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\begin{document}


\begin{center}
{\bf REGULARITY RESULTS VIA DUALITY \\
FOR MINIMIZERS  OF DEGENERATE  FUNCTIONALS }
\end{center}

\medskip

\begin{center}
{ Menita Carozza* Gioconda Moscariello** Antonia Passarelli di
Napoli***}
\end{center}

\medskip

\begin{center}


{\small *Universit\`{a} del Sannio, Piazza Arechi II- Palazzo De Simone - 82100
Benevento}

{\footnotesize e-mail:carozza@unisannio.it}


\end{center}

\begin{center}
{\small **Dipartimento di Matematica e Applicazioni "R. Caccioppoli"}

{\small Universit\`{a} di Napoli ``Federico II", via Cintia - 80126
Napoli}

{\footnotesize e-mail:gmoscari@unina.it}


\end{center}

\begin{center}
{\small ***Dipartimento di Matematica e Applicazioni "R. Caccioppoli"}

{\small Universit\`{a} di Napoli ``Federico II", via Cintia - 80126
Napoli}

{\footnotesize e-mail:
antonia.passarelli@unina.it\qquad (Corresponding author)}


\end{center}

\vskip2cm
\bigskip
\bigskip

\noindent {\footnotesize  ABSTRACT. We prove a regularity  
result for local minimizers of  degenerate variational
integrals, whose model arises in the study of mappings with finite 
distortion.

The degeneracy function ${\cal K}(x)$  
 lies in the exponential class, i.e. 
$exp (\lambda {\cal K}(x)) $ is integrable for some $\lambda>0$.

The right space of the gradient of a local minimizer $u$ turns out to 
be the Zygmund class $L^p\log^{-1} L$.
Our result states that if $\lambda$ is sufficiently large, then $Du$  belongs to the 
Zygmund space $L^p\log^\alpha L$, \ $\alpha\geq 1$ and $\alpha$ 
encreases with $\lambda$.}

\bigskip

\noindent {\footnotesize {\bf AMS Classifications.}  49N15; 49N60; 
49N99.}

\bigskip

\noindent {\footnotesize {\bf Key words.} {\it Degenerate variational 
integrals; Duality Theory;
Mappings with finite distortion.}}
\bigskip
\bigskip
\bigskip



\section
{Introduction}
\bigskip

In this paper we establish a regularity result for local minimizers
of  variational integrals satisfying a degenerate ellipticity
condition.

The variational integrals we  treat here are of  the form
$$I (\Om, v)\,=\, \int_{\Om} f(x,Dv)dx \eqno (1.1)$$
where $\Om$ is an open bounded subset of $\RR^{ N} $, $v:\Om\subset \RR^{ N}\to
\RR$ and  $f:\,\Om\times \RR^{ N}\to \RR $ is a Caratheodory function.

We assume that there exits a function $1\le k(x)<\infty$ such that
\medskip


\noindent (H1) \qquad\qquad\qquad\qquad $\frac{1}{ k(x)}|\xi|^p\leq
f(x,\xi)\leq
k(x)|\xi|^{p}$


\noindent for $1<p<\infty$, almost every $x\in \Om$ and all $\xi,\eta
\in\RR^{N}$ and

\medskip


\noindent (H2) \qquad\qquad\qquad\qquad $\xi\to
f(x,\xi)$ is strictly convex  a. e. $x\in \Om$.


Remark that, even in the case $p=2$, we are dealing with genuine
degenerate functionals
since the ratio between the eigenvalues  is unbounded.

\smallskip

\begin{defn}
  A function $u\in W^{1,1}_{loc}(\Om)$  is a
local
  minimizer of functional  (1.1) if $I(\Om',u)<\infty$ for every 
  $\Om'\subset\subset \Om$ and
$$
\int_{supp \varphi}f(x,Du) dx \le \int_{supp \varphi} f(x,Du+ D\varphi) 
dx\eqno (1.2)
$$
for every function $\varphi\in C^\infty_0(\Om)$.
\end{defn}

\smallskip




Our basic assumption is that the growth coefficient  $k(x)$,
which measures the degree of degeneracy of our variational integral,
belongs to the Orlicz space $EXP(\Om)$. Precisely, setting ${\cal
K}(x)=\max\{k(x),k^{1\over p-1}(x)\}$, we  assume
that
$$\int_{\Om}e^{\lambda {\cal K}(x)}dx<\infty \eqno (1.3)$$
for some $\lambda>0$.
\medskip
\begin{rem}
By Definition 1.1   a minimizer $u$ satisfy $$I(\Om', u)\,=\, \int_{\Om'} 
f(x,Du)dx <\infty$$
for every $\Om'\subset\subset \Om$.
Then assumption (H1) and relation (1.3) immediately imply, via 
H\"older inequality, that
$$\int_{\Om'}|Du|^{p}\log^{-1}(e+|Du|)$$
is finite, i.e.
the gradient of $u$ belongs to the Zygmund class $L^{p}\log^{-1}L_{loc}(\Om)$.
\end{rem}
Roughly speaking our result shows that the integrability of $Du$
increases when $\lambda$ gets larger.

\noindent Indeed, we have the following
\medskip
\begin{thm}\label{thm1}
Let $u$ be a local minimizer of the variational
integral (1.1) such that  $Du\in L^{p}\log^{-1}L_{loc}(\Om)$. Assume that
$f(x,\xi)$  verifies assumptions (H1) and (H2).
For every $\alpha>0$ there exists
$\lambda_{\alpha}=\lambda_{\alpha}(N)$ such that whenever ${\cal
K}(x)$
satifies (1.3) with $\lam>\lam_{\alpha}$ then $Du\in
L^{p}\log^{\alpha}L_{loc}(\Om)$ and
$$||Du||_{L^{p}\log^{\alpha}L(\Om'')}\le C_{\alpha}(N,\Om')\int_{\Om'} f(x,Du)dx $$
for every domain $\Om''\subset\Om'\subset\subset\Om$.
\end{thm}
\medskip
 From the proof of Theorem 1.1 one can easily deduce that
$\lambda_{\alpha}$ increases with $\alpha$.

