\input amstex
\documentstyle{amsppt}
\loadmsbm
\magnification=\magstephalf
\hcorrection{1cm}
\nologo \TagsOnRight \NoBlackBoxes
\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{EJDE--2001/02\hfil $p$-harmonic  systems
\hfil\folio}
\def\leftheadline{\folio\hfil Bianca Stroffolini
  \hfil EJDE--2001/02}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
  Electronic Journal of Differential Equations,
Vol. {\eightbf 2001}(2001), No. 02, pp. 1--7.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break
ftp ejde.math.swt.edu (login: ftp)\bigskip} }

\topmatter
\title
A stability result for  $p$-harmonic  systems with discontinuous coefficients
\endtitle

\thanks
{\it Mathematics Subject Classifications:} 35J60, 47B47.\hfil\break\indent
{\it Key words:} Bounded mean oscillation, Linear and Nonlinear Commutators, 
\hfil\break\indent
Hodge Decomposition.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted November 27, 2000.  Published January 2, 2001.
\endthanks

\author Bianca Stroffolini   \endauthor
\address  Bianca Stroffolini \hfill\break
Dipartimento di Matematica e Applicazioni \lq\lq R. Caccioppoli", \hfill\break
via Cintia, 80126 Napoli, Italy
\endaddress
\email stroffol\@matna2.dma.unina.it
\endemail
\def\diver{\operatorname{div}}

\abstract
The present paper is concerned with $p$-harmonic systems
$$ \diver (\langle A(x)  Du(x), Du(x) \rangle ^{{p-2}\over 2} A(x)
Du(x))=\diver ( \sqrt{A(x)} F(x)),$$
where $A(x)$ is a positive definite matrix whose entries have 
bounded mean oscillation (BMO),
$p$ is a real number greater than 1 and $F\in L^{r\over {p-1}}$
is a given matrix field.
We find a-priori estimates for a very weak solution of class $W^{1,r}$,
provided $r$ is close to $2$, depending  on the BMO norm of
$\sqrt{A}$, and $p$ close to $r$.
This result is achieved using the corresponding existence and
uniqueness result for linear systems with BMO coefficients
\cite{St}, combined with nonlinear commutators.
\endabstract
\endtopmatter
\document

\head 0. Introduction\endhead
Consider the $p$-harmonic system
$$\diver (|Du(x)|^{{p-2}} Du(x))=0 \eqno (0.1)$$
in a regular domain $\Omega\subset {\Bbb R}^n$.

A vector field $u$ in the Sobolev space
$W^{1,r}_{\text{loc}}(\Omega,{\Bbb R}^n)$,
$r>\max \{1,p-1\}$,
{\sl is a very weak  $p$-harmonic vector} \cite{IS1},\cite{L}
if it satisfies
$$\int_{\Omega}|Du|^{p-2}\langle Du, D\phi\rangle dx=0  \quad
\forall \phi \in C^{\infty}_o (\Omega, {\Bbb R}^n)\,.
$$
This definition was first introduced by Iwaniec and Sbordone in [IS1],
they were able to prove, using commutator results, that there exists a
range of exponents, close to $p$,
$1<r_{1}<p< r_{2}<\infty$, such that if $u \in
W^{1,{r_{1}}}_{\text{loc}}(\Omega,{\Bbb R}^n)$ is very weak 
$p$-harmonic, then $u$
belongs to $W^{1,{r_{2}}}_{\text{loc}}(\Omega,{\Bbb R}^n)$, so, in 
particular, is
$p$-harmonic. J. Lewis \cite{L}, using  that the maximal functions
raised to a small positive power is an $A_{p}$ weight in the sense of
Muckenhoupt, was able to obtain
similar results.
Kinnunen and Zhou \cite{KZ} gave a partial answer to a conjecture posed
by Iwaniec and Sbordone;
they proved that $r_{1}$ can be chosen arbitrarly close to $1$, if $p$
is close to $2$. Later Greco and Verde developed the same result for
$p$-harmonic
equations with $VMO\cap  L^{\infty}$ coefficients, using
estimates for linear elliptic equations with $VMO$ coefficients
\cite{D},\cite{IS2}.
Our result is concerned with $p$-harmonic systems with BMO
coefficients:
$$\diver ( \langle A(x) Du(x),Du(x)\rangle ^{{p-2}\over 2} A(x)
Du(x) )=\diver(\sqrt{A(x)} F(x))\eqno(0.2)$$
where $A(x)=(A_{ij}(x))$ is a symmetric, positive definite matrix
with entries in BMO, $F$ is a given matrix field in $L^{r\over
{p-1}}$. Our definition of very weak $p$-harmonic vector
a priori requires that the energy functional is finite along a
solution, that is:
$$\int_{\Omega} |\sqrt{A} Du|^r dx<\infty$$
a closed subspace of $W^{1,r}_{0}(\Omega,{\Bbb R}^n)$,
in addition for every $\phi\in C^{\infty}_{0}(\Omega, {\Bbb R}^n)$
$$\int_{{\Omega}}|\sqrt{A}Du|^{p-2} \langle \sqrt{A} Du, \sqrt{A}
D\phi \rangle dx=\int_{{\Omega}} \langle F(x), \sqrt{A} D\phi \rangle dx
\eqno(0.3)$$


