\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
Sixth Mississippi State Conference on Differential Equations and
Computational Simulations,
{\em Electronic Journal of Differential Equations},
Conference 15 (2007), pp. 377--385.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document} \setcounter{page}{377}
\title[\hfilneg EJDE-2006/Conf/15\hfil Sub- and supersolutions]
{Revisiting the method of sub- and supersolutions for
nonlinear elliptic problems}
\author[K. Schmitt\hfil EJDE/Conf/15 \hfilneg]
{Klaus Schmitt}
\address{Klaus Schmitt \newline
Department of Mathematics,
University of Utah,
155 South 1400 East,
Salt Lake City, UT 84112, USA}
\email{schmitt@math.utah.edu}
\thanks{Published February 28, 2007.}
\thanks{This paper here presents the text of a
lecture presented at the Mississippi State/UAB conference on
differential equations and computational simulations during May
2005. The author expresses his sincere gratitude to those involved
in organizing the conference and also to all those who attended,
particularly to several of his former students and
collaborators.}
\subjclass[2000]{35B45, 35J65, 35J60}
\keywords{Sub- and supersolutions; general boundary conditions;
\hfill\break\indent
variational inequalities}
\begin{abstract}
We discuss some of the historical highlights of the theory
of sub- supersolutions for boundary value problems
for nonlinear elliptic equations and variational inequalities.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
Sub- and supersolutions (upper- and lower
solutions) have played an important role in the study of nonlinear
boundary value problems for elliptic partial differential equations
for a long time. While some of the underlying principles are already
present in the Perron process for obtaining harmonic functions satisfying
(in a generalized sense) given boundary data (see,
e.g. \cite{gilbarg:epd01}), Scorza-Dragoni's paper \cite{scorza:ipd31}
was one of the earliest works, where the existence of an ordered pair
of solutions of differential inequalities was used to establish the
existence of a solution of a given boundary value problem for a
nonlinear second order ordinary differential equation. This was
followed by some fundamental work of Nagumo
\cite{nagumo:udd37, nagumo:udr42} which inspired much work on such problems
subject to Dirichlet boundary conditions for both
ordinary and partial differential equations during the decade of the
sixties
(see, e.g.
\cite{ako:dpq61, ako:sod69,jackson:ctn67,schmitt:bvp68, schmitt:bvp78}).
Using Cesari's method, Knobloch \cite{knobloch:enm63}
introduced the sub-supersolution method to
the study of periodic boundary value problems for nonlinear second
order ordinary differential equations. Using somewhat different
techniques,
similar problems were subsequently studied in
\cite{schmitt:psn67, mawhin:nfa74}, among others.
In all of the above cited papers sub- and supersolutions are assumed
to be smooth
solutions of differential inequalities (i.e., solutions in a classical
sense); such smooth sub- and supersolutions were also used to study
Dirichlet and/or Neumann boundary value problems for semilinear elliptic
problems in \cite{amann:set78,sattinger:mmn72},
for general (nonlinear) boundary value problems in
\cite{erbe:nbv70, erbe:esb82,mawhin:uls84}, and also for systems
of nonlinear ordinary
differential equations in
\cite{bebernes:pbv73,habets:bvp83,knobloch:bvp77}.
Deuel and Hess \cite{deuel:ive74} and Hess \cite{hess:sne76} were the first to
formulate concepts of weak sub- and supersolutions and obtained existence
results for weak solutions of semilinear elliptic Dirichlet problems.
This subject was subsequently continued by several authors (see, e.g.,
\cite{carl:ssm04,dancer:emw89,kura:wss89,le:bvp98,le:sst04,
le:sgc05, le:sep98, le:ssm01, le:sse01,schmitt:pss04}).
In some sense the weak approach gives a unifying way of treating such
problems
which is the theme of this paper. Further we shall discuss a
formulation of weak sub- and supersolutions which allows, by proper
choices of certain convex sets involved, for existence results for all
the types of boundary conditions considered heretofore. The
development follows that given in the recent papers
\cite{le:sst04, le:sgc05,schmitt:pss04}.
We remark here that certain existence results are also possible in the
presence of non ordered pairs of sub- and supersolutions and refer
to \cite{decoster:oml01, decoster:far02,gossez:nlu94} for
some such results.