Our main idea to get this result is to {\it approximate} the integrand $f$
by more regular ones. More precisely, we shall approximate $f$ by
strictly convex and
uniformly elliptic functions $f_{\e}$, $\e>0$,  growing as $\xi^{p}$ in the gradient.
To every minimizer $h_{\e}$ of such regular problems, according to
the duality theory in the sense of Ekeland and Temam (see section 2),
we can associate a suitable divergence free vector field denoted by
$B_{\e}$. For the pair $(Dh_{\e},B_{\e})$ we are entitled to
  apply an a priori estimate proved in \cite{immp}, which is
the basic tool in our proof.

More important, these a priori estimates are preserved in passing to
the limit and thus the minimizers of the regular integrals
converge to a minimizer  of our original problem.

We also obtain the convergence in the weak 
topology of $L^{q}\log^{\alpha}L_{\rm loc}(\Om)$ of the solutions of 
the dual problems  $ 
B_{\e}\in\partial f(x,Dh_{\e})$. In the following we will refer to $ 
B_{\e}$ as ``momenta'' of $h_{\e}$. 

We can summarize our result as follows:

\bigskip

i) $Dh_{\e}$ converges weakly in $L^{p}\log^{\alpha}L_{\rm loc}(\Om)$ 
to $Du$

\bigskip


ii)$\int_{\Om'}f_{\e}(x,Dh_{\e})dx$ converges to $\int_{\Om'}f(x,Du)dx$ 
for every $\Om'\subset\subset \Om$

\bigskip

iii) $B_{\e}$ converges weakly in $L^{q}\log^{\alpha}L_{\rm loc}(\Om)$ 
to $B$, such that $B\in \partial f(x,Du)$, for a.e. $x\in \Om$.

\bigskip


When $f$ is a  Gateaux differentiable integrand the minimizers
of (1.1) are solutions to the corresponding Euler - Lagrange equation.
Note that in this case the divergence free
vector field $ B$ coincides with the differential of $f$ in the sense
of Gateaux. In any case, when $f$  degenerates, the associated Euler
Lagrange
equation degenerates too. In connection with the study of mappings
with finite distortion
  the regularity for solutions to equations in which the
ellipticity bounds degenerate has been extensively studied 
 (\cite{cmp},
\cite{ikms}, \cite{immp}, \cite{is}, \cite{mo}).

In particular in \cite{ikms} and  \cite{mo} higher integrability results for finite energy solutions of
equation
$${\rm div}(A(x)|Du|^{p-2}Du)=0\eqno (1.4)$$
have been established  in the cases $p=n$ and $2\le p<n$ respectively. 

It is worth pointing out that here we do not assume differentiability of
the integrand $f$. Therefore our regularity result cannot be deduced
by the above mentioned papers. 

As an application of Theorem 1.1  we conclude with a continuity result
for minizers of (1.1), when $p$ is sufficiently close $N$ (see section 6).

Here we deal with question of regularity for $Du$,  but it is worth 
pointing out that under assumptions (H1) and (H2) a minimizer $u$ of 
$I(\Om, v)$ exists in $W^{1,r}$, $1\le r<p$ (see \cite{g}).
\bigskip
\section
{p-Harmonic couples }

Let $\Om$ be an open subset of $\RR^N$. A pair $[B,E]$ of vector
fields such that
$${\rm div}B(x)=0\qquad\qquad {\rm curl}E(x)=0$$
will be called a  $p$-harmonic couple, $1<p<\infty$, of distortion
${\cal K}(x)\ge 1$ if

$$
\frac{|E|^p}{p}+\frac{|B|^q}{q}\leq {\cal K}(x)\langle
B,E\rangle
\quad{\rm in}\quad \Om \eqno (2.1)
$$

We will be interested in the distortion ${\cal K}={\cal K}(x)$ of the
exponential class $EXP(\Om)$.
Precisely, we shall assume that

$$\int_{\Om}e^{\lambda {\cal K}(x)}dx<\infty \eqno (2.2)$$
for some $\lambda>0$.

It has to be noted (see \cite{is})  that the distortion function  ${\cal
K}
(x)$ admits a $BMO$-majorant. Precisely we can majorize ${\cal K}(x)$
point-wise by a function $K(x)\in BMO(\RR^N)$ such that
$$
e^{\lambda(K-K_0)}-1 \in L^1(\RR^N)
$$
for some $K_0\in L^\infty(\RR^N)$, with $1\leq K_0(x)\leq K(x)$.
Moreover, we can also ensure a bound for the BMO-norm of $K(x)$ in
terms of the exponent $\lambda$, i.e.
$$||K||_{BMO}\le {c(N)\over \lambda}.\eqno (2.3)$$
The following quantity will be used in the sequel
$$ [K]=||K_{0}||_{\infty}+{1\over \lam
}\int_{\RR^{N}}(e^{\lambda(K-K_0)}-1)\eqno (2.4)$$


The right spaces for  $p$-harmonic couples of exponential integrable
distortion are  the Orlicz-Zygmund
spaces
$L^s\log^\alpha L$, $1\leq s<\infty$, $\alpha\in \RR$, equipped by
the Luxemburg norm (\cite{rr}).