   We will use the existence and uniqueness result for linear
systems with bounded mean oscillation (BMO)
coefficients to derive a new Hodge decomposition for matrix fields
and, then, using commutators, we will prove
a continuity result for $p$ close to $2$, depending on the BMO-norm
of $\sqrt{A}$.

The method of proof is different from the linear case; in fact, there
we have at our disposal two commutator results:
one is a powertype perturbation of the kernel of a linear bounded
operator, the other is the Coifman-Rochberg-Weiss result
about the linear commutator of a Calderon-Zygmund operator with a BMO
matrix. In the nonlinear case,
we do not know of a result for nonlinear commutators
with a BMO function, so we can only use the commutator result of
powertype, applied to the natural Hodge decomposition coming from the
linear case. The statement is the following:
\proclaim{Main Theorem}
For $r$ given in such a way that $|r-2|<\varepsilon$, determined by
the BMO-norm of $\sqrt{A}$, there exists $\delta >0$ such that if
$|p-r|<\delta$ and u is a {\sl very
   weak $p$-harmonic vector}, then
$$\|\sqrt{A} Du\|_{r}^{r}\leq C\|F\|_{r \over {p-1}}^{r \over {p-1}}
\eqno(0.4)$$
\endproclaim
Further developments are presented considering some new spaces, the
so-called {\sl grand $L^q$ spaces}, in the spirit of \cite{GIS}.

\head{1. Definitions and preliminary results  }\endhead

  \definition{Definition 1}
Let $\Omega$ be a cube or the  entire space ${\Bbb R}^n$.
The John-Nirenberg space $BMO(\Omega)$ \cite{JN}
consists of all functions $b$
which are integrable on every cube $Q\subset \Omega$ and satisfy:
$$\|b\|_{\ast}=\sup \Big \{ \frac1{|Q|}\int_Q |b-b_Q|\, dx :
Q\subset \Omega\Big\}
<\infty$$
where $b_Q={1\over |Q|} \int_Q b(y) dy$.\enddefinition

\definition{Definition 2} For $1<q<\infty$ and $0\leq \theta <\infty$
the grand $L^q$-space, denoted by $L^{\theta,q)}(\Omega,{\Bbb 
R^{n\times n}} )$,
consists of matrices $F\in\bigcap_{0<\varepsilon\leq q-1}
L^{q-\varepsilon}(\Omega,{\Bbb R^{n\times n}})$
such that
$$\| F\|_{\theta,q)}=\sup_{0<\varepsilon\leq q-1}\varepsilon^{\theta
\over q} \|F\|_{q-\varepsilon}<\infty$$
\enddefinition

These spaces are Banach spaces, they  were introduced for $\theta=1$
in the study of
integrability properties of the Jacobian \cite{IS1} and were used
in \cite{GISS} to establish a degree formula for maps with
non-integrable Jacobian.

