\boldHuge {\boldmathHuge p}-Laplacian Type Equations Involving Measures
\boldnormal Abstract
\boldnormal Abstract
1104.tex
\input amstex
\input Amstex-document.sty
\widestnumber\key{41}
\nologo
%\NoBlackBoxes
\TagsAsMath
\redefine\div{\operatorname{div}}
\define\osc#1{\operatorname{osc} (#1)}
\define\Ah{$\Cal A$-harmonic}
\define\sh{$\Cal A$-superharmonic}
\define\psh{$p$-superharmonic}
\define\ph{$p$-harmonic}
\define\psu{$p$-superharmonic}
\define\A{\Cal A}
\define\rn{\bold R^n}
\define\re{\bold R}
\redefine\O{\Omega}
\define\dome{\partial\O}
\define\clome{\overline{\O}}
\define\xo{x_0}
\define\kapu{\operatorname{cap}_{p}}
\define\vep{\varepsilon}
\define\loc{\operatorname{loc}}
\define\wo#1{W^{1,p}(#1)}
\define\lwo#1{W^{1,p}_{\loc}(#1)}
\define\woo#1{W_0^{1,p}(#1)}
\define\wome{\wo{\O}}
\define\lwome{W^{1,p}_{\loc}(\O)}
\define\woome{\woo{\O}}
\define\caa#1{C^{\infty}(#1)}
\define\cao#1{C_0^{\infty}(#1)}
\define\vf{\varphi}
\define\wL#1#2{\operatorname{weak}-L^{#1}(#2)}
\define\wLl#1#2{\operatorname{weak}-L^{#1}_{\loc}(#2)}
\font\boldmathHuge = cmmib10 scaled 1748
\font\boldmathlarge = cmmib10 scaled 1200

\def\firstpage{167}
\def\lastpage{176}
\pageno 167
\def\volumeno{III}
\def\shorttitle{$p$-Laplacian Type Equations Involving Measures}
\def\shortauthor{T. Kilpel\"ainen}

\topmatter %
\title\nofrills{\boldHuge {\boldmathHuge p}-Laplacian Type Equations Involving Measures}
\endtitle

\author \Large T. Kilpel\"ainen* \endauthor

\thanks *Department of Mathematics \& Statistics, University of
Jyv\"askyl\"a, PO Box 35, FIN-40014 Jyv\"askyl\"a, Finland.
E-mail: terok\@math.jyu.fi \endthanks

\abstract\nofrills \centerline{\boldnormal Abstract}


\vskip 4.5mm

{\ninepoint This is a survey on problems involving equations $-\operatorname{div}{\Cal A}(x,\nabla u)=\mu$, where
$\mu$ is a Radon measure and ${\Cal A}:\bold {R}^n\times\bold R^n\to \bold R^n$ verifies Leray-Lions type
conditions. We shall discuss a potential theoretic approach when the measure is nonnegative. Existence and
uniqueness, and different concepts of solutions are discussed for general signed measures.


\vskip 4.5mm

\noindent {\bf 2000 Mathematics Subject Classification:} 35J60,
31C45.

\noindent {\bf Keywords and Phrases:} $p$-Laplacian, Quasilinear equations with measures, Nonlinear potential
theory, Uniqueness.}
\endabstract
\endtopmatter

\document

\baselineskip 4.5mm \parindent 8mm

\specialhead \noindent \boldLARGE 1. Introduction \endspecialhead

Throughout this paper we
let $\O$ be an open set in $\rn$ and  $1<p<\infty$ a fixed number.
We shall consider equations
$$
 -\div\A(x,\nabla u)=\mu\,,\tag{1.1}
$$
where $\mu$ is a Radon measure. We suppose that the mapping
$\A\:\rn\times\rn\to\rn$, $(x,\xi)\mapsto \A(x,\xi)$,
is measurable in $x$ and continuous in $\xi$ and that it verifies
the structural conditions:
$$
\gathered
\Cal A(x,\xi)\cdot \xi\,\ge\lambda|\xi|^p  \,,  %\tag{1.2}
\quad
|\Cal A(x,\xi)|\,\le\,\Lambda|\xi|^{p-1}, \quad\text{ and }\\
\left(\Cal A(x,\xi)-\Cal A(x,\zeta)\right)\cdot(\xi-\zeta)\,>\,0\,,
\endgathered
\tag{1.2}
$$
for  a.e\. $x\in \rn$ and all $\xi\ne \zeta\in\rn$.
A prime example of the operators is the $p$-Laplacian
$$
- \Delta_p u=-\div(|\nabla u|^{p-2}\nabla u).
$$

In Section 2 we discuss how nonlinear potential theory is
related to equations like (1.1); it corresponds to nonnegative
measures.
Then in
Section 3
we discuss the existence and uniqueness for  (1.1)
with general measures.