Let us recall that for $\alpha\geq 0$ the non-linear functional
\[
[f]_{s,\alpha}=\left[\int|f|^s\log^\alpha\left(e+\frac{|f|}{\|f\|_s}\right)\right]^{1\over s}
\]
is comparable with the Luxemburg norm in the sense that 
$$||f||_{L^{s}\log^{\alpha}L}\le [f]_{s,\alpha}\le 
2||f||_{L^{s}\log^{\alpha}L},\eqno (2.5)$$
see \cite{immp}.

If $\Om$ is a measurable subset of $\RR^{N}$, with positive and
finite
measure, if $s>1$ the following relations are straightforward
$$L^s\log^{-1} L(\Om)\subset L^{s-\e}(\Om) \qquad\quad \forall \e>0$$
and
$$f\in L^s\log^{-1} L(\Om)\,\, {\rm implies}\,\, \lim_{\e\to
0}\e\int_{\Om}|f|^{s-\e}dx=0$$


A technical but useful estimate is contained in the following lemma
proved  in \cite{immp}.

\medskip

\begin{lem}\label{lemma2.1}
For a function $F\in L^p\log^{\alpha} L$, $\alpha \ge 1$ and $K\in BMO$
satisfying (2.3) we have
$$||KF||_{L^{p}\log^{\alpha-p}L}\le {C\over
\lambda}||F||_{L^{p}\log^{\alpha}L}+C([K])||F||_{L^{p}\log^{\alpha-1}L}$$
with $C=C(N)$ and $[K]$ is the quantity defined at (2.4).
\end{lem}

Let us recall now some  results about  div-curl couples, that  we
shall use  in the following. We begin with the Sobolev-Poincar\'e
lemma (\cite{is})

\begin{lem}\label{lemma2.2}
Let $ F=[B,E]$ a div-curl couple in
$L^s_{loc}(\Omega,
\RR^{N}\times\RR^{N})$, $1<s<\infty$ and $\varphi\in
C_0^\infty(\Omega)$. Then there exists a div-curl couple $
F_0=(B_0,E_0)\in L^s(\RR^{N},\RR^{N}\times
\RR^{N})$ such that
\[
\|D(\varphi B-B_0)\|_s\leq c(n,s)\| | \nabla\varphi| \
|F| \|_s
\]
\[
\|D(\varphi E -E_0)\|_s\leq c(n,s)\| | \nabla\varphi| \
|F|
\|_s.
\]
\end{lem}


Concerning integrability properties of  $p$-harmonic
couples in $\RR^N$, the following basic a priori estimate was
proved in \cite{immp}.

\begin{thm}\label{thm2.3}
Let   $F=[B,E]$ be a p-harmonic couple of
distortion ${\cal K}$ with $
E\in L^p\log^{\alpha} L$ and $
B\in L^q\log^{\alpha} L$, where $1<p,q<\infty$, $p+q=pq$.
For any $\alpha\geq 0$ there exists $\lambda_\alpha\geq 1$ such that,
whenever $\cal K$ satisfies (2.2) with
$\lambda\geq\lambda_\alpha$, we have
\[
\|  \  |E|^p+|B|^q\|_{L\log^\alpha L(\RR^N)}\leq
c([K])\|  \  |E|^p+|B|^q\|_{L\log^{\alpha-1} L(\RR^N)} .
\]
\end{thm}

Combining Lemma 2.1 and Theorem 2.3, one gets the following a priori
estimate for the curl free vector field $E$ of a p-harmonic
couple.
\medskip

\begin{cor}\label{cor2.4}
Let   $ F=[ B, E]$ be a p-harmonic couple of
distortion ${\cal K}$   with $
E\in L^p\log^{\alpha} L$ and $
B\in L^q\log^{\alpha} L$, where $1<p,q<\infty$, $p+q=pq$.
For any $\alpha\geq p+1$ there exists $\lambda_\alpha\geq 1$ such that,
whenever $\cal K$ satisfies (2.2) with
$\lambda\geq\lambda_\alpha$, we have
\[
\|  \  |E|^p\|_{L\log^\alpha L}\leq
c(K)\|  \  |E|^p\|_{L\log^{\alpha-1} L} .
\]
\end{cor}
\bigskip

{\bf Proof}:\qquad By distortion inequality (2.1)   there
exists a positive constant $c_{p}$ such that
$$ |B|^{q}\leq c_{p}|{\cal K}E|^{p}\leq c_{p}| K
E|^{p}$$
where $K$ is a BMO-majorant of ${\cal K}$ verifying (2.3).
Fix $\alpha \ge p+1$ and  an integer $p\le n\le p+1$. We apply Theorem 2.3
n-times,  getting
\begin{eqnarray*}
\| | E|^p+| B|^q\|_{L\log^\alpha L} &\leq &
[c(K)]^{n}\|  | E|^p+| B|^q\|_{L\log^{\alpha-n} L}\cr\cr
& \leq  &
2[c(K)]^{n}\|  \  |K E|^p\|_{L\log^{\alpha-n} L}  \hskip4cm
(2.6)
\end{eqnarray*}
Last term in (2.6) can be estimate using Lemma 2.1 obtaining
$$\|    |KE|^p\|_{L\log^{\alpha-n} L}\leq {c\over \lambda}
\|  \  | E|^p\|_{L\log^{\alpha-n+p} L}+ C_{K}\|  \  |
E|^p\|_{L\log^{\alpha-n+p-1} L}\eqno (2.7)$$
Putting together estimates (2.6) and (2.7) and using the assumption
$p-n\le 0$ we get
$$ \| | E|^p\|_{L\log^\alpha L}\leq {2[c(K)]^{n}\over \lambda}
\|  \  | E|^p\|_{L\log^{\alpha} L}+ C_{K}\|  \  |
E|^p\|_{L\log^{\alpha-1} L}$$
The proof of the Corollary is completed by choosing
$\lambda>2[c(K)]^{n}$.
\bigskip

In what follows we shall use a local version of Corollary 2.4.


