\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 46, pp. 1--30.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/46\hfil Anisotropic nonlinear elliptic systems]
{Anisotropic nonlinear elliptic  systems \\ with measure data and
 anisotropic \\ harmonic maps into spheres}
\author[M. Bendahmane, K. H. Karlsen\hfil EJDE-2006/46\hfilneg]
{Mostafa Bendahmane, Kenneth H. Karlsen}  % in alphabetical order

\address{Mostafa Bendahmane \newline
         Centre of Mathematics for Applications, University of Oslo\newline
         P.O. Box 1053, Blindern, N--0316 Oslo, Norway}
\email{mostafab@math.uio.no}

\address{Kenneth Hvistendahl Karlsen \newline
         Centre of Mathematics for Applications, University of Oslo\newline
         P.O. Box 1053, Blindern, N--0316 Oslo, Norway}
\email{kennethk@math.uio.no}
\urladdr{http://www.math.uio.no/$\sim$kennethk/}

\date{}
\thanks{Submitted February 7, 2005. Published April 6, 2006.}
\subjclass[2000]{35A05, 35J70, 58E20}
\keywords{Elliptic system; anisotropic; nonlinear; measure data;
\hfill\break\indent
weak solution; existence; harmonic map}

\begin{abstract}
 We prove existence results for distributional solutions of
 anisotropic nonlinear elliptic systems with a
 measure valued right-hand side. The functional setting
 involves anisotropic Sobolev spaces as
 well as weak Lebesgue (Marcinkiewicz) spaces.
 In a special case we also prove maximal regularity and uniqueness
 results. Some of the obtained results are applied, along with
 an anisotropic variant of the div-curl lemma in the Hardy one space,
 to prove that the space of anisotropic harmonic maps into spheres
 is compact in the weak topology of the relevant anisotropic
 Sobolev space.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{defi}{Definition}[section]
\newtheorem{rem}{Remark}
\allowdisplaybreaks

\newcommand{\norm}[1]{\big\|#1\big\|}
\newcommand{\abs}[1]{\big|#1\big|}
\newcommand{\Set}[1]{\{#1\}}


\section{Introduction} \label{intro}

Let $\Omega$ be a bounded open set
in $\mathbb{R}^N$ ($N\ge 2$) with Lipchitz
boundary $\partial \Omega$. Our aim is to prove the
existence of at least one distributional
solution $u=(u_1,\dots,u_m)^\top$ ($m\ge 1$) to
the anisotropic nonlinear elliptic system
\begin{equation}
   \label{E1_tmp}
   \begin{aligned}
       -\sum_{l=1}^N  \frac{\partial}{\partial x_l}
      \sigma_l\big(x,\frac{\partial u}{\partial x_l} \big) &=\mu,\quad
      \text{in $\Omega$},\\
      u & =0,\quad \text{on $ \partial \Omega$},
   \end{aligned}
\end{equation}
where  the right-hand side $\mu=(\mu_1,\dots,\mu_m)^\top$ is a
given vector-valued Radon measure on $\Omega$ of finite mass.

We assume that the vector fields $\sigma_l:\Omega\times \mathbb{R}^m\to \mathbb{R}^m$,
$l=1,\dots,N$,
satisfy the following conditions concerning
continuity, coercivity, growth, and
strict monotonicity:
\begin{equation}
   \label{estimation:prob:1}
   \begin{split}
      &\text{$\sigma_l(x,\xi)$
      is measurable in $x\in \Omega$ for every
      $\xi\in \mathbb{R}^m$ and}
      \\ &\text{$\sigma_l(x,\xi)$
      is continuous in $\xi\in\mathbb{R}^m$ for a.e.~$x\in\Omega$};
      \\ &\sigma_l(x,\xi)\cdot \xi\ge c_1\abs{\xi}^{p_l}-c_2,
      \quad \forall (x,\xi)\in \Omega\times\mathbb{R}^m;
      \\ &\abs{\sigma_l(x,\xi)}\le c_1'\left|\xi \right|^{p_l-1}+c_2',
      \quad \forall (x,\xi)\in \Omega\times\mathbb{R}^m;
   \end{split}
\end{equation}
and for all $x\in \Omega$, and all $\xi,\xi'\in \mathbb{R}^m$,
\begin{equation}\label{monotonicity:1}
   \left(\sigma_l(x,\xi)-\sigma_l(x,\xi')\right)\cdot
    \left(\xi-\xi'\right)\ge
    \begin{cases}
       c_3\abs{\xi-\xi'}^{p_l}, &
       \text{if $p_l\ge 2$},\\
       c_4
       \frac{\abs{\xi-\xi'}^{2}}{\left(\abs{\xi}
       +\abs{\xi'}\right)^{2-p_l}},
       &\text{if $1<p_l< 2$},
    \end{cases}
\end{equation}
for some positive constants $c_1,c_2,c_1',c_2',c_3,c_4$.

We assume that the exponents $p_1,\dots,p_N>1$ satisfy
\begin{equation}
   \label{ass:pl_ellip_ass}
   \frac{\overline{p}(N-1)}{N(\overline{p}-1)}<p_l < \frac{\overline{p}(N-1)}{N-\overline{p}},
   \quad \overline{p}<N.
   \quad l=1,\dots,N,
\end{equation}
where $\overline{p}$ denotes the harmonic mean of
$p_1,\dots,p_N$, i.e.,
\begin{equation}
   \label{eq:harmonic_av}
   \frac{1}{\overline{p}}=\frac{1}{N}\sum_{l=1}^N  \frac{1}{p_l}.
\end{equation}

The relevance of \eqref{ass:pl_ellip_ass}
is discussed in Remark \ref{rem:discussion}. Here it
suffices to say that the lower bound
implies that solutions belong at least to
$W^{1,1}$, so that we can understand the partial
derivatives in \eqref{E1_tmp}
in the distributional sense.

