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\def\rightheadline{EJDE--1997/25\hfil  A note on very weak solutions
\hfil\folio}
\def\leftheadline{\folio\hfil Juha Kinnunen \&  Shulin Zhou
 \hfil EJDE--1997/25}

\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
 Electronic Journal of Differential Equations,
Vol.\ {\eightbf 1997}(1997), No.~25, pp.\ 1--4.\hfil\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break 
ftp (login: ftp) 147.26.103.110 or 129.120.3.113\bigskip} }

\topmatter
\title A note on very weak p--harmonic mappings
\endtitle

\thanks \noindent
{\it 1991 Mathematics Subject Classifications:} 35J60, 35B45.\hfil\break
{\it Key words and phrases:} Very weak p--harmonic mapping.
\hfil\break
\copyright 1997 Southwest Texas State University  and
University of North Texas.\hfil\break
Submitted November 27, 1997. Published December 19, 1997.\hfil\break
The second author would like to thank NSF of China for their support, 
and the University of Helsinki for its hospitality. 
\endthanks
\author  Juha Kinnunen  \& Shulin Zhou  \endauthor 

\address  Juha Kinnunen\hfil\break
Department of Mathematics, P.O.Box 4, FIN-00014 University of Helsinki,
Finland.
\endaddress
\email  Juha.Kinnunen\@ Helsinki.Fi
\endemail 
\address Shulin Zhou \hfil\break
Department of Mathematics, Peking University, Beijing 100871,
P. R. China
\endaddress
\email  zsl\@sxx0.math.pku.edu.cn
\endemail

\abstract
We prove a new {\it a priori\/} estimate for 
very weak $p$--harmonic mappings when $p$ is close to two.
This sheds some light on a conjecture posed by Iwaniec and Sbordone.
\endabstract
\endtopmatter

\define\dive{\operatorname{div}}
\define\loc{\operatorname{loc}}

\document
\head 1. Introduction \endhead

Let $\varOmega$ be a bounded regular domain 
(Definition 2.1 in [IS]) in $\bold R^n$
and $1<p<\infty$.
We are interested in the $p$--harmonic system
$$
\dive (|\nabla u|^{p-2}\nabla u)=0 \tag 1.1
$$
under minimal assumptions on $u$.
System (1.1) is understood in the weak sense:
a weak $p$--harmonic mapping is defined as 
$u\in W^{1,p}_{\loc}(\varOmega,\bold R^m)$ satisfying
$$
\int_\varOmega|\nabla u|^{p-2}\nabla u\cdot\nabla\phi
\,dx=0
\tag 1.2
$$
for every $\phi\in C^\infty_0(\varOmega,\bold R^m)$.
However, equation (1.2) makes sense when
$u\in W^{1,r}_{\loc}(\varOmega,\bold R^m)$ with
$r\ge\max\{1,p-1\}$.
Such solutions are called {\it very weak $p$--harmonic mappings}.

Iwaniec and Sbordone showed in [IS] that there exist  
$r_1=r_1(p,m,\varOmega)$ and 
$r_2=r_2(p,m,\varOmega)$ satisfying 
$$
1<r_1<p<r_2<\infty
\tag 1.3
$$
such that every very weak $p$--harmonic
mapping $u\in W^{1,r_1}_{\loc}(\varOmega,\bold R^m)$ belongs
to $W^{1,r_2}_{\loc}(\varOmega,\bold R^m)$.
They conjectured (Conjecture 1 in [IS]) that every
$r_1>\max\{1,p-1\}$ would do for the $p$--harmonic system, 
but their estimate for $r_1$ is very close to $p$.

The objective of our note is to study 
the case when $p$ is close to two.
We show that $r_1$ in (1.3) can be chosen 
arbitrarily close to one if $p$ is close to two.
This does not solve the conjecture by Iwaniec and Sbordone, but
we are able to obtain estimates when $r_1$ is arbitrarily
close to $\max\{1,p-1\}$.