\smallskip
\begin{thm}
  For any $\alpha\geq p+1$ there exists $\lambda_\alpha\geq 1$ such that,
whenever $\cal K$ satisfies (2.2) with $\lambda\geq
\lambda_\alpha$, we have

\medskip

$$\|  \ |\varphi^q E|^p\|_{L\log^\alpha L}
\leq c([K])\| \ |D\varphi^q| |E|^{p}
\ \|_{L^\frac{N}{N-1}} \eqno (2.8)
$$
for any function $\varphi\in C^\infty_0(\Om)$.
\end{thm}

\noindent The proof of estimate $(2.8)$ can be easily achieved using
Corollary  2.4 and Lemma 2.2.









\bigskip

\bigskip

\section
{The dual problem}

\bigskip


Let $V,\,V^{*}$ two locally convex topological vector spaces
(l.c.t.v.s) in duality and let $(\cdot,\cdot)$ denote the duality between them.
For any $G:V\to \RR$ we indicate by $G^{*}$ its Young-Fenchel
transform, i.e.
$$G^{*}(v^{*})=\sup_{v\in V}\{(v,v^{*})-G(v)\}\qquad\quad v^{*}\in
V^{*}$$
In case of integral functionals the expression of the Young-Fenchel
transform can be obtained  by using the following (see \cite{et})

\smallskip

\begin{prop}\label{proposition3.1}
     If $g(x,z):\Om\times \RR^{N}\to \RR$ is a Caratheodory function
     and $G(u)=\int_{\Om}g(x,u(x))dx$ , $u\in L^{s}(\Om)$, is finite
     on some $u_{o}\in L^{s}(\Om)$ then
     $$G^{*}(u^{*})=\int_{\Om}g^{*}(x,u^{*}(x))dx$$
     where $g^{*}$ denote the Young-Fenchel
transform of $g$ and is defined by
$$ g^{*}(x,\eta)= \sup_{\xi\in \RR^{N}}\{\langle \eta,\xi\rangle-g(x,\xi)\}$$
where $\langle \cdot,\cdot\rangle$ is the scalar product between vectors
in $\RR^{N}$.
\end{prop}

\smallskip
Let  $W,\,W^{*}$ be other two l.c.t.v.s in duality and let $\Lambda
:V\to W$ be a linear continuous operator. We shall denote by $\Lambda^{*}$
the adjoint operator of $\Lambda$.

Consider the problem
$$\inf_{v\in V}\{F(v)+G(\Lambda v)\}\leqno (\cal P)$$
where $F:\, V\to (-\infty,+\infty]$ and $G:\, W\to
(-\infty,+\infty]$, $$G(\Lambda u)=\int_{\Om}g(x,\Lambda u(x))dx$$ are convex 
and lower semicontinuous functionals.

The dual problem to $\cal P$ is
$$\sup_{p\in W^{*}}\{-F^{*}(\Lambda^{*} p)-G^{*}(-p)\}\leqno ({\cal P}^{*})$$
where $F^{*},G^{*}$ are the Young-Fenchel
transforms of $F,G$ respectively.

The relationship between $\cal P$ and ${\cal P}^{*}$ is given (see
\cite{et}) by the following

\smallskip

\begin{thm}\label{Theorem3.2}
     Let $V$ be a reflexive Banach space. If $G(\Lambda v)$ is finite
     for every $v\in V$ then problem ${\cal P}^{*}$ admits a solution
     $p$ and
     $$\inf {\cal P} \,=\, \sup {\cal P}^{*}$$
     Moreover if the problem ${\cal P}$ admits a solution $u$ the
     following extremality relations
     $$F(u)+F^{*}(\Lambda^{*} p)=(\Lambda^{*} p,u)\eqno (3.1)$$
     $$g(x,\Lambda u)+g^{*}(x,-p)=\langle u,-p\rangle \eqno (3.2)$$
     hold for a.e. $x\in \Om$.
\end{thm}


\smallskip

If $\Phi=t^{p}\log^{p-1}(e+t)$, $p>1$,  we will consider the following 
Orlicz-Sobolev spaces:
$$W^{1,\Phi}(\Om)=\{v\in W^{1,1}(\Om):\quad Dv\in L^{\Phi}(\Om)\}$$

$$W_{o}^{1,\Phi}(\Om)=\{v\in W_{o}^{1,1}(\Om):\quad Dv\in L^{\Phi}(\Om)\}$$
equipped with the seminorm
$$||v||_{W^{1,\Phi}(\Om)}=||Dv||_{L^{\Phi}(\Om)}.$$

Let us fix a function $\psi \in W^{1,\Phi}(\Om)$ and consider the following
problem
$$\inf_{v\in \psi+W_{o}^{1,\Phi}(\Om) }\int_{\Om}f(x,Dv(x))dx \leqno 
({\cal P}_{\Phi})$$
where the integrand $f(x,\xi)$ is a Caratheodory function verifying
assumption (H1). Following the arguments of \cite{fm}, we have