There are three recent monographs which cover this theory in
considerable depth and provide a wealth of applications; they are \cite{carl:nvp06}, \cite{decoster:tpb06},
and \cite{du:ost06}.
\section{Sub- and supersolutions}
\subsection{General remarks}
As said, we are interested here in sub-supersolution results for boundary value problems with second order principal operators and
general boundary conditions, where
the problems may or may not contain obstacles or constraints. We
shall, following \cite{le:sst04, le:sgc05}, give the weak (variational) formulation
of the problem, and deduce that the boundary conditions
(or at least parts of them) may usually be encoded into the set of test
(admissible) functions.
With this in mind one can show that in several cases (covering those
that have been studied in the literature), by formulating the problem
as a variational inequality,
even if it is a smooth equation, simple, unified, and general definitions of
sub- and supersolutions are possible. These concepts
of sub- and supersolutions extend the classical definitions for
equations subject to Dirichlet, Neumann, Robin, or No-Flux
(periodic boundary conditions for the one space dimensional problem)
boundary conditions (see e.g. \cite{schmitt:bvp78}).
The definitions are motived by the recent definitions of sub-supersolutions
for variational inequalities in \cite{le:sst04, le:ssm01, le:sse01}.
Another byproduct of the unified approach is that one can demonstrate
the existence of solutions and extremal solutions between
sub- and supersolutions and other
properties of the solution sets when sub- and supersolutions exist.
Since the problems considered are variational inequalities with a
closed convex constraint set $K$, by suitably chosing $K$ one may even
deduce sub-and supersolution results for finite difference
equations. This topic appears worth pursuing.
\subsection{Definitions}
Let $\Omega\subset \mathbb{R}^N$ be a bounded domain with Lipschitz boundary
and $W^{1,p}(\Omega)$ be the usual first-order
Sobolev space with the norm
\begin{equation}
\label{1}
\| u\| = \| u\|_{W^{1,p}(\Omega)} = \left(\| u\|_{L^p(\Omega)}^p
+ \| |\nabla u| \|_{L^p(\Omega)}^p\right)^{1/p},\quad u\in W^{1,p}(\Omega) .
\end{equation}
Assume that $K$ is a closed, convex subset of $W^{1,p}(\Omega)$,
and consider the following variational inequality on
$K$:
\begin{equation}
\label{2}
\begin{gathered}
\int_\Omega A(x,\nabla u)\cdot (\nabla v-\nabla u) dx + \int_\Omega f(x,u)(v-u) dx \\
+ \int_{\partial\Omega} g(x,u) (v-u)\,dS \ge 0,\quad \forall v\in K \\
u\in K .
\end{gathered}
\end{equation}
To simplify the notation we use $u$ and $v$ instead of $u|_{\partial\Omega}$ and $v|_{\partial\Omega}$ for the trace of
$u$ and $v$ on $\partial\Omega$ in the surface integral in (\ref{2}).
In the variational inequality (\ref{2}), $A$ is an elliptic
operator, $f$ is the lower order term, and $g$ is a boundary term.
As remarked above, different boundary conditions require different definitions of
sub- and supersolutions. As a consequence, separate arguments and calculations are needed to study the existence
and properties of solutions between sub- and supersolutions. In what follows, we show that common, unified
definitions of sub- and supersolutions may be given for various types of boundary conditions (including unilateral constraints).
Thus a common, comprehensive general existence theorem is possible for many different types of boundary value problems.
The discussion to follow is motivated by and also generalizes the work in the papers \cite{le:sst04, le:ssm01, le:sse01}.