\definition{Definition 3}The grand Sobolev space
$W^{\theta,p}_{0}(\Omega ,{\Bbb R}^n)$ consists of all vector fields $u$
belonging to $\bigcap_{0<\varepsilon \leq p-1}
W^{1,p-\varepsilon}_{0}(\Omega,{\Bbb R}^n)$ such that $Du\in
L^{\theta,p}(\Omega,{\Bbb R^{n\times n}})$; a norm on this space is 
$\|Du\|_{\theta,p)}$.
\enddefinition

Next, we recall a stability result for nonlinear perturbation of a
kernel of a bounded linear operator; namely:
$T^{-\delta}f=T(|f|^{-\delta}f)$, where
   $$T: L^p(\Omega,E)\longmapsto L^p(\Omega,E)$$ is a bounded linear
   operator and $E$ is a Hilbert space.

\proclaim {Theorem 1} Let $T: L^p(\Omega,E)\longmapsto L^p(\Omega,E)$
be a bounded linear operator for all
$p_1\leq p\leq p_2$; then for $1-{{p_2}\over p}\leq \delta \leq
1-{{p_1}\over p}$
   there is a constant $C=C(\|T\|_{p_1},\|T\|_{p_2})$ such that if f belongs
   to the kernel of T, we get
$$\|T(|f|^{-\delta}f)\|_{p\over {1-\delta}}\leq
C|\delta|\|f\|_p^{1-\delta}
\eqno(1.1)$$
\endproclaim


\subhead A new Hogde decompostion \endsubhead
   Consider a linear system with BMO coefficients:
   $$\diver (B(x) Du(x)) =\diver F(x)$$
   where $B(x)$ is a symmetric, positive definite matrix whose entries
are in BMO, $F$ is a given
   matrix field. We state the following  existence and uniqueness result
   for the solution of the Dirichlet problem:

\proclaim{Theorem 2 } \cite{St} There exists $\varepsilon>0$ , depending on the
BMO-norm of $B$, such that for $|r-2|<\varepsilon$  the Dirichlet problem:
$$\gathered \diver (BDu)=\diver F  \\
  F\in L^r (\Omega ,{\Bbb R}^{n\times n}), \quad
u\in W^{1,r}_o (\Omega ,{\Bbb R}^n)\endgathered \eqno(1.2)$$
admits a unique solution. In particular the energy functional
$$\int_{\Omega} |Du|^{-\varepsilon} \langle B(x) Du,Du\rangle\, dx$$
is finite and the following a-priori estimate holds
$$\|Du\|_{r}\leq C \|F\|_{r} \eqno (1.3)$$
   \endproclaim
\remark{Remark}
Note that, taking into account the uniform estimate $(1.3)$ for
exponents in a range determined by the BMO-norm of $B$,
we have actually existence and uniqueness in the grand Sobolev space
$W^{\theta,2)}_{0}(\Omega, {\Bbb R}^n)$.
\endremark

This Theorem can be rephrased in terms of a new Hodge decomposition.
More precisely,
\proclaim{Theorem 2' } There exists $\varepsilon>0$ , depending on the
BMO-norm of $B$, such that for $|r-2|<\varepsilon$
a matrix field $F\in L^r (\Omega ,{\Bbb R}^{n\times n})$
   can be decomposed  uniquely as it follows:
$$F=B D\phi +L$$
with $\diver L=0$ and $\phi \in W^{1,r}_o (\Omega ,{\Bbb R}^n)$.
Therefore, there exists a bounded linear operator
$$S:L^r (\Omega ,{\Bbb R}^{n\times n}) \to
L^r(\Omega ,{\Bbb R}^{n\times n})$$
given by $S(F)=B D\phi$.
\endproclaim
It is sufficient to solve the
linear system
$$\diver (B D \phi )=\diver F$$


   We will apply Theorem 1
to the operator $T=I-S$ with $B=\sqrt{A}$.
Notice that the square root operator acting on matrices with minimum
eigenvalue far from zero, for example greater or equal than $1$,
is Lipschitz, therefore the square root of $A$ is  still in BMO. The
kernel of the operator $T$
consists of matrix fields of the form
$\sqrt{A} D\phi$.

