2.1).

######  Theorem 6.5.

The point at $\infty$ is regular for the mixed boundary value problem (2.1) in
$G\setminus F$ if and only if the following condition holds

$\int_{1}^{\infty}\operatorname{cap}_{p,G_{t-1}}(F\cap({\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}_{t}\setminus G_{2t}))^{1/(p-1)}\,dt=\infty.$ (6.4)

The proof follows from [4, Section 8], with Proposition 6.2 playing the role
of Lemma 8.2 in [4]. See Examples 8.7 and 8.8 in [4] for concrete sets
satisfying or failing the Wiener condition (6.4).

##  7 General behaviour of the solutions at $\infty$

Our aim in this section is to show a Phragmén–Lindelöf type trichotomy for the
solutions of the equation $\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in
$G$ with zero conormal derivative $C_{p}$-quasieverywhere on $\partial G\cap
G_{t}$. We start by showing that sets of Sobolev $C_{p}$-capacity zero are
removable for the solutions. Recall that $F$ is a closed subset of
${\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}$.

For compact subsets of $G$, the following removability result is just [10,
Theorem 7.36]. Using the transformation $T$, we can remove also parts of the
lateral Dirichlet boundary and change them into the Neumann boundary. This
will make it possible to study the behaviour of the solutions at $\infty$.

Recall that $G_{t}:=\\{x\in{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}:x_{n}>t\\}$ is the truncated cylinder containing its lateral
boundary but not its base $B^{\prime}\times\\{t\\}$.

######  Lemma 7.1.

Assume that for some $t\geq 0$, the set $E:=F\cap G_{t}$ satisfies
$C_{p}(E)=0$. Let $u\in W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}\setminus
F)$ be a continuous weak solution of $\operatorname{div}{\mathcal{A}}(x,\nabla
u)=0$ in $G\setminus F$ with zero conormal derivative on $\partial G\setminus
F$. Assume that $u$ is bounded in the set $(G_{t}\setminus G_{\tau})\setminus
F$ for every $\tau>t$.

Then $u$ can be extended to $E$ as a continuous weak solution in $(G\setminus
F)\cup E$ with zero conormal derivative on $\partial G\setminus(F\setminus
G_{t})$.

  * Proof.

Set $\widetilde{E}:=T(E)\cup PT(E)$. By Lemma 5.2, the set $T(E)$ has Sobolev
$(p,\widetilde{w})$-capacity zero. By reflection, we have
$C_{p,\widetilde{w}}(\widetilde{E})=0$. As in (4.3), define

$\tilde{u}(\xi^{\prime},\xi_{n})=\begin{cases}(u\circ
T^{-1})(\xi^{\prime},\xi_{n})&\text{if }\xi\in T({\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}\setminus
F),\\\ (u\circ T^{-1})(\xi^{\prime},-\xi_{n})&\text{if }\xi\in PT(G\setminus
F).\end{cases}$ (7.1)

Then by Theorem 4.7, $\tilde{u}$ is a continuous weak solution of
$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u})=0$ in $D$, which is
bounded in $D\cap(B_{r}\setminus B_{\rho})$ for every $\rho>0$, where
$r=e^{-\kappa t}$. Note that $\widetilde{E}\setminus B_{\rho}$ is relatively
closed in $B_{r}\setminus B_{\rho}$.

Since $C_{p,\widetilde{w}}(\widetilde{E})=0$, we have by
Heinonen–Kilpeläinen–Martio [10, Theorem 7.36] that $\tilde{u}$ can be
extended to $\widetilde{E}$ so that it is a continuous weak solution in
$B_{r}\setminus B_{\rho}$ for every $\rho>0$, and thus also in
$D\cup(B_{r}\setminus\\{0\\})$. The desired extension of $u$ is then given by
$\tilde{u}\circ T$. ∎

Removability and behaviour of the solutions at $\infty$ are addressed in the
rest of the section. The following two lemmas provide suitable lower and upper
bounds for the solutions.

######  Lemma 7.2.

Assume that for some $t\geq 0$, the set $E:=F\cap G_{t}$ is empty. Let $u\in
W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$ be a weak continuous solution of
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus F$ with zero
conormal derivative on $\partial G\setminus F$. Then $u$ is bounded in the set
$G_{t^{\prime}}\setminus G_{\tau}$ for every $\tau>t^{\prime}>t$.