\specialhead \noindent \boldLARGE 2. Potential theoretic approach
\endspecialhead

A continuous solution $u\in\lwome$ of
$ -\div\A(x,\nabla u)=0$
is called $\A$-{\it harmonic} in $\O$. An  $\A$-{\it superharmonic}
function  in $\O$ is a lower semicontinuous function
$u\:\O\to\re\cup\{\infty\}$ that is not identically $\infty$ in any
component of $\O$ and that obeys the following comparison property:
{\sl for each open $D\subset\subset\O$ and each $h\in C(\bar D)$,
\Ah\ in $D$, the inequality $u\ge h$ on $\partial D$ implies
$u\ge h$ in $D$.}

For $k>0$ and $s\in\re$, let
$$
  T_k(s)=\max\big(-k,\min(s,k)\big)
$$
be the truncation operator. Then we have

{\bf 2.1. Theorem.} \cite{12, 26, 15}
{\it If $u$ is \sh\ in $\O$, then
$$
 T_k(u)\in\lwome\quad\text{ for all }k>0\,.
$$}

This enables us to show  that \sh\ functions have a ``gradient'':
Suppose that a function $u$ that is finite a.e\. has the property that
$T_k(u)\in\lwome$ for all $k>0$. Then we define the (weak) {\it gradient}
of $u$ as
$$
\nabla u(x)= DT_k(u)\quad\text{ if } |u(x)|<k\,.
$$
Here $ DT_k(u)$ is the distributional gradient of the Sobolev
function $T_k(u)\in\lwome$. Then $\nabla u$ is well defined. Observe that
if $\nabla u$ is locally integrable, then it is the distributional
derivative of $u$. However, it may happen that $u$ or $\nabla u$ fails to
be locally integrable and so $\nabla u$ is not always distributional
derivative, see \cite{15}, \cite{8}; this is a real issue only for
$p<2-1/n$.


{\bf 2.2. Theorem.} \cite{26, 12, 15, 10, 1}
{\it Suppose that $u$ is \sh\ in $\O$.
\roster
\item"i)" If $1<p<n$, then
$$
u\in \wLl{n(p-1)/(n-p)}{\O}\quad \text{ and }\quad \nabla u\in
\wLl{n(p-1)/(n-1)}{\O}.
$$
\item"ii)" If $p=n$, then $u$ is locally in $BMO$ and hence
$u\in L^q_{\operatorname{loc}}(\O)$ for all $q>0$;
moreover
$$
\nabla u\in \wLl{n}{\O}.
$$
 \item"iii)" If $p>n$, then
$$
u\in \lwome
$$
and hence $u$ is (H\"older) continuous.
\endroster
}


If $p>n$ \sh\ functions are continuous and locally in $W^{1,p}$, moreover
Radon measures then are in the dual of Sobolev space $W^{1,p}$. This makes
the cases $p>n$ very special and quite simple by the classical
results of Leray and Lions. Henceforth we shall be
concerned mainly with cases $1<p\le n$.

For \sh\ $u$ we have by Theorem
2.2 that $|\nabla u|^{p-1}$ is locally integrable. So the
distribution
$$
-\div\A(x,\nabla u)(\vf):=\int_{\O}\A(x,\nabla u)\cdot\nabla\vf\,dx
\,,\quad\vf\in \cao{\O}\,,
$$
is well defined.
\sh\ functions give rise to equation (1.1).