Fundamentally different from the
scalar case ($m=1$), it is well-known
\cite{Landes:89,Landes:test,Dolz-Hung-Mull,Dolz-Hung-Mull:1,Dolz-Hung-Mull:2,Bensoussan-Boccardo:02} that
an additional structure condition is needed to have
existence of solutions to elliptic systems with
$L^1$ or measure data. Here we shall mainly use the
following anisotropic version of
the so-called (right-)angle condition
(but see Section \ref{sec:newstruct} for a different condition):
\begin{equation}
   \label{estimation:prob:5}
   \begin{gathered}
      \text{$\forall x\in \Omega$, $\forall
      \xi\in \mathbb{R}^{m}$, and $\forall a \in \mathbb{R}^m$ with $\abs{a}\le 1$,}
      \\ \sigma_l(x,\xi)\cdot \left[\left(I-a\otimes a\right)\xi\right]\ge 0,
      \quad l=1,\dots,N,
   \end{gathered}
\end{equation}
where $(I-a\otimes a)$ is the rank $m-1$ orthogonal projector onto
the space orthogonal to the unit vector $a\in \mathbb{R}^m$. If
$\sigma_{i,l}$, $i=1,\dots,m$, denotes the components of the
vector $\sigma_l$, then the angle condition can be stated more
explicitly as 
$$\sum_{i,j=1}^m\sigma_{i,l}(x,\xi)\xi_j(\delta_{i,j}-a_i a_j)\ge 0. 
$$
Clearly, condition \eqref{estimation:prob:5} is void in the scalar case.

A prototype example that is covered by our
assumptions is the anisotropic
$p$-harmonic, or $(p_1,\dots,p_N)$-harmonic, system
\begin{equation}
   \label{E1_tmp:example}
   -\sum_{l=1}^N  \frac{\partial}{\partial x_l}
   \Big(\abs{\frac{\partial u}{\partial x_l}}^{p_l-2}\frac{\partial u}{\partial x_l}  \Big)=\mu.
\end{equation}

We prove herein the existence of a solution to \eqref{E1_tmp}. The
proof is based on the usual strategy of deriving a priori
estimates for a sequence of suitable approximate solutions
$(u_\varepsilon)_{0<\varepsilon\le 1}$ (for which existence is straightforward to
prove) and then to pass to the limit as $\varepsilon\to 0$. Introduce the
numbers
\begin{equation}
   \label{eq:def:q-qs}
 q=\frac{N(\overline{p}-1)}{N-1}, \quad
      q^\star=\frac{Nq}{N-q}= \frac{N(\overline{p}-1)}{N-\overline{p}}.
\end{equation} We derive a priori estimates for $u_\varepsilon$ and the
partial derivatives $\frac{\partial u_\varepsilon}{\partial x_l}$  in the
weak Lebesgue spaces $\mathcal{M}^{q^\star}$ and
$\mathcal{M}^{p_lq/\overline{p}}$, respectively (see Section
\ref{sec:prelim} for the definition of weak Lebesgue spaces). To
prove the weak Lebesgue space estimates we employ an anisotropic
Sobolev inequality \cite{Troisi}. Having derived the weak priori
estimates, we then prove a.e.~convergence of the partial
derivatives $\frac{\partial u_\varepsilon}{\partial x_l}$, which can be turned into strong $L^1$
convergence thanks to the $\mathcal{M}^{p_lq/\overline{p}}$ estimates and,
by \eqref{ass:pl_ellip_ass}, $p_lq/\overline{p}>1$. Equipped with this
convergence we pass to the limit in the strong $L^1$ sense in the
nonlinear vector fields $\sigma_l (x,\frac{\partial u_\varepsilon}{\partial x_l})$, and finally
conclude that the approximate solutions $u_\varepsilon$ converge to a
solution of \eqref{E1_tmp}.

Our existence result and the method of proof rely heavily
on previous work by Dolzmann, Hungerb\"uhler, and M\"uller
\cite{Dolz-Hung-Mull} (see also \cite{Dolz-Hung-Mull:1,Dolz-Hung-Mull:2,Fuchs-Reuling:95,Zhou:00,DalMaso-Murat:98})
dealing with the isotropic $p$-harmonic system
\begin{equation}
   \label{p-harmonic}
   -\mathop{\rm div} \big(\abs{D u}^{p-2} Du \big)=\mu.
\end{equation}
Under the assumption $2-\frac{1}{N}<p<N$, the
work \cite{Dolz-Hung-Mull} proves
existence and regularity
results for distributional solutions of the $p$-harmonic system.
These solutions satisfy $u\in \mathcal{M}^{q^\star}$ and
$Du\in \mathcal{M}^q$, where $q$, $q^\star$ are defined
as in \eqref{eq:def:q-qs} but with $\overline{p}$ replaced by $p$.
The lower bound on the exponent $p$ is
known to be optimal (also in the scalar case).
Regarding the anisotropic
system \eqref{E1_tmp}, note that
\eqref{ass:pl_ellip_ass} implies
$2-\frac{1}{N}<\overline{p}<N$.