\head 2. Proof of the Main Theorem \endhead

Consider a very weak $p$-harmonic vector $u\in W^{1,r}$, with $r$
determined by Theorem 2 and with finite energy.
Decompose $|\sqrt{A} Du|^{r-p}\sqrt{A}Du$
using the new Hodge decomposition:
$$|\sqrt{A} Du|^{r-p}\sqrt{A}Du= \sqrt{A} D\phi+L,\quad \diver L=0$$

Let us observe that $T(\sqrt{A} Du)=0$;
therefore  $L$ is a nonlinear perturbation of the kernel of
a bounded linear operator; we can apply Theorem 1  with
$\delta=p-r$ to get the following estimate
$$\|L\|_{ r\over {1-\delta}}\leq C|\delta |\|\sqrt{A}
Du\|_{r}^{1-\delta} \eqno(2.1)$$
Using the above equality
we find
$$\int_{{\Omega}}|Du|^rdx \leq \int_{\Omega}|\sqrt{A} Du|^r=
\int_{{\Omega}}|\sqrt{A} Du|^{p-2}\langle \sqrt{A} Du, L\rangle dx+
\int_{{\Omega}} \langle F, \sqrt{A}D\phi \rangle dx\,.$$

Using H\"{o}lder's inequality on the last two terms of the above
expression and $(2.1)$,
$$\aligned
\int_{{\Omega}}|\sqrt{A}Du|^r dx \leq&
\|\sqrt{A}Du\|_{r}^{p-1} \|L\|_{r\over {r-p+1}}+\|F\|_{r\over
{p-1}}\|\sqrt{A}D\phi\|_{r\over{r-p+1}} \\
\leq& C|r-p|\|\sqrt{A}Du\|_{r}^{r}+C\|F\|_{r\over
{p-1}}\|\sqrt{A}Du\|^{r\over{r-p+1}}_{r}
\endaligned$$
Using Young's inequality and choosing $r$ such that
$C|r-p|<1$, we get the assertion.


We will prove also  the uniqueness of the very weak $p$-harmonic
vector in a space larger than $W^{1,r}$, refining estimate $(0.4)$.
We begin with establishing
the following Theorem, that for the $p$-harmonic case was established
in \cite{GIS}.


\proclaim{Theorem 3} For $r$ given in such a way that
$|r-2|<\varepsilon$ , determined by Theorem 2, there exists $\delta$
such that if $|p-r|<\delta$ and $u, v \in W^{1,r}(\Omega,{\Bbb R}^n)$ are
very weak $p$-harmonic vectors respectively with data $F,G\in
L^{r \over {p-1}}(\Omega,{\Bbb R^{n\times n}})$ with finite energy, 
the following
estimate holds:
$$\aligned
  &\|\sqrt{A}Du-\sqrt{A}Dv\|_{r}^{p-1} \\
&\leq C\varepsilon^{{p-1} \over {|p-2|}}
(\|F\|_{r\over{p-1}}+\|G\|_{r\over{p-1}})
+C\cases \|F-G\|_{r\over{p-1}} &  (p\geq 2) \\
\|F-G\|_{r\over{p-1}}^{p-1}(\|F\|_{r\over{p-1}}+\|G\|_{r\over{p-1}})^{2-p}
& (1<p<2) \endcases
\endaligned  \eqno(2.2)$$
\endproclaim