{\bf 2.3. Theorem.} \cite{18}
{\it If $u$ is \sh\ in $\O$, then $-\div\A(x,\nabla u) $
is represented by a nonnegative Radon
measure $\mu$.
}

As to the existence we have:


{\bf 2.4. Theorem.} \cite{18, 3}
{\it Given a finite nonnegative Radon measure $\mu$ on bounded $\O$,
there is an \sh\
function $u$ in $\O$ such that
$$
\cases
-\div\A(x,\nabla u)=\mu\\
T_k(u)\in\woome\text{ for all }k>0\,.
\endcases\tag{2.5}
$$
}

If $\O$ is undounded, then there also is an \sh\ solution to (1.1).
This is rather easily seen if $1<p<n$. The case $p\ge n$ requires a
more careful analysis, see \cite{9, 17}.


In light of Theorem 2.2 we have that a solution $u$ to
(2.5) satisfies
$$
u\in W^{1,q}_0(\O)\quad\text{ for all } q<\frac{n(p-1)}{n-1}
$$
(for $p>2-1/n$).
One naturally asks if such a solution is unique. Unfortunately this is
not the case in general (except for $p>n$).
To see this consider the function
$$
 u(x)=\cases |x|^{\frac{p-n}{p-1}} -1&\text{ if }1<p<n ,\\
  -\log|x|&\text{ if }p=n\,.
\endcases
$$
In the $p$-Laplacian cases,  $u$ is then
a $p$-superharmonic solution to (2.5)
with $\mu$=0 in the punctured
ball $\O=B(0,1)\setminus\{0\}$, but $v\equiv 0$ is
another solution. This is rather artificial example but there are more
severe ones. The question
of uniqueness is a real issue to which we shall return in Section
3 below.

\specialhead \noindent \boldlarge Regularity and estimates
\endspecialhead


In the classical potential theory
the uniqueness can be solved by the aid of the Riesz decomposition theorem
which states that superharmonic functions are sums of a potential and
a harmonic function. No such decomposition is available in the nonlinear
world. However, this lack can be compensated for to an extent by estimating
in terms of the {\it Wolff potential} of the measure $\mu$,
$$
{\bold W}^{1,p}_{\mu}(x_0,r)=
\int_0^r\left(\frac{\mu(B(x_0,t))}{t^{n-p}}\right)^{\frac 1{p-1}}\frac{dt}{t}\,.
$$

{\bf 2.6. Theorem.} \cite{19}
{\it Let $u$ be a nonnegative \sh\ function in $B(\xo,3r)$ with
$\mu=-\div\A(x,\nabla u)$.
Then
$$
 c_1{\bold W}^{1,p}_{\mu}(x_0,r)
\le u(x_0)\le  c_2{\bold W}^{1,p}_{\mu}(x_0,2r)
 + c_3\inf_{B(x_0,r)}u\,,
$$
where $c_j=c_j(n,p,\lambda,\Lambda)>0$.
}

Theorem 2.6 was discovered by the author with Mal\'y
\cite{18, 19} and later generalized for equations depending also on
$u$  by Mal\'y and Ziemer \cite{30}. Mikkonen \cite{32} worked out
the argument for weighted operators, this was later written up in a
metric space setup in \cite{2}.
Recently a totally different proof that works for quasilinear subelliptic
operators was found by Trudinger and Wang \cite{39}. Labutin \cite{22}
gave a generalization for $k$-Hessian operators.


As the first major application
of the potential estimate in 2.6 the author
and Mal\'y
established the necessity of the Wiener test for the regularity
for the Dirichlet problem: We say that $\xo\in\dome$
is an $\A$-{\it regular} boundary point of bounded $\O$ if
$$
\lim_{x\to\xo}u(x)=\vf(\xo)
$$
whenever $\vf\in\caa{\rn}$ and $u$ is \Ah\ in $\O$ with
$u-\vf\in\woome$. Then it turns out that regularity is independent
of the particular operator and it depends only on its type $p$.
More precisely,
define the $p$-{\it capacity} of the set $E$ as
$$
\kapu(E):=\inf\int_{\rn}|v|^p+|\nabla v|^p\,dx\,,
$$
where the infimum is taken over all
$v\in W^{1,p}(\rn)$ such that
$v\ge 1$  on an open neighborhood of  $E$. Then