Even when $p_l\equiv p$ for all $l$, so that
\eqref{ass:pl_ellip_ass} implies $2-\frac{1}{N}<p<N$
and our results yield the existence of a solution $u$
to \eqref{E1_tmp:example} such that
$u\in \mathcal{M}^{q^\star}$, $\frac{\partial u}{\partial x_l}\in \mathcal{M}^q$ for all $l$,
\eqref{E1_tmp:example} does not coincide with \eqref{p-harmonic}.

While \eqref{p-harmonic} can be viewed
as the Euler-Lagrange system of the classical energy functional
\begin{equation}
   \label{eq:I_def_isotropic}
   I[w]:=\int_{\Omega}\frac{1}{p}\abs{Du}^p\,dx
\end{equation}
on the Sobolev space $W^{1,p}_0$, \eqref{E1_tmp:example}
can be viewed as the Euler-Lagrange system of the anisotropic
energy functional
\begin{equation}
   \label{eq:I_def}
   I[w]:=\int_{\Omega}\sum_{l=1}^N \frac{1}{p_l}
   \abs{\frac{\partial w}{\partial x_l}}^{p_l}\,dx
\end{equation}
on the anisotropic Sobolev space $W^{1,(p_1,\dots,p_N)}_0$.
This illustrates a key difference between \eqref{E1_tmp:example}
and \eqref{p-harmonic}, even when $p_l=p$ for all $l$.

We recall that in the scalar case ($m=1$), existence and regularity results
for distributional solutions with $L^1$ or measure data have
been obtained in \cite{BocGalMar:96,LiZhao:01,Li:01,Bendahmane-Karlsen} for a class
of anisotropic elliptic and parabolic equations.
For an anisotropic parabolic reaction-diffusion-advection
system similar results have been established in \cite{BenLanSaa:Anisotropic}.
These works can be viewed as extensions of
parts of the well known theory developed for distributional solutions of
isotropic elliptic and parabolic equations
with measure data, see, e.g., \cite{BocEtal:89,BocEtal:93,BocGalVaz:01}
and the references cited therein.

When $p\in \left(1,2-\frac{1}{N}\right]$ one
cannot expect solutions to belong to $W^{1,1}$, and hence the notions
of weak derivatives and distributional solutions break down.
This problem is dealt with in the literature on scalar equations
using the notion of entropy/renormalized solutions, see, e.g.,
\cite{BenEtal:95,BlaMur:97,BocEtal:93,DalMurOrsPrig:99,Lions:NSI,Rak:94}.
For isotropic elliptic systems (such as \eqref{p-harmonic}) Dolzmann,
Hungerb\"uhler, and M\"uller \cite{Dolz-Hung-Mull:1}
introduced a notion of solution based on replacing
the weak derivative $Du$ by the approximate
derivative ap\,$Du$. Moreover, existence
results for such solutions were proved.

In our anisotropic setting \eqref{E1_tmp}, we cannot expect
solutions to belong to $W^{1,1}$ as long as
$1<p_l\le\frac{\overline{p}(N-1)}{N(\overline{p}-1)}$, which implies $\overline{p}\in
(1,2-\frac{1}{N}]$. Although we are not going to pursue this here,
let us mention that it seems likely that one can adapt the notion
of solution as well as the arguments used in
\cite{Dolz-Hung-Mull:1}, together with the ideas used in the
present paper, to analyze \eqref{E1_tmp} also in the range
$1<p_l\le\frac{\overline{p}(N-1)}{N(\overline{p}-1)}$.

In \cite{Dolz-Hung-Mull:2}, Dolzmann, Hungerb\"uhler, and M\"uller
proved maximal regularity and uniqueness of
solutions to isotropic $N$-Laplace type
systems. We apply the machinery developed
in \cite{Dolz-Hung-Mull:2} to prove similar
results for anisotropic $N$-Laplace type
systems. A typical example of such a system
is \eqref{E1_tmp:example} with
$p_l=N$ for all $l$ (which does not coincide
with \eqref{p-harmonic} with $p=N$).

One of our motivations for studying \eqref{E1_tmp}
comes from applications to $(p_1,\dots,p_N)$-harmonic maps from $\Omega$ into
the sphere $\mathbb{S}^{m-1}\subset \mathbb{R}^m$  ($m\ge 2$),  sometimes simply
called anisotropic harmonic maps.

Let $b:\overline{\Omega}\to \mathbb{S}^{m-1}$ be a smooth function, and
consider the anisotropic
Dirichlet energy \eqref{eq:I_def} with $w$ belonging
to the admissibility class
\begin{equation}
   \label{eq:def_Aset}
   \mathcal{A} = \Set{w\in W^{1,(p_1,\dots,p_N)}(\Omega;\mathbb{S}^{m-1})\, :\,
   \text{$w=b$ on $\partial\Omega$ in the trace sense}}.
\end{equation}
The corresponding Euler-Lagrange system
is the anisotropic elliptic system
\begin{equation}
   \label{E1_map}
   -\sum_{l=1}^N  \frac{\partial}{\partial x_l}
   \Big(\abs{\frac{\partial u}{\partial x_l}}^{p_l-2}\frac{\partial u}{\partial x_l}  \Big)=
   \sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^{p_l} u,
\end{equation}
together with the constraint $\abs{u}=1$ a.e.~in $\Omega$. A vector-valued
map $u$ of class $W^{1,(p_1,\dots,p_N)}(\Omega;\mathbb{S}^{m-1})$ is
called $(p_1,\dots,p_N)$-harmonic if it satisfies
\eqref{E1_map} in the distributional sense.
Note that the critical growth right-hand side
of \eqref{E1_map} belongs to $L^1(\Omega;\mathbb{R}^m)$.