We first give the assumptions on the principal operator:
$$
A: \Omega\times \mathbb{R}^N \to \mathbb{R}
$$
is a Carath\' eodory function satisfying the growth condition
\begin{equation}
\label{3}
|A(x,\xi) | \le a_1 (x) + b_1 |\xi |^{p-1}, \quad
\mbox{ for a.e.\ $x\in \Omega$, all $\xi\in \mathbb{R}^N$},
\end{equation}
with $p\in [1,\infty)$ (fixed), $a_1\in L^{p'}(\Omega)$, $\frac{1}{p}+\frac{1}{p'}=1$, and $b_1 >0$. Moreover, $A$ is monotone, i.e.,
\begin{equation}
\label{4}
\left (A(x,\xi) - A(x,\xi ')\right )\cdot (\xi-\xi') \ge 0, \quad
\mbox{ for a.e.\ $x\in \Omega$, all $\xi , \xi '\in \mathbb{R}^N$} ,
\end{equation}
and $A$ is coercive in the following sense: There exist $a_2\in L^1(\Omega)$ and $b_2 >0$ such that
\begin{equation}
\label{5}
A(x,\xi)\cdot \xi \ge b_2 |\xi|^{p} - a_2(x), \quad
\mbox{ for a.e.\ $x\in \Omega$, all $\xi\in \mathbb{R}^N$}.
\end{equation}
We also suppose that $f : \Omega\times\mathbb{R} \to \mathbb{R}$
and
$g : \partial \Omega\times\mathbb{R} \to \mathbb{R}$
are Carath\'eodory functions subject to certain growth conditions to be
specified later.
We shall use the standard notation $u \wedge v = \min\{ u, v\}$, $u \vee v = \max\{ u, v\}$,
$U*V = \{ u*v : u\in U, v\in V\}$, and $u*V = \{ u\}* V$, where
$u,v\in W^{1,p}(\Omega)$, $U,V\subset
W^{1,p}(\Omega)$ and $*\in \{\wedge, \vee\}$.
The following are the definitions of sub- and supersolutions of (\ref{2}).
\begin{definition} \label{def6} \rm
A function $\underline{u}\in W^{1,p}(\Omega)$ is called a subsolution of (\ref{2})
if the following conditions are satisfied:
\begin{equation}
\label{7}
f(\cdot, \underline{u})\in L^q(\Omega) ,\quad
g(\cdot, \underline{u})\in L^{\tilde{q}}(\partial\Omega),
\end{equation}
where $q\in (1, p^*)$ and $\tilde{q}\in (1,\tilde{p}^*)$,
\begin{equation}
\label{8}
\underline{u}\vee K \subset K ,
\end{equation}
and
\begin{equation} \label{9}
\int_\Omega A(x,\nabla \underline{u})\cdot \nabla (v-\underline{u}) dx + \int_\Omega f(x,\underline{u}) (v-\underline{u}) dx +
\int_{\partial\Omega} g(x,\underline{u}) (v-\underline{u})\,dS \ge 0, \\
\end{equation}
for all $v\in \underline{u} \wedge K$.
\end{definition}
Here, $p^*$ is the Sobolev conjugate exponent of $p$
$$
{p}^* =\begin{cases}
\frac{Np}{N-p} & \mbox{if } N > p \mbox{ and } N>1 \\
\infty & \mbox{if } N \le p \mbox{ or } N = 1
\end{cases}
$$
and
$$
\tilde{p}^* =
\begin{cases}
\frac{(N-1)p}{N-p} & \mbox{if } N > p \mbox{ and } N>1 \\
\infty & \mbox{if } N \le p \mbox{ or } N = 1 .
\end{cases}
$$
We have a similar definition for supersolutions of (\ref{2}).
\begin{definition} \label{def6-super} \rm
A function $\overline{u}\in W^{1,p}(\Omega)$ is called a supersolution
of (\ref{2}) if the following conditions are satisfied:
\begin{equation}
\label{7-super}
f(\cdot, \overline{u})\in L^q(\Omega) , \quad
g(\cdot, \overline{u})\in L^{\tilde{q}}(\partial\Omega),
\end{equation}
where $q\in (1, p^*)$ and $\tilde{q}\in (1,\tilde{p}^*)$,
\begin{equation}
\label{8-super}
\overline{u}\wedge K \subset K ,
\end{equation}
and
\begin{equation}
\label{9-super}
\int_\Omega A(x,\nabla \overline{u})\cdot \nabla (v-\overline{u}) dx + \int_\Omega f(x,\overline{u}) (v-\overline{u}) dx +
\int_{\partial\Omega} g(x,\overline{u}) (v-\overline{u})\,dS \ge 0,
\end{equation}
for all $v\in \overline{u} \vee K $.