\demo{Proof}
Take $u\in W^{1,r}(\Omega,{\Bbb R}^n)$ with finite energy, a very weak
solution of the equation:
$$\diver(|\sqrt{A} Du|^{{p-2} }  A Du )
=\diver(\sqrt{A} F) \eqno(2.3)$$ and $v \in W^{1,r}(\Omega,{\Bbb R}^n)$
with finite energy, a very weak solution of
$$\diver(|\sqrt{A}
Dv|^{{p-2} }  A Dv ) =div(\sqrt{A} G) \eqno(2.4)$$
Consider the Hodge decomposition of
$$
|\sqrt{A}Du-\sqrt{A}Dv|^{r-p}(\sqrt{A}Du-\sqrt{A}Dv)=\sqrt{A}D\phi+L
$$
we have estimates:
$$\gathered
\|\sqrt{A}D\phi\|_{r\over{1-\delta}}\leq
C\|\sqrt{A}Du-\sqrt{A}Dv\|_{r}^{1-\delta}\cr
\|L\|_{r\over{1-\delta}}\leq C|\delta|
\|\sqrt{A}Du-\sqrt{A}Dv\|_{r}^{1-\delta}
\endgathered
$$ We can use
$\sqrt{A}D\phi$ as test function in (2.3) and (2.4) and subtract
the two equations, to obtain:
$$\gathered
\int_{{\Omega}}\langle
|\sqrt{A}Du|^{p-2}\sqrt{A}Du-|\sqrt{A}Dv|^{p-2}\sqrt{A}Dv
,|\sqrt{A}Du-\sqrt{A}Dv|^{r-p} (\sqrt{A}Du-\sqrt{A}Dv) \rangle \cr
=\int_{\Omega}\langle F-G,\sqrt{A}D\phi\rangle
+\int_{\Omega}\langle
|\sqrt{A}Du|^{p-2}\sqrt{A}Du-|\sqrt{A}Dv|^{p-2}\sqrt{A}Dv
,L\rangle\,,
\endgathered $$
and
$$ \gathered
\int_{\Omega}(|\sqrt{A}Du|+|\sqrt{A}Dv|)^{p-2}
|\sqrt{A}Du-\sqrt{A}Dv|^{2-p+r} \cr
\leq C(p)\int_{\Omega}|F-G||\sqrt{A}D\phi|+C(p)\int_{\Omega}(|\sqrt{A}Du|
+|\sqrt{A}Dv|)^{p-2} |\sqrt{A}Du-\sqrt{A}Dv||L|\,.
\endgathered $$
Now, using
H\"older's and Young's inequalities we get the assertion.
\enddemo

This Theorem is the key to prove uniqueness of the solution of
$(0.2)$ in the grand Sobolev space $W^{\theta,p}_{0}$ when the
right-hand side is in a
grand $L^{\theta,q)}$ space. We state the following uniqueness
Theorem.

\proclaim{Theorem 4} For each $F\in L^{\theta,q)}(\Omega,{\Bbb 
R^{n\times n}})$ with
$q$ the H\"older conjugate of $p$, and $p$ in the range determined
by Theorem 2, the $p$-harmonic system $(0.2)$ may have at most one
solution in the closed subspace of $W^{\theta,p}(\Omega,{\Bbb R}^n)$:
$${\Cal E}^{\theta,p}=\{u\in W^{\theta,p}(\Omega,{\Bbb R}^n) :
\|\sqrt{A} Du\|_{\theta,p)}<\infty \} $$
and we get the uniform estimate for the operator
${\Cal H}:L^{\theta,q)}(\Omega,{\Bbb R^{n\times n}}) \to {\Cal E}^{\theta,p}$
that carries $F$ into $\sqrt{A}Du$:
$$\|{\Cal H}F-{\Cal H} G\|_{\theta,p)}^{p-1} \leq C(n,p,\|A\|_{{\ast}})
\, \|F-G\|_{{\theta,q)}}^{\alpha
}(\|F\|_{{\theta,q)}}+\|G\|_{{\theta,q)}})^{1-\alpha} \eqno(2.5)$$
where
   $$\alpha=\cases {{p-\theta (p-2)}\over p } & \hbox
{if $p\geq 2$} \\
{{p+\theta (p-2)}\over q } & \hbox
{if $p\leq 2$}\endcases$$
If, in addition, $A$ is in $L^{\infty}$, we get existence.
\endproclaim