Although anisotropic harmonic maps
have been very little studied in the literature,
harmonic maps (between general manifolds)
have been intensively studied over the years in terms of their
compactness, existence, uniqueness, and regularity
properties. For an excellent introduction to the theory of harmonic maps,
we refer to the recent book by H{\'e}lein \cite{Helein:2002}.

In the final section of this paper we
study the question of compactness  of sequences of
$(p_1,\dots,p_N)$-harmonic maps with respect
to the weak topology of $W^{1,(p_1,\dots,p_N)}$, at least when $1<\overline{p}<N$.
If $(u_\varepsilon)_{0<\varepsilon\le 1}$ is
a sequence of such maps that converges
weakly to a limit map $u$ as $\varepsilon\to 0$, is it then true
that $u$ is $(p_1,\dots,p_N)$-harmonic?
This is a highly nontrivial question since the
system \eqref{E1_map} has a nonlinearity
of critical growth. Questions like this have been studied by
Chen \cite{Chen:89}, Shatah \cite{Shatah:88},
Evans \cite{Evans:harmonic,Evans:LN}, and  H{\'e}lein \cite{Helein:2002}
for harmonic maps, which are special cases ($p=2$) of $p$-harmonic maps
(see also earlier work by Schoen and Uhlenbeck \cite{SchoenUhlenbeck:82}
on minimizing maps).
A $p$-harmonic map $u$ from $\Omega$ into
$\mathbb{S}^{m-1}$ is a distributional solution of
$$
-\mathop{\rm div} \big(\abs{D u}^{p-2} Du \big)=\abs{Du}^p u.
$$
Compactness properties of $p$-harmonic
maps (between general manifolds) have been
studied by Toro and Wang \cite{ToroWang:95} (see also
Hardt and Lin \cite{HardtLin:87} and Luckhaus \cite{Luckhaus:93} for
earlier work on minimizing maps).
Inspired by Toro and Wang, we prove that
limits of weakly converging sequences of
$(p_1,\dots,p_N)$-harmonic maps are again
$(p_1,\dots,p_N)$-harmonic. This is done under the assumption
that the anisotropy $(p_1,\dots,p_N)>1$ satisfies
\begin{equation}
   \label{ass:pl_har_ass}
   \overline{p}<N, \quad \overline{p}^\star>p_{\rm max}.
\end{equation}
The important condition is the last one, which
requires that the anisotropy is not too much spread out.
The proof relies on some compactness arguments used
for \eqref{E1_tmp} and the important
fact that the right-hand side of \eqref{E1_map} belongs
to the local Hardy one space $\mathcal{H}^1_{{\rm loc}}(\Omega)$. To deduce
this compensated integrability property we rely on an anisotropic variant
of the Hardy space version of the
div-curl lemma due to Coifman, Lions, Meyer, and Semmes \cite{CLMS:93},
which we prove under assumption \eqref{ass:pl_har_ass}.

The remaining part of this paper is organized
as follows: Section \ref{sec:prelim} is devoted
to mathematical preliminaries, including, among other
things, a brief discussion of anisotropic Sobolev and
weak Lebesgue spaces. We also prove
a weak Lebesgue space estimate that will
be used later to obtain a priori estimates
for our approximate solutions.
The main existence result is stated
and proved in Section \ref{sec:results}.
In Sections \ref{nonlinear:case} and \ref{sec:newstruct}
we discuss some extensions. In Section \ref{uniq-results} we prove
maximal regularity and uniqueness results
for \eqref{E1_tmp} when $p_l=N$ for all $l$.
Finally, in Section \ref{sec:harmonic} we study
compactness properties of anisotropic
harmonic maps into spheres.


\section{Mathematical preliminaries}
\label{sec:prelim}

In this section real-valued functions on $\Omega$ are denoted by $g=g(x)$.
Let $1\le p_1,\dots,p_N <\infty$ be $N$ real
numbers. Denote by $\overline{p}$ the harmonic mean of these numbers, i.e.,
$\frac{1}{\overline{p}}=\frac{1}{N}\sum_{l=1}^N  \frac{1}{p_l}$,
and set $p_{\rm max}=\max(p_1,\dots,p_N)$, $p_{\rm min}=\min(p_1,\dots,p_N)$.
We always have $p_{\rm min}\le \overline{p}\le Np_{\rm min}$.
The Sobolev conjugate of $\overline{p}$ is denoted by
$\overline{p}^\star$, i.e., $\overline{p}^\star=\frac{N\overline{p}}{N-\overline{p}}$.


\subsection{Anisotropic Sobolev spaces}
Anisotropic Sobolev spaces were
introduced and studied by Nikol'ski\u{\i} \cite{Nik:58},
Slobodecki\u{\i} \cite{Slob:58}, Troisi \cite{Troisi}, and later by
Trudinger \cite{Trud:74} in the framework of Orlicz spaces.

Herein we need
the anisotropic Sobolev space
$$
W^{1,(p_1,\dots,p_N)}_0(\Omega)=\big\{g \in W^{1,1}_0(\Omega)\,:\, \frac{\partial
g}{\partial x_l}\in L^{p_l}(\Omega), \, l=1,\dots,N\big\}.
$$
This is a Banach space under the norm
$$
\norm{g}_{W^{1,(p_1,\dots,p_N)}_0({\Omega})} =
\norm{g}_{L^1(\Omega)}+
\sum_{l=1}^N  \norm{\frac{\partial g}{\partial x_l}}_{L^{p_l}(\Omega)}.
$$
We use standard notation for
the vector- and matrix-valued
versions of the space/ norm introduced above.
For example, the $\mathbb{R}^m$-valued
version of $W^{1,(p_1,\dots,p_N)}_0(\Omega)$ is denoted
by $W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m)$.