\end{definition}
The following is the main general existence theorem; its proof
is patterned after the arguments used in
\cite{le:bvp98, le:sst04, le:ssm01} and may be found in \cite{le:sgc05}.
\begin{theorem}\label{thm10}
Assume there exists a pair of sub- and supersolution of (\ref{2}) such
that $\underline{u}\le \overline{u}$ and that $f$ and $g$
satisfy the following growth conditions between $\underline{u}$ and $\overline{u}$:
\begin{equation}
\label{11}
|f(x,u) | \le a_3 (x), \quad | g(\xi , v) | \le \tilde{a}_3 (\xi) ,
\end{equation}
for almost all $x\in \Omega$, $\xi\in \partial\Omega$,
all $u\in [\underline{u}(x), \overline{u}(x)]$,
$v\in [\underline{u}(\xi), \overline{u}(\xi)]$, where $a_3\in L^{q'}(\Omega)$,
$\tilde{a}_3\in L^{\tilde{q}'}(\partial\Omega)$, $q\in (1, p^*)$,
$\tilde{q}\in (1, \tilde{p}^*)$, and $p^*$,
$\tilde{p}^*$ are defined as in Definition \ref{def6}.
Then, there exists a solution $u$ of (\ref{2}) such that
$\underline{u}\le u\le \overline{u}$.
\end{theorem}
The above result has the the following generalization. The
proof again follows ideas already used in
\cite{kura:wss89, le:bvp98} and is given in \cite{le:sgc05}.
\begin{theorem} \label{thm30}
Assume that $\underline{u}_1, \dots, \underline{u}_k$
(resp.\ $\overline{u}_1, \dots , \overline{u}_m$) are subsolutions
(resp. supersolutions) of (\ref{2}) such that
\begin{equation} \label{31}
\underline{u}_0 := \max\{\underline{u}_1, \dots, \underline{u}_k\}
\le \min\{\overline{u}_1, \dots , \overline{u}_m\} =: \overline{u}_0 ,
\end{equation}
and that $f$ and $g$ have the following growth conditions between
the sub- and supersolutions:
\begin{equation} \label{32}
|f(x,u)|\le a_3(x),\quad | g(\xi , v) | \le \tilde{a}_3 (\xi) ,
\end{equation}
for a.a.\ $x\in \Omega$, $\xi\in \partial\Omega$,
all $u\in [\min\{\underline{u}_1 (x), \dots, \underline{u}_k(x)\}$,
$\max\{\overline{u}_1(x), \dots , \overline{u}_m(x)\}]$,
all $v\in [\min\{\underline{u}_1 (\xi), \dots, \underline{u}_k(\xi)\}$,
$\max\{\overline{u}_1(\xi), \dots , \overline{u}_m(\xi)\}]$, where
where $a_3$ and $\tilde{a}_3$ are as in Theorem \ref{thm10}.
Then, there exists a solution $u$ of (\ref{2}) such
that $\underline{u}_0\le u\le \overline{u}_0$.
\end{theorem}
\begin{remark} \label{rem41a} \rm
(a) The above theorem suggests more general definitions of sub-
and supersolutions. Namely: An element $\alpha \in W^{1,p}(\Omega)$
is a subsolution if it is the supremum of a finite number of
functions each of which is a subsolution satisfying Definition
\ref{def6} and an element $\beta \in W^{1,p}(\Omega)$ is a
supersolution if it is the infimum of a finite number of
supersolutions each of which is a supersolution satisfying
Definition \ref{def6-super}. In this case the set of subsolutions
is closed with respect to the operation $\vee $ and the set of
supersolutions is closed with respect to the operation $\wedge $,
and, of course, Theorem \ref{thm30} is simply a restatement of
Theorem \ref{thm10}. Thus, if we let $\mathcal S$ be the set of
solutions of (\ref{2}) between $\underline{u}_0$ and $\overline{u}_0$. Theorem
\ref{thm30} means that $\mathcal S\not= \emptyset$ and
under the above assumptions, one can prove
(cf. \cite {le:bvp98, le:sse01}) that ${\mathcal S}$ is compact
and directed. As a consequence, ${\mathcal S}$ has greatest (the
supremum of all subsolutions) and smallest (the infimum of all
supersolutions) elements with respect to the standard ordering,
which are the extremal solutions of (\ref{2}) between $\underline{u}_0$
and $\overline{u}_0$. Such results also have a long history and likely
go back to \cite{ako:dpq61}, see also
\cite{dancer:emw89,kura:wss89, schmitt:bvp78}.