In fact, given $F\in L^{\theta,q)}(\Omega,{\Bbb R^{n\times n}})$, we consider a
convolution $F_{k}$ with a standard mollifier;
the approximations $F_{k}$ converge to $F$ in $ 
L^{\theta',q)}(\Omega,{\Bbb R^{n\times n}})$
for every $\theta'>\theta$. Next, solve the $p$-harmonic system:
$$\diver ( \langle A(x) Du_{k}(x),Du_{k}(x)\rangle ^{{p-2}\over 2} A(x)
Du_{k}(x) )=\diver(\sqrt{A(x)} F_{k}(x))$$
for $u_{k}\in W^{1,p}_{0}(\Omega,{\Bbb R}^n)$. We use estimate $(2.5)$
with $\theta'$ in place of $\theta$ to show that $u_{k}$
is a Cauchy sequence in $W^{\theta',p)}_{0}(\Omega,{\Bbb R}^n)$:
$$\|\sqrt{A} Du_{k}-\sqrt{A} Du_{j}\|_{\theta',p)}^{p-1} \leq
C(n,p,\|A\|_{{\ast}})
\, \|F_{k}-F_{j}\|_{{\theta',q)}}^{\alpha
}(\|F_{k}\|_{{\theta',q)}}+\|F_{j}\|_{{\theta',q)}})^{1-\alpha}$$
Passing to the limit in the integral identities:
$$\int_{{\Omega}}|\sqrt{A}Du_{k}|^{p-2} \langle \sqrt{A} Du_{k}, \sqrt{A}
D\phi \rangle dx=\int_{{\Omega}} \langle F_{k}(x), \sqrt{A} D\phi \rangle
dx$$
we then conclude that the limit $u$ is in $W^{\theta,p}_{0}(\Omega,
{\Bbb R}^n)$.

\Refs
\widestnumber\key{GISS}

\ref \key{D}\by G. Di Fazio \paper $L^p$ estimates for divergence
form elliptic equations with discontinuous coefficients
\jour Boll. Un. Mat. Ital. \vol 7 \yr 1996  \pages 409--420
\endref

\ref\key{GIS} \by L. Greco, T. Iwaniec and C. Sbordone \paper
Inverting the $p$-Harmonic Operator \jour Manuscripta Mathematica
\vol 92 \yr 1997 \pages 249--258
\endref

\ref\key{GISS} \by  L. Greco, T.Iwaniec, C. Sbordone and B. Stroffolini
\paper  Degree formulas for maps with nonintegrable Jacobian
\jour Topological Methods in Nonlinear Analysis \vol 6 \yr 1995
\pages 81--95
\endref

\ref\key{GV} \by  L. Greco and A. Verde
\paper  A regularity property of $p$-harmonic functions
\jour Annales
Academiae Scientiarum Fennicae (to appear)
\endref

\ref\key{I}\by T.Iwaniec \book Nonlinear Differential Forms  \publ
Lectures in Jyv\"askyl\"a,
\yr August 1998
\endref

\ref\key{IS1} \by T. Iwaniec and C. Sbordone
\paper Weak minima of variational integrals \jour J.Reine
Angew.Math.\vol 454 \yr 1994
   \pages 143--161
   \endref

\ref\key{IS2} \by T. Iwaniec and C. Sbordone
\paper Riesz transform and elliptic PDEs with
VMO-coefficients \jour Journal d'Analyse Math.\vol 74 \yr 1998
   \pages 183--212
   \endref

\ref\key{KZ} \by J. Kinnunen and S. Zhou \paper A note on very weak
   $P$-harmonic mappings \jour Electronic Journal of Differential
   Equations,
\vol 25 \yr 1997 \pages 1--4
\endref

\ref\key{JN} \by F. John and L. Nirenberg \paper On functions of
bounded mean oscillation
\jour Comm. Pure Appl. Math.\vol14 \yr1961 \pages 415--426
\endref

\ref\key{L} \by J. Lewis \paper On very weak solutions of certain
elliptic systems
\jour Comm. Part. Diff. Equ.\vol 18 \yr1993  \pages 1515--1537
\endref

\ref\key{S} \by  E. M. Stein
\book Harmonic Analysis \publ Princeton University
Press
\yr 1993
\endref

\ref\key{St} \by B.Stroffolini \paper Elliptic Systems of PDE with BMO
coefficients \jour Potential Analysis (to appear)
\endref
   
\endRefs\enddocument