We need the anisotropic
Sobolev embedding theorem.

\begin{thm}[Troisi \cite{Troisi}]
\label{thm:trois}
Suppose $g\in W^{1,(p_1,\dots,p_N)}_0(\Omega)$, and let
$$
\begin{cases}
   q=\overline{p}^\star, & \text{if $\overline{p}^\star<N$}, \\
   q\in [1,\infty), & \text{if $\overline{p}^\star\ge N$.}
\end{cases}
$$
Then there exists a constant $C$, depending
on $N$, $p_1,\dots,p_N$ if $\overline{p}<N$ and also on $q$
and $\abs{\Omega}$ if $\overline{p}\ge N$, such  that
\begin{equation}
   \label{trois}
   \norm{g}_{L^q(\Omega)}\leq
   C \prod_{l=1}^N
   \norm{\frac{\partial g}{\partial x_l}}_{L^{p_l}(\Omega)}^{1/N}.
\end{equation}
\end{thm}

We can replace the geometric mean on the right-hand
side of \eqref{trois} by an arithmetic mean. Indeed,
the inequality between geometric and arithmetic
means implies
$$
\norm{g}_{L^q(\Omega)}\leq
\frac{C}{N} \sum_{l=1}^N 
\norm{\frac{\partial g}{\partial x_l}}_{L^{p_l}(\Omega)},
$$
and thus there is in particular, when $\overline{p}<N$, a continuous
embedding of the space $W^{1,(p_1,\dots,p_N)}_0(\Omega)$ into $L^q(\Omega)$ for all
$q\in [1,\overline{p}^\star]$.

The exponent $\overline{p}^\star$, which is suggested by the usual
scaling argument, is critical if the numbers $p_1,\dots,p_N$ are close enough
to ensure $\overline{p}^\star\ge p_{\rm max}$. It may happen that $\overline{p}^\star<p_{\rm max}$ if
the anisotropy is too much spread out, in which case the true critical
exponent is $p_{\rm max}$ rather than $\overline{p}^\star$. However, this latter
case is excluded by our assumptions, see
\eqref{eq:ops_cond} below.

\subsection{Weak Lebesgue spaces and a technical lemma}
In this paper we will use the weak Lebesgue (Marcinkiewicz)
spaces $\mathcal{M}^q(\Omega)$ ($1<q<\infty$), which
belong to the scale of Lorentz
spaces. They contain the measurable
functions $g:\Omega\to\mathbb{R}$ for
which the distribution function
$$
\lambda_g(\gamma) = \abs{\left\{ x \in \Omega\,:\,
|g(x)|>\gamma\right\}}, \quad \gamma\ge 0,
$$
satisfies an estimate of the form
$$
\lambda_g (\gamma) \leq C \gamma^{-q},
\quad
\text{for some finite constant $C$.}
$$
The space $\mathcal{M}^q(\Omega)$ is a Banach space under the norm
$$
\norm{g}^*_{\mathcal{M}^q(\Omega)}= \sup_{t>0}\, t^{1/q}
\Big(\frac{1}{t}\int_0^t g^*(s)\,ds\Big),
$$
where $g^*$ denotes the nonincreasing
rearrangement of $f$:
$$
g^*(t)=\inf\{\gamma>0\,:\, \lambda_g(\gamma)\le t\}.
$$
We will in what follows use the pseudo norm
$$
\norm{g}_{\mathcal{M}^q(\Omega)} =\inf\{C \,:\, \lambda_g(\gamma)\le
C\gamma^{-q},\, \forall \gamma>0\},
$$
which is equivalent to the norm $\norm{g}^*_{\mathcal{M}^q(\Omega)}$.

It is clear that $L^{q}(\Omega ) \subset \mathcal{M}^{q}(\Omega )$, and
this inclusion is strict as the function
$g(x)=|x|^{-N/q}$ belongs to $\mathcal{M}^q(\Omega)$
but not $L^q(\Omega)$.

A useful property of weak Lebesgue spaces is
the following version of H\"older's inequality: Let
$E\subset \Omega$, $g\in \mathcal{M}^q(\Omega)$, $r<q$, then
$$
\norm{g}_{\mathcal{M}^r(E)}\le \big(\frac{q}{q-r}\big)^{1/r}
|E|^{\frac{1}{r}-\frac{1}{p}} \norm{g}_{\mathcal{M}^q(E)}.
$$
It is then immediate that $\mathcal{M}^{q}(\Omega ) \subset \mathcal{M}^{r}(\Omega)$
if $r< q$. Similarly to the anisotropic Sobolev spaces, we use
standard notation for the vector/matrix-valued versions of the
weak Lebesgue spaces.