\smallskip

\begin{lem}\label{Lemma3.3}
     Let $\tilde \Phi=t^{q}\log^{-1}(e+t)$, ${1\over p}+ {1\over q}=1$.
     The dual problem to ${\cal P}_{\Phi}$ is given by
     $$\sup_{
      p\in L^{\tilde \Phi}(\Om)\,\,
     {\rm div}\,p=0
     }\left \{\int_{\Om}\langle p,D\psi \rangle
     dx-\int_{\Om}f^{*}(x,p)dx\right \} \leqno({\cal P}_{\Phi}^{*})$$
     and admits a solution $\bar p\in  L^{\tilde \Phi}(\Om)$.
     Moreover $$\inf {\cal P}_{\Phi}=\sup {\cal P}_{\Phi}^{*}$$
     and if $\bar u$ is a solution of problem ${\cal P}_{\Phi}$ the 
following extremality relation
     $$f(x,D\bar u)+f^{*}(x,\bar p)=\langle D\bar u,\bar p\rangle \eqno (3.3)$$
     holds for a.e. $x\in \Om$.
     \end{lem}

\smallskip

\noindent {\bf Proof:}\quad Let us take
$$V=W^{1,\Phi}(\Om)\qquad\qquad\qquad V^{*}=(W^{1,\Phi}(\Om))^{*}$$
$$W=L^{\Phi}(\Om) \qquad\qquad\qquad W^{*}=L^{\tilde \Phi}(\Om) $$
and let $\Lambda \, :V\to W $ the operator defined by $\Lambda v=Dv$.
Moreover, we set
$$G(w)=\int_{\Om}f(x,w(x))dx \qquad\forall w\in L^{\Phi}(\Om)$$
$$
F(v)=\left\{ \begin{array}{ll}
0
& {\rm if}\quad v\in \psi+W_{o}^{1,\Phi}(\Om) \\
& \\
  +\infty & {\rm otherwise} \end{array} \right.
$$
The dual problem to ${\cal P}_{\Phi}$ is
$$\sup_{p\in L^{\tilde \Phi}(\Om)}\{-F^{*}(\Lambda^{*}p)-G^{*}(-p)\}.$$
By definition we have that
$$F^{*}(\Lambda^{*}p)=\sup_{v\in
\psi+W_{o}^{1,\Phi}(\Om)}\left(\int_{\Om}\langle p,Dv \rangle 
dx\right) =\int_{\Om}\langle
p,D\psi \rangle dx+\sup_{v\in
W_{o}^{1,\Phi}(\Om)}\left(\int_{\Om}\langle p,Dv \rangle dx\right)$$
But
$$\sup_{v\in
W_{o}^{1,\Phi}(\Om)}\left(\int_{\Om}\langle p,Dv \rangle
dx\right)=\sup_{v\in {\cal D}(\Om)}\{(-{\rm div}p,v)_{{\cal
D}(\Om)}\}$$
and then
$$F^{*}(\Lambda^{*}p)=\left\{ \begin{array}{ll}
\int_{\Om}\langle
p,D\psi \rangle dx
& {\rm if}\quad {\rm div}p=0 \\
& \\
  +\infty & {\rm otherwise} \end{array} \right.
$$
On the other hand by proposition 3.1  we have
$$G^{*}(p)=\int_{\Om}f^{*}(x,p)$$
 We observe that
$$G(Dv)=\int_{\Om}f(x,Dv)$$
is finite for every $v\in W^{1,\Phi}(\Om)$, thanks to the  growth assumption
(H1) and H\"older inequality in Orlicz spaces. Moreover, the 
convexity of $f$ tells us that ${\cal P}_{\Phi}^{*}$ has a solution 
$\bar p$ for which the extremality relation (3.3) holds (see \cite{et}, 
Propositions III 2.2, III 2.3 and III 2.4).

\bigskip



\noindent  {\bf Remark 3.5}\qquad The growth and coercitivity 
assumptions on  $ f(x,\xi) $ imply
that the $f^{*}$ satisfies the following

\medskip

$$c_{p}{1\over k^{1\over {p-1}}(x)}|\eta|^q\leq
f^{*}(x,\eta)\leq c_{p} k^{1\over {p-1}}(x)|\eta|^{q}\leqno (H3)$$

\noindent for almost every $x\in \Om$ and all $\eta\in\RR^{N}$ with 
$p,q$ H\"older conjugate exponents.




To  every solution $\bar u$ of ${\cal P}_{\Phi}$ correspond
two vector
fields,
the first one $ B(x)$, solution of ${\cal P}^{*}_{\Phi}$, is 
divergence free, whereas the second
$ E(x)=D\bar u(x)$ is curl free. The growth and coercitivity
assumptions on $f(x,\xi)$ and on $f^{*}(x,\eta)$ expressed by (H1)
and (H3) imply that $ B(x)$ and $ E(x)$
  are coupled by the   inequality
$$
\frac{|E|^p}{p}+\frac{|B|^q}{q}\leq {\cal K}(x)\langle
B,E\rangle
\quad{\rm in}\quad \Om
$$

\bigskip

One can easily check that the  function  ${\cal K}(x)$
coincide with $ k(x)$, in case $p\geq 2$ and with  $k^{1\over 
{p-1}}(x)$, in  case $1<p<2$.


This inequality tells us that
a  solution  of ${\cal P}_{\Phi}$ give rise to a p-harmonic
couple.

\section
  { Proof of Theorem 1.1}

\bigskip

\bigskip
Let us fix a ball $\Om'$ compactly contained in $\Om$.
We divide the proof in two steps. In the first one we construct a map $h:\Om'\to R^{N}$, 
whose gradient lies in
$L^{p}\log^{\a}L_{\rm loc}(\Om')$, for all $\a>0$, while
in the second one we show that the map $h$ has the 
same energy of
our minimizer $u$ on the ball $\Om'$.
Thus we conclude that $u=h$ a.e. using strict convexity of $f$.