(b) If only a subsolution (or a supersolution) of (\ref{2})
exists and $f$ and $g$ satisfy certain one-sided
growth conditions then we can also show the existence of solutions
of (\ref{2}) above the subsolution (or below the supersolution).
We can also show the existence of a minimal solution above that subsolution
(or a maximal solution below that supersolution) (see e.g. \cite{le:sse01}).
(c) The question of the structure of the set of all
solutions which lie between a given pair of sub-and supersolutions has
been addressed in the literature often with the latest results given
in \cite{le:pat05}. This paper's bibliography also provides a fairly
complete list of references for this problem.
\end{remark}
\section{Some examples}
\subsection{Problems subject to Dirichlet boundary conditions}
\quad\\ Consider the boundary-value problem
\begin{gather} \label{42}
-{\rm div}[A(x,\nabla u)] + f(x,u) = 0 \quad\mbox{in } \Omega ,
\\ \label{43}
u = 0 \quad\mbox{on } \partial\Omega ,
\end{gather}
the variational form of which is the inequality (\ref{2}) with $g=0$ and
\begin{equation} \label{44}
K = W_0^{1,p}(\Omega),
\end{equation}
which is equivalent to the variational equality:
\begin{gather*}
\int_\Omega A(x,\nabla u)\cdot \nabla v dx + \int_\Omega f(x,u) v dx = 0, \quad
\forall v\in W^{1,p}_0(\Omega) \\
u \in W^{1,p}_0(\Omega) .
\end{gather*}
In this case, for assumption (\ref{8})
(respectively (\ref{8-super})) to be fulfilled, we need that
\begin{equation} \label{45}
\underline{u}\le 0 \quad \mbox{on } \partial\Omega \quad
\mbox{(respectively $\overline{u}\ge 0$ on $\partial\Omega$).}
\end{equation}
Concerning condition (\ref{9}), it can be checked that the set
$\{v-\underline{u} : v\in \underline{u} \wedge W^{1,p}_0(\Omega)\}$
is dense in the negative cone of $W^{1,p}_0(\Omega)$:
$$
W^{1,p}_-(\Omega) := \{ w\in W^{1,p}_0(\Omega) : w \le 0 \quad\mbox{a.e.\ on } \Omega \}.
$$
Therefore, condition (\ref{9}), in this particular case,
becomes the following condition
\begin{equation}
\label{46}
\int_\Omega A(x,\nabla \underline{u})\cdot \nabla v dx + \int_\Omega f(x, \underline{u}) v dx \ge 0,
\quad \forall v\in W^{1,p}_0(\Omega), \; v\le 0 \mbox{ on } \Omega .
\end{equation}
In view of (\ref{45}) and (\ref{46}), we re-obtain the classical concept
of sub- and supersolution for
equations with homogeneous Dirichlet boundary condition
(cf.\ e.g.\ \cite{hess:sne76, dancer:emw89, kura:wss89, le:bvp98}).
For problems with nonhomogeneous Dirichlet conditions, we have equation (\ref{42}) together with
\begin{equation}
\label{47}
u = h \quad \mbox{on } \partial\Omega ,
\end{equation}
instead of (\ref{43}), where $h\in W^{1-\frac{1}{p}, p}(\partial\Omega)$ is
the trace of a function in $W^{1,p}(\Omega)$,
still denoted by $h$, for simplicity. In this case, problem
(\ref{42})--(\ref{47}) is, in the variational form,
the inequality (\ref{2}) with $g=0$ and
$$
K = \{h\}\oplus W^{1,p}_0(\Omega)=\{u\in
W^{1,p}(\Omega):u=h,~\mbox{on}~\partial \Omega\}.
$$
The condition $\underline{u}\vee K \subset K$ is satisfied if and only if $\underline{u}$ satisfies
the boundary condition $\underline{u} \le h$ a.e.\ on $\partial\Omega$. The set
$$
\{ v-\underline{u} : v\in\underline{u} \wedge [\{ h\}\oplus W^{1,p}_0(\Omega)]\}
= \{ w - (\underline{u} -h) : w\in (\underline{u} -h) \wedge W^{1,p}_0(\Omega)\}
$$
is dense in the negative cone $W^{1,p}_-(\Omega)$ (because $\underline{u}-h \le 0$ on $\partial\Omega$).