We now prove an ``anisotropic version''
of a weak Lebesgue space estimate that goes back
to Talenti \cite{Talenti} and Benilan \textit{et al.}~\cite{BenEtal:95}
for isotropic elliptic equations, and
Dolzmann, Hungerb\"uhler, and M\"uller \cite{Dolz-Hung-Mull,Dolz-Hung-Mull:1}
for isotropic elliptic systems.

\begin{lem}
\label{lempoint1}
Let $g$ be a nonnegative function in $W^{1,(p_1,\dots,p_N)}_0(\Omega)$.
Suppose $\overline{p}< N$, and that there exists a constant $c$ such that
\begin{equation}
   \sum_{l=1}^N \int_{\left\{g\le \gamma\right\}}
   \abs{\frac{\partial g}{\partial x_l}}^{p_l}
   \,dx \leq c(\gamma+1),\quad \forall \gamma>0.
   \label{ass:Lpl_der_est_I:or}
\end{equation}
Then there exists a constant $C$, depending on $c$, such that
\begin{equation*}
   \norm{g}_{\mathcal{M}^{\frac{N(\overline{p}-1)}{N-\overline{p}}}(\Omega)}\leq C.
\end{equation*}
\end{lem}

\begin{proof}
For any $\gamma>0$, the
standard scalar truncation function
$T_\gamma$ on $[0,\infty)$ (at height $\gamma$) is defined as
$$
T_{\gamma}(r):=
\begin{cases}
   r,  &\text{if $ r\leq  \gamma$},\\
   \gamma, &\text{if $r> \gamma$.}
\end{cases}
$$
Then, by \eqref{ass:Lpl_der_est_I:or}, for $\gamma\ge 1$
$$
\int_{\Omega} \abs{\frac{\partial T_{\gamma}(g)}{\partial
x_l}}^{p_l}\,dx = \int_{\{g\le \gamma\}}
\abs{\frac{\partial g}{\partial x_l}}^{p_l}\,dx \le C\gamma,\quad
l=1,\dots,N,
$$
so that the anisotropic Sobolev inequality \eqref{trois} gives
\begin{align*}
   \int_{\Omega}
   \abs{T_{\gamma}(g)}^{\overline{p}^\star}\,dx
   &\leq
   C_1 \Big[\prod_{l=1}^N\Big(\int_{\Omega}
   \abs{\frac{\partial T_{\gamma}(g)}{\partial x_l}}^{p_l}
   \,dx\Big)^{\frac{1}{p_lN}}\Big]^{\overline{p}^\star}
   \\ &\leq C_{2} \Big[\prod_{l=1}^N \gamma^{\frac{1}{p_l N}}\Big]^{\overline{p}^\star}
   = C_2\gamma^{\frac{\overline{p}^\star}{\overline{p}}}.
\end{align*}
Hence, for $\gamma\ge 1$,
\begin{equation*}
   \begin{split}
   \lambda_{g}(\gamma)\le \gamma^{-\overline{p}^\star}\int_{\Omega}\abs{T_{\gamma}(g)}^{\overline{p}^\star}\,dx
   \le C_2 \gamma^{-\overline{p}^\star+\frac{\overline{p}^\star}{\overline{p}}}
    =  C_2 \gamma^{-\frac{N(\overline{p}-1)}{N-\overline{p}}}.
    \end{split}
\end{equation*}
For $\gamma<1$, we have trivially that
$\lambda_{g}(\gamma)\le |\Omega|\le |\Omega|
\gamma^{-\frac{N(\overline{p}-1)}{N-\overline{p}}}$.
This shows that $g\in \mathcal{M}^{\frac{N(\overline{p}-1)}{N-\overline{p}}}(\Omega)$.
\end{proof}

\subsection{Truncation function}
For any $\gamma>0$, define the spherial (radially symmetric) truncation
function $T_{\gamma}:\mathbb{R}^m \rightarrow \mathbb{R}^m $ by
\begin{equation}
   \label{definition:trancation1}
   T_{\gamma}(r):=
   \begin{cases}
      r,  &\text{if $|r|\leq  \gamma$},\\

      \frac{r}{|r|}\gamma, &\text{if $|r|> \gamma$}.
   \end{cases}
\end{equation}
This function will be used repeatedly to derive
a priori estimates for our approximate
solutions.  Observe that
\begin{equation*}
   DT_{\gamma}(r)=
   \begin{cases}
      I,  &\text{if $ |r| < \gamma$,}\\
      \frac{\gamma}{|r|}\big(I-\frac{r \otimes r}{|r|^2}\big),
      &\text{if $|r| >\gamma$}.
   \end{cases}
\end{equation*}
In particular, \eqref{estimation:prob:5} implies
for all $\xi,r\in \mathbb{R}^m$ the crucial property
\begin{equation}
   \label{eq:trunc_prop}
   \sigma_l(x,\xi)\cdot DT_{\gamma}(r)\xi
   \ge \sigma_l(x,\xi)\cdot \xi\, \chi_{|r|< \gamma},
   \quad l=1,\dots,N.
\end{equation}

We refer to Landes \cite{Landes:test} for a discussion
of $T_\gamma$ and other test functions
for elliptic systems, which indeed is a delicate issue.