More precisely, we have the following:

\bigskip
{\bf Step 1}\quad {\it The approximation}

\medskip

Let $\e$ denote a sequence of positive real numbers converging to
zero. Define $u_{\e}$ as the $\e$-mollification of $u$
through $\varphi_{\e}$, where $\{\varphi_{\e}\}_{\e>0}$ is a family of
smooth mollifiers. Then $Du_{\e}\to Du $ in $L^{p}\log^{-1}L(\Om')$

\noindent  Then, for $\delta>0$, we set
$$f_{\e,\delta}(x,\xi)\, =\, {1+\e\delta \over 1+\e\delta+\e
k(x)}f(x,\xi)+{\e k(x)\over 1+\e\delta+\e
k(x)}\vert\xi\vert^{p} $$
One can easily check that

$${1\over k(x)}\vert \xi\vert^{p}\le f_{\e,\d}(x,\xi)\le k(x)\vert
\xi\vert^{p}\eqno (4.1)$$
$${\e\over 1+\e+\e\d}\vert \xi\vert^{p}\le f_{\e,\d}(x,\xi)\le
{1+\e+\e\d\over \e} \vert
\xi\vert^{p}\eqno (4.2)$$
Next,    consider the variational integrals
$$I_{\e,\d} [h]\,=\, \int_{\Om'} f_{\e,\d}(x,Dh)dx
$$
and the problems
$$\inf_{h\in u_{\e}+W^{1,p}_{0}(\Om')}\int_{\Om'} f_{\e,\d}(x,Dh)dx \leqno 
({\cal P}_{\e,\d})$$
that  are well defined since $u_{\e}\in W^{1,p}(\Om')$ for every
$\e>0$ .

\noindent Since  $f_{\e,\d}$ is strictly
convex by definition and it is coercive thanks to (4.2), each problem
${\cal P}_{\e,\d}$  admits a unique solution  $h_{\e}=h_{\e,\d}\in
W^{1,p}(\Om')$.
Setting
$$V=W^{1,p}(\Om')\qquad\qquad V^{*}=(W^{1,p}(\Om'))^{*}$$
$$W=L^{p}(\Om')\qquad\qquad W^{*}=L^{q}(\Om')$$
and $\Lambda u=Du$ in
Theorem 3.2, we find that the dual problem to ${\cal P}_{\e}$ is given by
$$\sup_{B\in L^{q},\, div B=0}\left\{\int_{\Om'}\langle
B,Du_{\e}\rangle dx -\int_{\Om'}f^{*}_{\e,\d}(x,
B)dx\right\}\leqno ({\cal P}^{*}_{\e,\d}) $$
and has a 
solution denoted by $B_{\e}=B_{\e,\d}$, which , by the extremality 
relations of Theorem 3.2,  satisfies
$$\langle Dh_{\e,\d},B_{\e,\d} \rangle=
f_{\e,\d}(x,Dh_{\e,\d})+f^{*}_{\e,\d}(x,B_{\e,\d})\eqno (4.3)$$

The div-curl couple
$(Dh_{\e,\d},B_{\e,\d})$ is a p-harmonic couple in $\Om'$
of distortion  ${\cal K}(x)$, thanks to (4.1).

\noindent Moreover,  (4.2)
implies that,  for every $\e$, $(Dh_{\e,\d},
B_{\e,\d})$
is also a p-harmonic couple of bounded distortion and then
$$|B_{\e,\d}|\le C_{\e,\d}|Dh_{\e,\d}|^{p-1}\eqno
(4.4)$$

Classical higher integrability results for minimizers of strictly
elliptic variational integrals tell us that $h_{\e}$ belongs to $W^{1,r}_{\rm
loc}(\Om')$ for some $r>p$
and then  $Dh_{\e}\in
L^{p}\log^{\a}L_{\rm loc}(\Om')$, for every $\a>0$ (see 
\cite{gm}, \cite{me}).

\noindent By (4.4),  $
B_{\e}\in
L^{q}\log^{\a}L_{\rm loc}(\Om')$, for every $\a>0$ where, as usual, $q$ is the
H\"older
conjugate
exponent of $p$.

Therefore, we are legitimate to apply Theorem 2.5 to
find , for every $\Om''$ compactly contained in $\Om'$, that

\begin {eqnarray*}
||Dh_{\e}||^{p}_{L^{p}\log^{\a}L(\Om'')} & \le &
C\int_{\Om'}f_{\e,\d}(x,Dh_{\e}(x))dx\cr\cr
& \le &
  C\int_{\Om'}f_{\e,\d}(x,Du_{\e}(x))dx\hskip 3cm (4.5)
  \end {eqnarray*}
where we also  used (4.1) and the minimality of $h_{\e}$. Here and in
what follows $C$ denotes a constant depending on $\a,N $ and $\Om''$.

Now, we choose
  $\d$  related to $\e$
via the equality
$$\d=\d_{\e}\,:= \left(\int_{\Om'}k(x)|Du_{\e}|^{p}dx\right)^{2}
\eqno (4.6)$$
Using the definition of $f_{\e,\d(\e)}$, Jensen inequality, dominated 
convergence Theorem and
  (4.6) we get
\begin{eqnarray*}
||Dh_{\e}||^{p}_{L^{p}\log^{\a}L(\Om'')} &\le &
C\int_{\Om'}f_{\e,\d(\e)}(x,Du_{\e}(x))dx
\cr\cr
& \le   & C\Bigg[\int_{\Om'}f(x,Du_{\e}(x))dx+\int_{\Om'}{\e
k\over
1+\e\d+\e k}|Du_{\e}|^{p}dx\Bigg]\cr\cr
& \le & C\Bigg[\int_{\Om'}f(x,Du(x))dx+o(\e)\Bigg]
\end{eqnarray*}
A routine diagonal argument gives us the sequence $h_{j}=h_{\e_{j}}$ weakly
converging.
The gradient of limit mapping $h$, $Dh$, actually belongs to
$L^{p}\log^{\a}L_{loc}(\Om')$, by  the lower semicontinuity of the
norms.