If, moreover, $u(x)\leq 0$ when $x_{n}=t$, then either $u\leq 0$ in $G_{t}$ or
there exist $A>0$ and $\tau_{0}>t$ such that for all $\tau\geq\tau_{0}$,

$\max_{x_{n}=\tau}u(x)\geq A(\tau-t).$

  * Proof.

Define $\tilde{u}$ as in (7.1). Then by Theorem 4.7, $\tilde{u}\in
H^{1,p}_{\rm loc}(D,\widetilde{w})$ is a continuous weak solution of
$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u}(\xi))=0$ in $D$. In
particular, since $E$ is empty, $\tilde{u}\in H^{1,p}_{\rm
loc}(B_{r}\setminus\\{0\\},\widetilde{w})$ is a continuous weak solution in
$B_{r}\setminus\\{0\\}$, where $r=e^{-\kappa t}$. This immediately implies the
boundedness of $u$ in $G_{t^{\prime}}\setminus G_{\tau}$ for every
$\tau>t^{\prime}>t$.

Next, if $u(x)\leq 0$ when $x_{n}=t$, then $\tilde{u}(\xi_{0})\leq 0$ for all
$\xi_{0}\in\partial B_{r}$. Hence, by [10, Theorem 7.40], we have that either
$\tilde{u}\leq 0$ in $B_{r}\setminus\\{0\\}$ or

$\liminf_{\rho\to
0}(\operatorname{cap}_{p,\widetilde{w}}(B_{\rho},B_{r}))^{1/(p-1)}\max_{\partial
B_{\rho}}\tilde{u}>0,$ (7.2)

where $\operatorname{cap}_{p,\widetilde{w}}$ is the variational capacity
associated with the weight $\widetilde{w}$, as in [10, Chapter 2]. Inequality
(7.2) reveals that there exist constants $a>0$ and $r_{0}>0$ such that for all
$0<\rho\leq r_{0}$, we have

$\max_{\partial B_{\rho}}\tilde{u}\geq
a(\operatorname{cap}_{p,\widetilde{w}}(B_{\rho},B_{r}))^{1/(1-p)}.$ (7.3)

By Björn–Mwasa [4, Lemma 7.6], we have for all $\rho<r$,

$(\operatorname{cap}_{p,\widetilde{w}}(B_{\rho},B_{r}))^{1/(1-p)}\gtrsim\log\frac{r}{\rho}=\kappa(\tau-t),\quad\text{where
}\tau=-\frac{1}{\kappa}\log\rho>t$

and the constant in $\gtrsim$ depends only on $n$ and $p$. Substituting in
(7.3) and using the definition (7.1) of $\tilde{u}$, concludes the proof. ∎

######  Lemma 7.3.

Assume that for some $t\geq 0$, the set $E:=F\cap G_{t}$ is empty. Let $u\in
W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle G}\kern
0.0pt}$\hss}{G}}}\setminus F)$ be a weak continuous solution of
$\operatorname{div}{\mathcal{A}}(x,\nabla u)=0$ in $G\setminus F$ with zero
conormal derivative on $\partial G\setminus F$. Assume that $u\geq 0$ in
$G_{t}$. Then there exists $A_{0}\geq 0$ such that for all $\tau\geq
t+\tfrac{1}{\kappa}\log 2$,

$\max_{x_{n}=\tau}u(x)\leq A_{0}(\tau-t).$

  * Proof.

Define $\tilde{u}$ as in (7.1). As in the proof of Lemma 7.2, $\tilde{u}$ is a
continuous weak solution of
$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u}(\xi))=0$ in
$B_{r}\setminus\\{0\\}$, where $r=e^{-\kappa t}$. For $0<\rho<\frac{1}{2}r$,
let $\tilde{v}$ be the potential of ${\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}$\hss}{B}}}_{\rho}$
in $B_{r}$, i.e. the continuous weak solution of
$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{v})=0$ in
$B_{r}\setminus{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{\rho}$ with boundary values $1$ on ${\mathchoice{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\displaystyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle B}\kern 0.0pt}$\hss}{B}}}_{\rho}$
and $0$ on $\partial B_{r}$. Then by Heinonen–Kilpeläinen–Martio [10, Lemma
6.21], there exists $c>0$ such that

$\tilde{v}(\xi)\geq
c\biggl{(}\frac{\operatorname{cap}_{p,\widetilde{w}}(B_{\rho},B_{r})}{\operatorname{cap}_{p,\widetilde{w}}(B_{r/2},B_{r})}\biggr{)}^{1/(p-1)}\quad\text{for
all }\xi\in{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{r/2}.$

From [4, Lemma 7.6] and [10, Theorem 2.18], we thus have for all
$\xi\in{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{r/2}$ that

$\tilde{v}(\xi)\gtrsim\operatorname{cap}_{p,\widetilde{w}}(B_{\rho},B_{r})^{1/(p-1)}\gtrsim\Bigl{(}\log\frac{r}{\rho}\Bigr{)}^{-1}=\frac{1}{\kappa(\tau-t)},$
(7.4)

where $\tau=-\frac{1}{\kappa}\log\rho>t$ and the constants in $\gtrsim$ depend
only on $c$, $n$ and $p$.