\section{Existence of a solution}
\label{sec:results}

\subsection{Statement of main theorem}
\begin{defi}
\label{definition1} \rm
A distributional solution of \eqref{E1_tmp}
is a vector-valued
function $u:\Omega\to \mathbb{R}^m$ satisfying
\begin{equation}
   \label{eq:weakreg}
    u\in W^{1,1}_0(\Omega;\mathbb{R}^m), \quad
    \sigma_l\big(x,\frac{\partial u}{\partial x_l} \big)\in L^1(\Omega;\mathbb{R}^m),
    \quad l=1,\dots,N,
\end{equation}
and for all $\varphi \in C^\infty_c(\Omega;\mathbb{R}^m)$,
\begin{align*}
   \int_{\Omega}
   \sum_{l=1}^N 
   \sigma_l\big(x,\frac{\partial u}{\partial x_l} \big)\cdot
   \frac{\partial \varphi}{\partial x_l}\,dx
   =\int_{\Omega}\varphi\, d\mu.
\end{align*}
\end{defi}

\begin{thm}\label{thm:theo1}
Suppose \eqref{estimation:prob:1}-\eqref{estimation:prob:5} hold.
Let $\mu=(\mu_1,\dots,\mu_m)^\top$ be a Radon measure on
$\Omega$ of finite mass. Then there exists
at least one distributional solution $u=(u_1,\dots,u_m)^\top$
of \eqref{E1_tmp}. Moreover,
\begin{equation}
   \label{eq:weak_est}
   u\in \mathcal{M}^{q^\star}(\Omega;\mathbb{R}^m),\quad
   \frac{\partial u}{\partial x_l} \in \mathcal{M}^{p_lq/\overline{p}}(\Omega;\mathbb{R}^m),
   \quad l=1,\dots,N,
\end{equation}
where the exponents $q$ and $q^\star$ are defined
in \eqref{eq:def:q-qs}.
\end{thm}

This theorem will
be an immediate consequence of the
results proved in the subsections that follow.

\begin{rem}
\label{rem:discussion} \rm
The fact that $\overline{p}>2-\frac{1}{N}$ (which is
a consequence of the lower bound in \eqref{ass:pl_ellip_ass})
yields $\overline{p}>\frac{2N}{N+1}>1$ (since $N\ge 2$). This in turn
implies $\frac{\overline{p}(N-1)}{N(\overline{p}-1)} <\frac{\overline{p}(N-1)}{N-\overline{p}}$ and
also $q^\star>1$. Moreover, the lower bound in
\eqref{ass:pl_ellip_ass} is equivalent to $p_lq/\overline{p}>1$ for all
$l$. The upper bound in \eqref{ass:pl_ellip_ass} is equivalent to
$p_lq/\overline{p}>p_l-1$ for all $l$, which is needed for proving strong
convergence of the nonlinear vector fields
$\sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)$, $l=1,\dots,N$. The upper bound is
also equivalent to having $q^\star>p_lq/\overline{p}$ for all $l$.

We do not know if the upper condition in \eqref{ass:pl_ellip_ass}
is optimal for having existence of a solution to \eqref{E1_tmp}, but note
that it is equivalent to having
\begin{equation}
   \label{eq:ops_cond}
   \overline{p}^\star > p_{\rm max} + \frac{\overline{p}}{N-\overline{p}}, \quad
   p_{\rm max}=\max(p_1,\dots,p_N).
\end{equation}
Roughly speaking, this condition requires
that the anisotropy $(p_1,\dots,p_N)$ is
not too much spread out. The case $\overline{p}^\star<p_{\rm max}$ (i.e., when the
anisotropy is highly spread out) seems
difficult to handle since the anisotropic Sobolev
inequality does not imply
$f\in L^{p_{\rm max}}$ when $f\in W^{1,(p_1,\dots,p_N)}_0$. On the other hand,
one may wonder if it is possible to prove existence
under the less restrictive condition $\overline{p}^\star \ge p_{\rm max}$ (but we do
not know how to do it).
In the scalar case \cite{BenLanSaa:Anisotropic,Li:01,LiZhao:01,Bendahmane-Karlsen},
conditions similar to \eqref{ass:pl_ellip_ass} have also been imposed in order
to have existence of a solution. We recall
that there are well known examples of minimizers
of anisotropic integral functionals
that are unbounded when the anisotropy is too spread out
\cite{Gia:87}, see also \cite{AcerFus:94,BMS:90,StrofI,StrofII} for
regularity results for minimizers
of anisotropic integral functionals
that hold under the assumption that the anisotropy
is not too spread out.
\end{rem}