Condition (\ref{9}) is again equivalent to (\ref{46}).
\subsection{Problems with Neumann and Robin boundary conditions}
In the case where $K = W^{1,p}(\Omega)$, (\ref{2}) reduces to the
variational equality
\begin{equation} \label{48}
\begin{gathered}
\int_\Omega A(x,\nabla u)\cdot \nabla v dx + \int_{\Omega} f(x,u) v dx
+ \int_{\partial\Omega} g(x,u) v\,dS = 0,
\quad \forall v\in W^{1,p}(\Omega) \\
u\in W^{1,p}(\Omega) ,
\end{gathered}
\end{equation}
which is the weak form of the boundary value problem
\begin{gather*}
-{\rm div} [A(x,\nabla u)] + f(x,u) = 0 \quad\mbox{in } \Omega \\
A(x,\nabla u) \cdot n = - g(x,u) \quad\mbox{on } \partial\Omega .
\end{gather*}
When $g=0$ on $\partial \Omega$, we have a homogeneous Neumann boundary condition.
Otherwise, one has a nonhomogeneous Neumann boundary condition which
also may depend on $u$. It is clear that condition (\ref{8}) always
holds. Also, for any $\underline{u}$ in $W^{1,p}(\Omega)$, we have
$ \underline{u}\wedge W^{1,p}(\Omega) = \{ v\in W^{1,p}(\Omega) : v\le \underline{u}
\mbox{ a.e.\ on } \Omega \}$.
Therefore, (\ref{9}) is equivalent to the inequality
$$
\int_\Omega A(x,\nabla\underline{u})\cdot \nabla w dx + \int_\Omega f(x,\underline{u}) w dx +
\int_{\partial\Omega} g(x,\underline{u}) w\,dS \ge 0,
$$
for all $w\in W^{1,p}(\Omega)$ such that $w\le 0$ a.e.\ on $\Omega$, which is,
in its turn, equivalent to
\begin{equation} \label{49}
\begin{gathered}
\int_\Omega A(x,\nabla\underline{u})\cdot \nabla w dx + \int_\Omega f(x,\underline{u}) w dx +
\int_{\partial\Omega} g(x,\underline{u}) w\,dS \le 0, \\
\forall w\in W^{1,p}(\Omega), \; w \ge 0 \quad \mbox{a.e.\ on }\Omega .
\end{gathered}
\end{equation}
We have a similar condition for supersolutions of (\ref{48}).
These concepts of sub- and supersolutions here
coincide with the classical ones for sub- and supersolutions in Neumann
problems. Our definitions here also cover the cases
where the Neumann conditions also depend on $u$. In fact,
when $g(x,u) = a u$, we have a Robin boundary
condition.
\subsection{Mixed boundary conditions}
By choosing $K=\{u \in W^{1,p}(\Omega) : u = h \mbox{ on } \Gamma\}$, where $\Gamma$ is a measurable
subset of $\partial\Omega$, we have the equation (\ref{42}) with a mixed boundary
condition consisting of a Dirichlet condition on $\Gamma$ and a
Neumann/Robin condition on $\partial\Omega\setminus \Gamma$.
\subsection{Obstacle problems}
Let $K$ be a convex subset of $W^{1,p}_0 (\Omega)$. The inequality (\ref{2}),
in this case, formulates problems
with unilateral constraints (such as obstacle problems) and
homogeneous Dirichlet boundary conditions, which were
discussed in \cite{le:ssm01}. Many results in that paper are particular
cases of those discussed here. In fact, the general definitions of
sub- supersolutions presented here are motivated in part by
the concepts and arguments in \cite{le:ssm01}.
\subsection{Zero flux and periodic problems}
Let us consider the choice
$$K = \{ u\in W^{1,p}(\Omega) : u = \mbox{const on } \partial\Omega\}.
$$
For $u\in W^{1,p}(\Omega)$, we note that $u\vee K \subset K$ (resp.\ $u\wedge K \subset K$)
if and only if $u\in K$.