\medskip

{\bf Step 2}\quad {\it The uniqueness}

\medskip



Thank to the strict convexity of $f$, to complete the proof  we need  to show that
$$\int_{\Om'}f(x,Du)=\int_{\Om'}f(x,Dh) \eqno (4.7)$$
Fix an arbitrary compact subset $\Om''$ of $\Om'$ and a number $0<s<1$.
Observe that
$$\lim_{s\to 1}f^{s}(x,Dh_{\epsilon}(x))=f(x,Dh_{\epsilon}(x))$$
and
$$f^{s}(x,Dh_{\epsilon}(x))\leq {k}^{s}|Dh_{\epsilon}(x)|^{sp}$$
Since
$$\int_{\Om'}{k}^{s}|Dh_{\epsilon}(x)|^{sp}dx\le ||
k||_{exp}^{s}||Dh_{\epsilon}||^{sp}_{L^{p}\log^{m}L}\le
C\left (\int_{\Om'}f(x,Du)dx\right)^{s}$$
we are legitimate to apply dominated  convergence Theorem and obtain
$$\lim_{s\to
1}\int_{\Om'}f^{s}(x,Dh_{\epsilon}(x))dx=\int_{\Om'}f(x,Dh_{\epsilon}(x))dx
\eqno (4.8)$$

The lower semicontinuity of the integral, the definition of $f_{\e,\d}$
and (4.8) imply, up to a not relabelled subquence, that

\begin{eqnarray*} & &\int_{\Om''} f(x, Dh)dx  \le  \lim_{\e\to
0}\int_{\Om'}f(x, Dh_{\e})dx\cr\cr
& = &\lim_{\e\to 0}
\lim_{s\to 1}\int_{\Om'}f(x, Dh_{\e})^{s}dx\cr\cr
& \le & \lim_{\e\to 0}
\lim_{s\to 1}\int_{\Om'}f_{\e,\d}(x, Dh_{\e})^{s}\Big({1+\e\d+\e k\over
1+\e\d}\Big)^{s}\hskip4cm (4.9)\cr\cr
& \le & \lim_{\e\to 0}
\lim_{s\to 1}\Big(\int_{\Om'}f_{\e,\d}(x,
Dh_{\e})\Big)^{s}\Big[\int_{\Om'}\Big({1+\e\d+\e k\over
1+\e\d}\Big)^{s\over 1-s}\Big]^{1-s}\cr\cr
& \le & \lim_{\e\to 0}\Big[\int_{\Om'}f_{\e,\d}(x, Dh_{\e})\lim_{s\to
1}\Big(|\Om'|^{1-s}+\e||k||_{exp}^{s}\Big)\Big]\cr\cr
& = &\lim_{\e\to 0}\int_{\Om'}f_{\e,\d}(x, Dh_{\e})
\end{eqnarray*}
where we used H\"older inequality. Now, using in (4.9) the minimality
of
$h_{\e}$, choosing $\d$ given by (4.6), again by Jensen
inequality and dominated  convergence Theorem we get

\begin{eqnarray*}
  & &\int_{\Om''} f(x, Dh)dx\cr\cr
\le \lim_{\e\to
0}& &\int_{\Om'}f_{\e,\d}(x, Dh_{\e})\cr\cr
\le \lim_{\e\to
0}& &\int_{\Om'}f_{\e,\d}(x, Du_{\e})\cr\cr
\le & & \int_{\Om'}f(x, Du)
\end{eqnarray*}
Since $\Om''$ is an arbitrary subset of $\Om'$ previous inequality 
yields
that
$$\int_{\Om'}f(x, Dh(x))dx \le \int_{\Om'}f(x, Du(x))dx $$
The opposite inequality follows by the minimality of $u$, once we
observe that the limit map $h$ coincides with $u$ on the boundary of
$\Om'$.



\bigskip

\bigskip
\section
{\bf The convergence of the momenta}

\bigskip

\bigskip
Theorem 1.1 tells us that  $u$ is a solution of the problem ${\cal 
P}_{\Phi}$ defined in Section 3. Namely, if $\Om'\subset\subset \Om$, we have
$$\int_{\Om'}f(x, Du(x))dx=\inf_{v\in u+W_{o}^{1,1}(\Om') }\int_{\Om'}f(x,Dv(x))dx 
=\min_{v\in u+W_{o}^{1,\Phi}(\Om') }\int_{\Om'}f(x,Dv(x))dx
$$
and also 
$$\inf {\cal 
P}_{\Phi}=\sup {\cal 
P}^{*}_{\Phi}$$

We have the following
\medskip

\begin{thm}
    Let  the assumptions of Theorem 1.1 hold. For every $\a>0$  there 
    exists $\lambda_{\a}=\lambda_{\a}(N)$ such that whenever ${\cal 
    K}(x)$ satisfies (1.3) with $\lambda>\lambda_{\a}$ then ${\cal 
P}^{*}_{\Phi}$ admits a solution  
    $B\in 
    L^{q}\log^{\a}L_{loc}(\Om,\RR^{N})$ , i.e.
$$f^{*}(x, B)+f(x, Du)=\langle Du, 
B \rangle$$
for a.e. $x\in \Om$.
\end{thm}

\medskip



\noindent {\bf Proof.}\qquad
Observe that, by the extremality relations (4.3), we have that
the sequence $\{B_{\e}\}$ of the solutions of problems ${\cal 
P}^{*}_{\e}$ is bounded in $L^{q}\log^{\a}L(\Om')$, for every $\a>0$ 
and every $\Om'$ compactly contained in $\Om$.