Let $m_{\rho}$ be the minimum of $\tilde{u}$ on the sphere $\partial
B_{\rho}$. Using the boundary values of $\tilde{v}$ and $\tilde{u}$ on both
$\partial B_{\rho}$ and $\partial B_{r}$, it follows from the comparison
principle [10, Lemma 3.18] and from (7.4) that

$\tilde{u}\geq m_{\rho}\tilde{v}\gtrsim\frac{m_{\rho}}{\tau-t}\quad\text{in
}{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{r/2}\setminus{\mathchoice{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\displaystyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\textstyle B}\kern
0.0pt}$\hss}{B}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptstyle B}\kern 0.0pt}$\hss}{B}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptscriptstyle B}\kern
0.0pt}$\hss}{B}}}_{\rho}.$

The Harnack inequality for $\tilde{u}$ on the sphere $\partial B_{\rho}$ then
gives for any fixed $\xi_{0}\in\partial B_{r/2}$ that

$\max_{x_{n}=\tau}u(x)=\max_{\partial B_{\rho}}\tilde{u}(x)\lesssim
m_{\rho}\lesssim\tilde{u}(\xi_{0})(\tau-t),$

where the constants in $\lesssim$ depend only on ${\mathcal{A}}$, $\kappa$,
$n$ and $p$. ∎

We are now ready to conclude the paper with the following trichotomy for the
solutions of the mixed problem at $\infty$, when $F$ is negligible near
$\infty$.

######  Theorem 7.4.

Assume that for some $t\geq 0$, the set $E:=F\cap G_{t}$ is such that
$C_{p}(E)=0$. Let $u\in W^{1,p}_{\rm loc}({\mathchoice{\hbox to0.0pt{$\kern
0.0pt\kern 0.0pt\overline{\phantom{\displaystyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\textstyle G}\kern 0.0pt}$\hss}{G}}{\hbox
to0.0pt{$\kern 0.0pt\kern 0.0pt\overline{\phantom{\scriptstyle G}\kern
0.0pt}$\hss}{G}}{\hbox to0.0pt{$\kern 0.0pt\kern
0.0pt\overline{\phantom{\scriptscriptstyle G}\kern 0.0pt}$\hss}{G}}}\setminus
F)$ be a weak continuous solution of $\operatorname{div}{\mathcal{A}}(x,\nabla
u)=0$ in $G\setminus F$ with zero conormal derivative on $\partial G\setminus
F$. Assume that $u$ is bounded in the set $(G_{t}\setminus G_{\tau})\setminus
F$ for every $\tau>t$.

Then there exist constants $\tau_{0}>t$, $M$, $M_{0}$ and $A_{0}\geq A\geq 0$,
such that exactly one of the following holds.

  1. (i)

The solution $u$ is bounded in $G_{\tau_{0}}$ and the limit

$\lim_{G\setminus F\ni x\to\infty}u(x)=:u(\infty)$

exists and is finite. Moreover, for some $\alpha\in(0,1]$ and all $x\in
G_{\tau_{0}}$,

$|u(x)-u(\infty)|\lesssim e^{-\kappa\alpha x_{n}}.$ (7.5)

  2. (ii)

The solution $u$ tends roughly linearly either to $\infty$ or to $-\infty$,
i.e. either

$M+A\tau\leq u(x^{\prime},\tau)\leq M_{0}+A_{0}\tau\quad\text{for all
$x^{\prime}\in B^{\prime}$ and $\tau\geq\tau_{0}$,}$ (7.6)

or (7.6) holds for $-u$ in place of $u$.

  3. (iii)

The solution changes sign and approaches both $\infty$ and $-\infty$. More
precisely,

$\displaystyle\max_{x_{n}=\tau}u(x)\geq M+A\tau\text{ and
}\displaystyle\min_{x_{n}=\tau}u(x)\leq M_{0}-A\tau\quad\text{for all
$\tau\geq\tau_{0}$.}$

  * Proof.