{\bf 2.7. Theorem.} \cite{31, 19}
{\it A boundary point $\xo\in\dome$ is regular if and only if
$$
\int_0^1\left(\frac{\kapu(\complement\O\cap
B(x_0,t))}{t^{n-p}}\right)^{\frac 1{p-1}}
\frac{dt}{t}=\infty\,.
$$
}

Maz'ya \cite{31} introduced the Wiener type test in 2.7 and
proved its sufficiency. Gariepy and Ziemer \cite{11} generalized the result
by a different argument. Lindqvist and Martio \cite{27}
proved the necessity for $p>n-1$; the general case was treated in \cite{19}.
For generalizations see \cite{30, 32, 39, 22}.

Theorem 2.6 can also be used to characterize singular solutions
and (H\"older) continuity of \sh\ functions (see
\cite{17, 19, 21}). Recall that \Ah\ functions are locally
H\"older continuous: there is a constant $\varkappa\in (0,1]$ depending
only on the structure such that
$$
\osc{u,B(x,r)}\le c(\frac rR)^{\varkappa}
\osc{u,B(x,R)}\tag{2.8}
$$
whenever $u$ is \Ah\ in $B(x,R)$, $r<R$ \cite{37, 15}.
For the $p$-Laplacian $\varkappa=1$.
We have

 {\bf 2.9. Theorem.} \cite{21}
{\it Suppose that $u\in\lwome$ is \sh\ with $\mu=-\div\A(x,\nabla u)$. If $u$
is $\alpha$-H\"older continuous, then
$$
 \mu\big(B(x,r)\big)\le c\,r^{n-p+\alpha(p-1)}\quad
\text{ whenever }\quad B(x,2r)\subset\O.
$$
The converse holds if $\alpha<\varkappa$.
}

The case $\alpha=\varkappa$
is quite different and it is not yet well understood.

Rakotoson and Ziemer \cite{36} proved that the condition of Theorem
2.9 for the measure gives the H\"older continuity of
the solution with some exponent. Lieberman \cite{25} showed that
for smooth operators like the $p$-Laplacian  the condition
$ \mu\big(B(x,r)\big)\le c\,r^{n-1+\vep}$ for some $\vep>0$
implies that the solution is in $C^{1,\beta}$.



Theorem 2.9 can be employed to establish the following
removability result which is due to Carleson \cite{5} in the Laplacian
case.


{\bf 2.10. Theorem.} \cite{21}
{\it Suppose that $u\in C^{0,\alpha}(\O)$ is \Ah\ in $\O\setminus E$.
If $E$ is of $n-p+\alpha(p-1)$ Hausdorff measure zero, then $u$ is \Ah\
in $\O$.

If $E$ is of positive $n-p+\alpha(p-1)$ Hausdorff measure and
$\alpha<\varkappa$, then there is $u\in C^{0,\alpha}(\O)$ that is \Ah\ in
$\O\setminus E$ but not
in the whole $\O$.
}

\specialhead \noindent \boldLARGE 3. General Radon measures
\endspecialhead


In this section we let $\mu$ be any signed Radon measure and consider equation
(1.1). More specifically, we shall discuss the problem
$$
\cases
-\div\A(x,\nabla u)=\mu\\
u=0\text{ on } \dome\,
\endcases\tag{3.1}
$$
where $\O$ is a bounded domain in $\rn$. Here the equation is understood in the
distributional sense, i.e\.
$$
 \int_{\O}\A(x,\nabla u)\cdot\nabla\vf\,dx=\int_{\O}\vf\,d\mu\,,\quad\vf\in\cao{\O},
$$
where we, of course,
assume that $x\mapsto \A(x,\nabla u)$ is locally integrable.
The boundary values $u=0$ are assumed in a weak Sobolev space sense.
The existence of the solution to this problem is known:

{\bf 3.2. Theorem.} \cite{3, 8, 9, 10}
{\it For each Radon measure $\mu$ of finite total variation,
there is a solution $u$ to (3.1) such that
\roster
\item"i)" the truncations
$$
T_k(u)\in\woome\text{ for all }k>0\,,
$$
\item"ii)" $$
u\in \wL{n(p-1)/(n-p)}{\O}\text{ if }1<p<n \quad
\text{ and }\quad
u\in\operatorname{BMO}\text{ if }p=n\,,
$$
\item"iii)"
 $$
\nabla u\in \wL{n(p-1)/(n-1)}{\O}.
$$
\endroster
}



We next discuss the uniqueness of such a solution.
There are examples of mappings $\A$ for which there is a solution $u$
of equation
$\A(x,\nabla u)=0$
such that $u$ satisfies ii) and iii) of Theorem 3.2, but
fails to be \Ah, see \cite{33, 38, 16, 29}.
See also the nonuniqueness example in an irregular domain
after Theorem 2.4 above.