Then, there exists a divergence free vector field $B$ such that

$$B_{\e}\to B \qquad {\rm weakly\,\,in}\,\, L^{q}\log^{\a}L(\Om')$$
The definition of $f_{\e}$ implies that
$$f^{*}(x,\eta)\, \le \, {1+\e\delta+\e
k(x)\over 1+\e\delta}f^{*}_{\e}(x,\eta)+c_{p}{\e k(x)\over 
1+\e\delta}\vert\eta\vert^{q} \eqno (5.1) $$
The lower semicontinuity  of $\int_{\Om'}f^{*}(x,\eta)$, (5.1) and the 
extremality relations (4.3), up to a not relabelled subsequence,  yield
\begin{eqnarray*}
    & &\int_{\Om'} f^{*}(x, B)dx  \le  \lim_{\e\to
0}\int_{\Om'}f^{*}(x, B_{\e})dx\cr\cr
& \le &\lim_{\e\to 0}
\int_{\Om'}\Big({1+\e\delta+\e
k(x)\over 1+\e\delta}\Big)f^{*}_{\e}(x, B_{\e})dx+c_{p}\int_{\Om'}\Big({\e k(x)\over 
1+\e\delta}\Big)|B_{\e}|^{q}\cr\cr
& = & \lim_{\e\to 0}
\int_{\Om'}f^{*}_{\e}(x, B_{\e}) dx\hskip7cm (5.2)\cr\cr
& = & \lim_{\e\to 0}
\int_{\Om'}(\langle Dh_{\e}, 
B_{\e}\rangle-f_{\e}(x, Dh_{\e})dx
\cr\cr
& = &\int_{\Om'}(\langle Du, 
B \rangle- f(x, Du))dx
\end{eqnarray*}
where the last equality follows by compensated compactness theorem 
(see \cite{t}).
Inequality (5.2) implies 
$$\int_{\Om'} (f^{*}(x, B)+f(x, Du))dx\le \int_{\Om'}\langle Du, 
B \rangle$$
Opposite inequality follows by the definition of Young-Fenchel 
transform.
\section
{\bf The continuity}

\bigskip

When $p>n-1$, Theorem 1 implies continuity of finite energy local
minimizers of the variational integral (1.1).
We begin by proving that every finite energy local
minimizers of the variational integral (1.1) is weakly monotone in
the sense of Manfredi (see \cite{m}).
  To this aim let us give the following

\begin{defn}
A real valued function $u\in W_{loc}^{1,p}(\Omega)$ is said to be
weakly monotone if for every relatively compact subdomain
$\Omega^{\prime}$ of $\Omega$ and all constants $m\leq M$ such that
\[
(m-u)^{+}\in W_0^{1,p}(\Omega^{\prime})
\]
and
\[
(u-M)^{+}\in W_0^{1,p}(\Omega^{\prime})
\]
we have
\[
m\leq u(x)\leq M.
\]
\end{defn}

\smallskip

\begin{cor}
Let $u$ be a finite energy local minimizer of integral (1.1).
  There exists $\lambda_0=\lambda_0(n,p)$ such that if ${\cal K}(x)$
satisfies (1.3) for some $\lambda\geq \lambda_0$, then $u$ is weakly
monotone.
\end{cor}

\noindent {\bf Proof.}\qquad  Fix $\alpha>0$. From Theorem 1.1, there exists
$\lambda_\alpha=\lambda_\alpha(n,p)$ such that if ${\cal K}$
satisfies (1.3)
  for $\lambda\geq\lambda_\alpha$ then $Du\in
L^p\log^{\alpha}L(\Omega^{'})$ for every
$\Omega^{'}\subset\subset\Omega$.
  Then, if $m\leq M$ are two constants such that  $(u-M)^{+}$ and $(m-u)^{+}$
  are in the space  $W_0^{1,p}(\Omega^{\prime})$, we obtain
$$\int_{\Om'}f(x,Du)dx\le
\int_{\Om'}f(x,Du-D(u-M)^{+})dx=\int_{\Om'\cap
\{x:\,u(x)\le M\}}f(x,Du)dx$$
This inequality yields
$$\int_{\Om'\cap
\{x:\,u(x)> M\}}f(x,Du)dx=0$$
Then we have that $Du(x)=0$ for a.e. $x\in \Om' \cap\{x:\,u(x)> M\}$.
This
implies that $u\le M$ for a.e. $x\in \Om'$.
Arguing in the same way we prove that $m\le u$ for a.e. $x\in \Om'$
and then the result.

\bigskip

Now, we are in position to apply a result of \cite{hm} to obtain continuity
of local minimizer of integral (1.1), when $n-1<p\le n$. Namely, we
have

\bigskip

\begin{cor}
Let $u$ be a finite energy local minimizer of integral (1.1).
  There exists $\lambda_0=\lambda_0(n,p)$ such that if ${\cal K}(x)$
satisfies (1.3) for some $\lambda\geq \lambda_0$, then $u$ is locally
bounded and continuous outside a set $\cal Z$ with ${\cal H}^{n-p}({\cal
Z})=0$.
Moreover, $u$ is differentiable a.e. .
\end{cor}


\vfill\eject


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\end{document}





