By Lemma 7.1, we can assume that $E$ is empty. Define $\tilde{u}$ as in (7.1).
Then by Theorem 4.7, the function $\tilde{u}$ is a continuous weak solution of
$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u})=0$ in
$B_{r}\setminus\\{0\\}$, where $r=e^{-\kappa t}$. Let
$r^{\prime}\leq\tfrac{1}{2}r$. By the continuity of the solution $\tilde{u}$
in $B_{r}\setminus\\{0\\}$, there exist constants $m^{\prime},M^{\prime}$ such
that $m^{\prime}\leq\tilde{u}(\xi)\leq M^{\prime}$ for all $\xi\in\partial
B_{r^{\prime}}$. Hence, it follows from the definition of $\tilde{u}$ that
$m^{\prime}\leq u(x)\leq M^{\prime}$ when
$x_{n}=t^{\prime}:=-\frac{1}{\kappa}\log r^{\prime}>t$.

Applying the second part of Lemma 7.2 to $G_{t^{\prime}}$ with $u$ replaced by
$u-M^{\prime}$ and $m^{\prime}-u$, respectively, shows that there exist
$A^{\prime}>0$ and $\tau_{0}>t^{\prime}$ such that the following two
statements hold:

    1. (a)

$u\leq M^{\prime}$ in $G_{t^{\prime}}$ or
$\displaystyle\max_{x_{n}=\tau}u(x)\geq
M^{\prime}+A^{\prime}(\tau-t^{\prime})$ for all $\tau\geq\tau_{0}$,

    2. (b)

$u\geq m^{\prime}$ in $G_{t^{\prime}}$ or
$\displaystyle\min_{x_{n}=\tau}u(x)\leq
m^{\prime}-A^{\prime}(\tau-t^{\prime})$ for all $\tau\geq\tau_{0}$.

Combining the first two alternatives in (a) and (b) gives (i), while the
second alternatives give (iii). Since $C_{p,\widetilde{w}}(\\{0\\})=0$, we can
in the bounded case (i) use [10, Theorem 7.36] and extend $\tilde{u}$ to $0$
so that it becomes a continuous weak solution of
$\operatorname{div}{\mathcal{B}}(\xi,\nabla\tilde{u})=0$ in $B_{r}$. By [10,
Theorem 6.6], $\tilde{u}$ is Hölder continuous at the origin, which shows that
(7.5) holds.

The remaining alternatives will lead to case (ii). If $u\geq m^{\prime}$ in
$G_{t^{\prime}}$ and

$\max_{x_{n}=\tau}u(x)\geq
M^{\prime}+A^{\prime}(\tau-t^{\prime})\quad\text{for all $\tau\geq\tau_{0}$,}$

then using the Harnack inequality for $\tilde{u}-m^{\prime}$ on the sphere
$\partial B_{\rho}$ with $\rho=e^{-\kappa\tau}$, we see that for some $C>0$
independent of $\tau\geq\tau_{0}$,

$\min_{x_{n}=\tau}(u-m^{\prime})\geq C\max_{x_{n}=\tau}(u-m^{\prime})\geq
C(M^{\prime}-m^{\prime}+A^{\prime}(\tau-t^{\prime})).$

This proves the lower bound in (7.6), while the upper bound follows from Lemma
7.3 applied to $u-m^{\prime}$. The second case of (ii), i.e. (7.6) for $-u$,
follows in a similar way by combining $u\leq M^{\prime}$ with the second
alternative in (b). ∎

We finish by giving a concrete example illustrating the cases in Theorem 7.4.

######  Example 7.5.

Let $G=(-1,1)\times(0,\infty)\subset\mathbf{R}^{2}$ and
$F=[-1,1]\times\\{0\\}$. The linear function $u(x_{1},x_{2})=ax_{2}+b$, where
$a,b\in\mathbf{R}$, satisfies $\Delta u=0$ in $G$ and $\partial
u/\partial\nu=0$ on the lateral boundary $\partial G\setminus F$. The cases
(i) and (ii) follow if $a=0$, $a>0$ and $a<0$, respectively. Also, consider
the function

$u(x_{1},x_{2})=e^{\frac{\pi}{2}x_{2}}\sin\frac{\pi}{2}x_{1},$

which is easily verified to satisfy $\Delta u=0$ in $G$. Then (iii) is
achieved when $\sin\tfrac{\pi}{2}x_{1}$ attains its maximum and minimum at
$[-1,1]$, respectively.

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