There are various approaches trying to treat the uniqueness problem
by attaching additional attributes to the solution. To formulate these
we need to recall a decomposition of measures.
The $p$-capacity, defined in Section 2, is an outer measure.
Hence the usual proof of the Lebesgue decomposition theorem gives us
that any Radon measure $\mu$ can be decomposed as
$$
  \mu =\mu_0+\mu_s\,,
$$
where $\mu_0$ and $\mu_s$ are Radon measures such that
$\mu_0$ is {\it absolutely continuous} with respect to the
$p$-capacity (i.e., $\mu_0(E)=0$ whenever $\kapu(E)=0$) and
$\mu_s$ is {\it singular} with respect to the
$p$-capacity (i.e., there is a Borel set $B$ such that  $\kapu(B)=0$
and $\mu_s(E\setminus B)=0$ for all $E$).


Let  $u$ be a solution to (3.1) described in Theorem
3.2. We say that
\roster
\item"-" $u$ is an {\it entropy solution} of (3.1) if $u$ is Borel
measurable and
$$
\int_{\O}\A(x,\nabla u)\cdot\nabla T_k(u-\vf)\,dx
\le\int_{\O}  T_k(u-\vf) \,d\mu
$$
for all $\vf\in\cao{\O}$ and $k>0$.
\item"-"  $u$ is a {\it renormalized solution} of (3.1)
if for all $h\in W^{1,\infty}(\re)$ such that $h'$ has
compact support we have
$$
\align
\int_{\O}&\A(x,\nabla u)\cdot\nabla u h'(u)\vf\,dx
+\int_{\O}\A(x,\nabla u)\cdot h(u)\nabla \vf\,dx
\\&=\int_{\O}  h(u)\vf \,d\mu_0
+h(\infty)\int_{\O}  \vf \,d\mu_s^+-
h(-\infty)\int_{\O}  \vf \,d\mu_s^-
\endalign
$$
whenever
$\vf\in W^{1,r}(\O)\cap L^{\infty}(\O)$ with $r>n$ is such that
  $h(u)\vf\in \woome$;
here
$$
h(\infty)=\lim_{t\to\infty} h(t)\,,\quad
h(-\infty)=\lim_{t\to-\infty} h(t)\,,
$$
and
$\mu_s^+$ and $\mu_s^-$ are the positive and negative parts
of the singular measure $\mu_s$.
\endroster

Observe that we assume here that $u$ satisfies equation (3.1)
in the distributional sense. The entropy condition in this context was
first used in \cite{1}; the renormalized solution was introduced by
Lions and Murat \cite{28} and in the refined form in \cite{7, 8}.
The artificial function we had as a counterexample for the uniqueness
(after Theorem 2.4) is not entropy nor renormalized solution.
Existence is known; most existence proofs follow a similar idea to that
in \cite{3}. The uniqueness can be established for certain measures since
one can use truncated solutions as test functions.


{\bf 3.3. Theorem.} \cite{4, 20, 8, 40}
{\it Suppose that $\mu$ is a finite Radon measure. Then
there is a renormalized and an entropy solution of
(3.1). Moreover such a solution is unique if
$\mu$ is absolutely continuous with respect to the $p$-capacity.
}


The uniqueness with $\mu\in L^1$ was proved in \cite{1} and \cite{28}
see also \cite{33}, \cite{34}. For measures absolutely continuous
wrt $p$-capacity, the uniqueness is established e.g. in
 \cite{4, 20, 8, 40}.



A renormalized solution is always an entropy solution,
whence the concepts coincide at least if $\mu$ is
absolutely continuous wrt $p$-capacity; see \cite{8, 6}.