In fact, it is clear that if $u\in K$, then $u\vee K, u\wedge K \subset K$. Conversely, assume that
$u\vee K \subset K$. For any constant function $c$, we have
$ u \vee c = \max\{ u, c\} = $ constant on $\partial\Omega$. Therefore, either $u\le c$ a.e.\ on $\partial\Omega$ or
$u\ge c$ a.e.\ on $\partial\Omega$ (with respect to the Hausdorff measure). Since this is true for any $c\in \mathbb{R}$,
we must have $u=$ constant on $\partial\Omega$, that is,
$u\in K$. For $\underline{u}\in K$, we have
\begin{align*}
\underline{u}\wedge K & = \{v\in K : v\le \underline{u} \mbox{ a.e.\ on } \Omega \} \\
& = \{\underline{u}-w : w\in K , w\ge 0 \mbox{ a.e.\ on } \Omega\}.
\end{align*}
Therefore, inequality (\ref{9}) is equivalent to
$$
\int_\Omega A(x,\nabla\underline{u})\cdot \nabla v dx + \int_\Omega f(x,\underline{u}) v dx +
\int_{\partial\Omega} g(x,\underline{u}) v\,dS \le 0,
$$
for all $v\in K$ such that $v\geq 0$ on $\Omega$.
One has a similar equivalence for supersolutions.
Note that in this
particular case, the definitions for sub- and supersolutions here reduce to those in \cite{le:sst04}.
We note that in the case that $g\equiv 0$ the problem considered here is the boundary value problem
\begin{gather*}
-\mathop{\rm div}A(x,\nabla u) +f(x,u)=0\\
u|_{\partial \Omega}=\mbox{constant},\quad
\int _{\partial \Omega} A(x,\nabla u)\cdot n\,dS=0,
\end{gather*}
where the constant boundary data are not specified.
This problem in dimension $N=1$
(the periodic boundary value problem) was first studied by sub- and
supersolution methods by Knobloch \cite{knobloch:enm63}. (See also
\cite{amster:esn04},
\cite{schmitt:pss04}, where free boundary problems of this type are studied.)
\subsection{Unilateral problems}
For another example, let us consider the boundary value problem consisting of
(\ref{42}) and the following
unilateral boundary condition on the boundary:
\begin{equation} \label{50}
\begin{gathered}
u \ge \psi ,\\
A(x,\nabla u) \cdot n \ge 0 , \\
(u - \psi) [A(x,\nabla u) \cdot n] = 0 \quad\mbox{on } \partial\Omega ,
\end{gathered}
\end{equation}
($\psi$ is a measurable function on $\partial\Omega$) which occur in problems
with semi-permeable media. The problem can be formulated as the
variational inequality (\ref{2}) with $g = 0$ and
$$
K = \{ u\in W^{1,p}(\Omega) : u \ge \psi \mbox{ a.e.\ on } \partial\Omega\}.
$$
It is worth noting that in this case, there is a non-symmetry
concerning conditions (\ref{8}) and (\ref{8-super})
in the definitions of sub- and supersolutions. In fact, it is easy to
see that for $\underline{u}, \overline{u}\in W^{1,p}(\Omega)$,
$\underline{u}$ always satisfies (\ref{8}), while (\ref{8-super}) holds if and
only if $\overline{u} \ge \psi$ a.e.\ on $\partial\Omega$,
that is $\overline{u}\in K$.
\section*{Some Remarks}
Problems for which the domain $\Omega $ is unbounded may be tackled in a
similar way by using classical approaches (see e.g. \cite{kura:wss89}).
The above definitions and approach may be extended in a straightforward
manner to problems with lower terms
depending also on the gradient of $u$, i.e., $f = f(x, u , \nabla u)$.
We can also extend them to problems with locally
Lipschitz constraints together with convex constraints
(variational hemivariational inequalities) such as those considered,
for example, in \cite{carl:ssm04} and the references therein.
For some recent applications of the sub- supersolution method in the
study of zero flux problems we refer to \cite{amster:esn04} and
Ambrosetti-Prodi type problems for the p-Laplacian to \cite{koizumi:app05}.
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\end{document}