\proclaim{Theorem 1.4}
Let $0<\eta<1$.
For every exponent $r_1\ge1+\eta$ there are 
$\delta=\delta(\eta,m,\varOmega)>0$
and $r_2=r_2(p,m,\varOmega)>p$
such that every very weak $p$--harmonic mapping 
$u\in W^{1,r_1}_{\loc}(\varOmega,\bold R^m)$ 
belongs to $W^{1,r_2}_{\loc}(\varOmega,\bold R^m)$ 
provided $|p-2|<\delta$.
\endproclaim

Our method is a sharpening of [IS].
If $u$ belongs to the Sobolev space with the natural 
exponent $p$, then we may use the classical method of reverse
H\"older inequalities, see [ME]. 
However, if $u$ is assumed to belong to a Sobolev space
below the natural exponent, then we cannot choose
$\eta u$, where $\eta$ is a cut-off function, as a 
test function in (1.2) and hence it is difficult
to obtain any {\it a priori}\/ estimates.
Iwaniec and Sbordone used the Hodge decomposition 
ingeniously to build a test function for equation (1.2).
Another argument has been given by Lewis in [L].
The main new feature in our proof is that we
use the Hodge decomposition twice and also employ
the fact that the matrix field involved
is divergence free. 
This trick works only for $p$--harmonic operator.
As Serrin's example in [S] shows, our result is not true 
for $\Cal A-$harmonic operators studied by [IS] and [L].

\head
2. Main results
\endhead

For the convenience of the reader we recall 
the formulation of the Hodge decomposition (Theorem 3 in [IS]).

\proclaim{Theorem 2.1}
Let $\varOmega$ be a regular domain in $\bold R^n$ and
$w\in W^{1,r}_0(\varOmega,\bold R^m)$ with $r>1$, and let
$-1<\varepsilon<r-1$.
Then there exist
$\phi\in W^{1,r/(1+\varepsilon)}_0(\varOmega,\bold R^m)$
and a divergence free matrix field
$H\in L^{r/(1+\varepsilon)}(\varOmega,\bold R^{m\times n})$ such that
$$
|\nabla w|^\varepsilon\nabla w
=\nabla\phi+H,
\tag 2.2
$$
and
$$
\|H\|_{r/(1+\varepsilon),\varOmega}
\le C_r(\varOmega,m)|\varepsilon|
\,\|\nabla w\|_{r,\varOmega}^{1+\varepsilon}.
\tag 2.3
$$
\endproclaim

An examination of [IS] reveals that all their results
are based on Theorem 5.1 in [IS].
Instead of rewriting all results in [IS],
we only prove a sharpening of Theorem 5.1 in [IS]
when $p$ is close to two.

We consider the very weak solutions  
$w\in W^{1,r}_0(\varOmega,\bold R^m)$ 
with $r>\max\{1,p-1\}$ of
the non homogeneous system
$$
\dive\big(|g+\nabla w|^{p-2}(g+\nabla w)\big)
=\dive  h
\tag 2.4
$$
where $g\in L^r(\varOmega,\bold R^{m\times n})$ 
and $h\in L^{r/(p-1)}(\varOmega,\bold R^{m\times n})$
are matrix fields.
Equation (2.4) is understood in the weak sense, that is,
$$
\int_\varOmega|g+\nabla w|^{p-2}(g+\nabla w)\cdot\nabla\phi\,dx
=\int_\varOmega h\cdot\nabla\phi\,dx
\tag 2.5
$$
for every $\phi\in W^{1,r/(r-p+1)}_0(\varOmega,\bold R^m)$.

\proclaim{Theorem 2.6}
Let $0<\eta<1$.
Suppose that $w\in W^{1,r}_0(\varOmega,\bold R^m)$ with
$r\ge1+\eta$ satisfies {\rm (2.3)}. 
Then there is $\delta=\delta(\eta,m,\varOmega)>0$ 
such that if $|p-2|<\delta$, then
$$
\int_\varOmega|\nabla w|^r\,dx
\le C(\eta,p,m,\varOmega)\int_\varOmega\big(|g|^r+|h|^{r/(p-1)}\big)\,dx.
\tag 2.7
$$
\endproclaim