In case of a general measure
the uniqueness of renormalized solution appears to be an open
problem. There are some partial results: Assume that the following strong
monotoneity assumption holds. For all $0\not=\xi,\eta\in\rn$
$$
\big(\A(x,\xi)-\A(x,\eta)\big)\cdot(\xi-\eta)
\ge\cases
\beta|\xi-\eta|^p&\text{ if }p\ge 2,\\
\beta\dfrac{|\xi-\eta|^2}{(|\xi|+|\eta|)^{2-p}}&\text{ if }p< 2.
\endcases\tag{3.4}
$$
Assume also the H\"older continuity:
$$
|\A(x,\xi)-\A(x,\eta)|
\le\cases
\gamma\big(b(x)+|\xi|+|\eta|\big)^{p-2}|\xi-\eta|^2&\text{ if }p\ge 2,\\
\gamma |\xi-\eta|^{p-1} &\text{ if }p< 2\,,
\endcases\tag{3.5}
$$
where $b\in L^p$ is nonnegative. For instance,
the $p$-Laplacian satisfies these assumptions.
Then we have:

{\bf 3.6. Theorem.} \cite{7, 8, 14}
{\it Suppose the additional assumptions (3.4) and
(3.5) hold. If $u$ and $v$ are two renormalized solutions of
(3.1) with measure $\mu$ such that either
$\nabla u-\nabla v\in L^p(\O)$ or $u-v$ is bounded from one side,
then $u=v$.
}



Rakotoson \cite{35} proved that a continuous renormalized solution is
unique in smooth domains; continuity requires the measure be rather special.

\specialhead \noindent \boldlarge Borderline case {\boldmathlarge p} = {\boldmathlarge n}
\endspecialhead

We close this paper by considering the special case when $p=n$.
Then the uniqueness can be reached:

{\bf 3.7. Theorem.} \cite{41, 13, 10}
{\it Suppose that $\A$ verifies
additional assumption (3.4)
with $p=n$ and that $\O$ is bounded and regular.
For each Radon measure $\mu$ of finite total variation,
there is a unique solution $u$ to (3.1) such that
\roster
\item"i)" the truncations
$$
T_k(u)\in W^{1,n}_0(\O)\text{ for all }k>0\,,
$$
\item"ii)" $u$ is in $\operatorname{BMO}$, and
 $$
\nabla u\in \wL{n}{\O}.
$$
\endroster
}

The regularity of $\O$ refers to the fact that the complement of $\O$
needs to be thick enough to exclude counterexamples we had in Section
2. Zhong \cite{41} formulated a weak condition for this by
requiring that the complement of $\O$ is {\it uniformly} $p$-{\it thick},
i.e\., $  \kapu(\complement\O\cap B(x,r))\approx r^{n-p}$
for all small $r>0$;
see \cite{23, 15, 32} for more information
about uniform thickness.

In fact stronger uniqueness properties than in Theorem 3.7 hold:
by using a Hodge decomposition argument
Greco, Iwaniec, and Sbordone \cite{13} proved that for $p$-Laplacian
the solution is unique
in the {\it grand Sobolev space}  $W^{1,n)}(\O)$, i.e\.
$$
u\in{\tsize\bigcap\limits_ {q<n}}
            W^{1,q}_0(\O)\quad
\text{ and }\quad\sup_{\vep>0}\vep\int_{\O}|\nabla u|^{n-\vep}\,dx<\infty\,.
$$

The regularity $\nabla u\in \wL{n}{\O}$
in Theorem 3.7  (proved in \cite{10}) is better than
$u\in W^{1,n)}(\O)$.

By using a maximal function argument similar to that introduced
by Lewis \cite{24},
Zhong \cite{41} proved a stronger uniqueness result: the solution is
unique in
$$
u\in{\tsize\bigcap\limits_ {q<n}}
            W^{1,q}_0(\O)\,.
$$
Even stronger result appeared in \cite{10}: there is a $\vep>0$ depending on
the structural assumptions and on $\O$ such that any solution in
$W^{1,n-\vep}_0(\O)$ is actually the unique solution declared in Theorem
3.7. A similar result follows from the estimates in \cite{13}.

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\enddocument