\demo{Proof}
Using Theorem 2.1 with $\varepsilon=r-p$ we obtain 
functions $\phi_1\in W^{1,r/(r-p+1)}_0(\varOmega,\bold R^m)$
and $H_1\in L^{r/(r-p+1)}(\varOmega,\bold R^{m\times n})$ such that
$$
\align
|\nabla w|^{r-p}\nabla w
&=\nabla\phi_1+H_1,
\tag 2.8
\\
\int_\varOmega H_1\cdot\nabla\varphi\,dx
&=0,
\qquad\text{for every $\varphi\in W^{1,r/(p-1)}_0(\varOmega,\bold R^m)$,}
\tag 2.9
\endalign
$$
and
$$
\|H_1\|_{r/(r-p+1),\varOmega}
\le c_1|r-p|
\,\|\nabla w\|_{r,\varOmega}^{r-p+1},
\qquad c_1=C_r(\varOmega,m).
\tag 2.10
$$
In particular, we have
$$
\|\nabla\phi_1\|_{r/(r-p+1),\varOmega}
\le (c_1+1)
\,\|\nabla w\|_{r,\varOmega}^{r-p+1}.
\tag 2.11
$$
Since $\phi_1$ can be used as a test function in (2.5), 
we obtain
$$
\int_\varOmega|\nabla w+g|^{p-2}(\nabla w+g)
\cdot\nabla\phi_1\,dx
=\int_\varOmega h\cdot\nabla\phi_1\,dx.
$$
Inserting (2.8) we arrive at
$$
\aligned
\int_\varOmega|\nabla w|^r\,dx
=&\int_\varOmega|\nabla w|^{p-2}\nabla w\cdot H_1\,dx
+\int_\varOmega h\cdot\nabla\phi_1\,dx
\\
&+\int_\varOmega\big(|\nabla w|^{p-2}\nabla w
-|\nabla w+g|^{p-2}
(\nabla w+g)\big)
\cdot\nabla\phi_1\,dx
\\
=&I_1+I_2+I_3. 
\endaligned
\tag 2.12
$$
We begin with estimating $I_1$.
This is the crucial step of our argument.
By using Theorem 2.1 again with $\varepsilon=p-2$, we obtain 
$\phi_2\in W^{1,r/(p-1)}(\varOmega,\bold R^m)$
and $H_2\in L^{r/(p-1)}(\varOmega,\bold R^{m\times n})$
such that
$$
\align
|\nabla w|^{p-2}\nabla w
&=\nabla\phi_2+H_2
\tag 2.13
\\
\int_\varOmega H_2\cdot\nabla\varphi\,dx
&=0,
\qquad\text{when $\varphi\in W^{1,r/(r-p+1)}_0(\varOmega,\bold R^m)$,}
\tag 2.14
\endalign
$$
and
$$
\|H_2\|_{r/(p-1),\varOmega}
\le c_1|p-2|
\,\|\nabla w\,\|_{r,\varOmega}^{p-1},
\qquad c_1=C_r(\varOmega,m).
\tag 2.15
$$
Using (2.13), (2.14), (2.8), (2.9) and (2.15) 
we have
$$
\aligned
I_1
&=\int_\varOmega\big(\nabla\phi_2+H_2\big)
\cdot H_1\,dx
=\int_\varOmega H_1\cdot H_2\,dx
\\
&=\int_\varOmega\big(|\nabla w|^{r-p}\nabla w-\nabla\phi_1
\big)\cdot H_2\,dx\\
&=\int_\varOmega|\nabla w|^{r-p}\nabla w\cdot H_2\,dx
\\
&\le c_1|p-2|\,\|\nabla w\|_{r,\varOmega}^r.
\endaligned
$$
The same reasoning shows that
$$
I_1\le c_1|r-p|\,\|\nabla w\|_{r,\varOmega}^r
$$
and hence
$$
I_1\le c_1\min\big\{|p-2|,|r-p|\big\}\|\nabla w\|_{r,\varOmega}^r.
\tag 2.16
$$
By virtue of (2.11) we may estimate $I_2$ and $I_3$ 
in the same way as in [IS]. 
That is
$$
\align
I_2+I_3
\le & C\big(\|\nabla w\|_{r,\varOmega}^{r-p+1}
\|h\|_{r/(p-1),\varOmega}
\\
&+\|g\|_{r,\varOmega}^{p-1}
\|\nabla w\|^{r-p+1}_{r,\varOmega}
+\|g\|_{r,\varOmega}
\|\nabla w\|^{r-1}_{r,\varOmega}
\big).
\endalign
$$
Recalling Young's inequality we conclude that, 
for every $\theta>0$,
$$
I_2+I_3
\le\theta\,\|\nabla w\|_{r,\varOmega}^r
+C_\theta\big(\|g\|_{r,\varOmega}^r
+\|h\|_{r/(p-1),\varOmega}^{r/(p-1)}\big).
\tag 2.17
$$
Using (2.12), (2.16) and (2.17) we obtain
$$
\big(1-c_1\min\{|p-2|,|r-p|\}-\theta\big)\|\nabla w\|^r_{r,\varOmega}
\le C_{\theta}\big(
\|g\|^r_{r,\varOmega}
+\|h\|^{r/(p-1)}_{r/(p-1),\varOmega}
\big). 
$$
In particular, if $c_1|p-2|<1$, then (2.5) holds.
Estimates for the constant $c_1=C_r(\varOmega,m)$ can be found
in [I] and formula (2.10) in [IS].
Using these estimates it is easy to see that we may choose
$c_1=c(\eta,m,\varOmega)$.
This completes the proof.   
\enddemo

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\enddocument