\subsection{Approximate solutions}
To prove existence of a solution
to \eqref{E1_tmp} we introduce approximating problems for
which existence is easy to prove.
To this end, let $(f_\varepsilon)_{0<\varepsilon\le 1}\subset C_c^\infty(\Omega;\mathbb{R}^m)$
be a sequence defined by $f_\varepsilon=\mu\star\omega_{\varepsilon}$,
where $\omega_{\varepsilon}(x)=\frac{1}{\varepsilon^N}\omega_0\big(\frac{x}{\varepsilon}\big)\ge 0$
and $\omega_0$ is a nonnegative function in $C_c^{\infty}(B(0,1))$
with $\int \omega_0\,dx=1$. It is always understood that $\varepsilon$
takes values in a sequence in $(0,\infty)$
tending to zero.  Clearly,
\begin{equation}
   \label{Zu}
   \begin{gathered}
      |f_\varepsilon|\leq C(\varepsilon)
      \quad \text{and}\quad
      \int_{\Omega} |f_\varepsilon| \,dx\leq |\mu|,\\
      f_{\varepsilon} \overset{\star}\rightharpoonup  
      \mu\mbox{ in the sense of measures as $\varepsilon \to 0$}.
   \end{gathered}
\end{equation}
For $u,v\in W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m)$, we denote
by {\bf A} the operator
$$
{\bf A}:u\mapsto \Big(v\mapsto
\int_\Omega \sum_{l=1}^N  \sigma_l\big(x,\frac{\partial u}{\partial x_l} \big)\cdot
\frac{\partial v}{\partial x_l} \,dx \Big).
$$
Clearly, ${\bf A}$ is well-defined and monotone. We recall
that monotone means 
$$
\langle {\bf A}(u)-{\bf A} (v),u-v\rangle \ge 0
$$ 
for all $u,v \in W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m)$. Here
$\langle \cdot,\cdot\rangle$ denotes the duality pairing between
$W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m)$ and
$\sum_{l=1}^{N}W^{-1,p_l'}(\Omega;\mathbb{R}^m)$ ($p_l'=\frac{p_l}{p_l-1}$).
It is not difficult to deduce from the coercivity condition in
\eqref{estimation:prob:1} that ${\bf A}$ is coercive. The
growth condition of our operator ${\bf A}$ implies that
${\bf A}$ is hemicontinuous, i.e., the mapping $\lambda \to
\langle{\bf A}(u+\lambda v),w\rangle$
is continuous on the real axis for $u,v,w \in W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m)$.
On the other hand, by \eqref{estimation:prob:1},
\begin{equation*}
      \abs{\langle {\bf A} u,v \rangle} \le
      c\sum_{l=1}^N  \Big(\int_{\Omega}\Big(\abs{\frac{\partial u}{\partial x_l}}^{p_l-1}+1
     \Big)^{\frac{p_l}{p_l-1}}\,dx \Big)^{\frac{p_l-1}{p_l}}
     \Big(\int_{\Omega} \abs{\frac{\partial v}{\partial x_l}}^{p_l} \,dx \Big)^{1/p_l},
\end{equation*}
which implies the boundedness of ${\bf A}$.
Then, using a standard theorem for monotone operators
(see, e.g., \cite[Theorem 2.1/Chapter 2]{Lions:Book69}),
it follows that ${\bf A}$ is bijective, and hence
there exists a sequence of functions
$$
(u_\varepsilon)_{0<\varepsilon\leq 1}  \subset W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m),
$$
each of them satisfying the weak formulation
\begin{equation}
   \label{E1_approx:weak}
   \int_\Omega \sum_{l=1}^N 
   \sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)\cdot
   \frac{\partial \varphi}{\partial x_l}\,dx=
   \int_{\Omega}f_\varepsilon \cdot\varphi\,dx,
   \quad \forall \varphi \in W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m).
\end{equation}
Now the proof of Theorem \ref{thm:theo1} consists of two
main steps. First, we prove $\varepsilon$-uniform
a priori estimates in weak Lebesgue
spaces for $u_\varepsilon$ and $\frac{\partial u_\varepsilon}{\partial x_l}$.
Second, we pass to the limit in \eqref{E1_approx:weak}
as $\varepsilon\to 0$.

\subsection{A priori estimates}
\label{priori}

\begin{lem}
\label{lem:main_est}
There exists a constant
$c$, not depending on $\varepsilon$, such that
\begin{equation}
   \sum_{l=1}^N \int_{\left\{|u_\varepsilon|\le \gamma\right\}}
   \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l}
   \,dx \leq c(\gamma+1), \quad \forall \gamma>0.
   \label{ass:Lpl_der_est_I}
\end{equation}
\end{lem}

\begin{proof}
Inserting $\varphi=T_\gamma(u_\varepsilon)$
into \eqref{E1_approx:weak} gives
\begin{equation*}
   \begin{split}
      & \int_{\Omega} \sum_{l=1}^N 
       \sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l} \big) \cdot DT_\gamma(u_\varepsilon)\frac{\partial u_\varepsilon}{\partial x_l} \,dx
      =\int_{\Omega} f_\varepsilon\cdot T_\gamma(u_\varepsilon)\,dx.
   \end{split}
\end{equation*}
Using \eqref{eq:trunc_prop} and the coercivity condition
in \eqref{estimation:prob:1}, we obtain \eqref{ass:Lpl_der_est_I}.
\end{proof}

\begin{lem}
\label{lem:weak-apriori}
There exists a constant
$C$, not depending on $\varepsilon$, such that
\begin{equation}
   \label{eq:weak_estI}
   \norm{u_\varepsilon}_{\mathcal{M}^{q^\star}(\Omega;\mathbb{R}^m)}\le C
\end{equation}
and
\begin{equation}
   \label{eq:weak_estII}
   \norm{\frac{\partial u_\varepsilon}{\partial x_l}}_{\mathcal{M}^{p_lq/\overline{p}}
   (\Omega;\mathbb{R}^m)}\leq C,
   \quad l=1,\dots,N.
\end{equation}
where the exponents $q$ and $q^\star$ are defined in \eqref{eq:def:q-qs}.
\end{lem}

\begin{proof}
Let $ a=\frac{N(\overline{p}-1)}{N-\overline{p}}$. By Lemma \ref{lem:main_est} and
$\abs{\frac{\partial}{\partial x_l}|u_\varepsilon|}
\le\abs{\frac{\partial}{\partial x_l} u_\varepsilon}$,
$$
\sum_{l=1}^N \int_{\{|u_\varepsilon|\le \gamma\}}
\abs{\frac{\partial |u_\varepsilon|}{\partial x_l}}^{p_l}
\,dx \leq c(\gamma+1).
$$
Applying Lemma \ref{lempoint1} to
$|u_\varepsilon|$ gives $\norm{\,|u_\varepsilon|\,}_{\mathcal{M}^{a}(\Omega)}\le C$,
which also proves \eqref{eq:weak_estI}.
By \eqref{ass:Lpl_der_est_I} and \eqref{eq:weak_estI}, we have
for any $\alpha,\gamma\ge 1$
\begin{align*}
   \lambda_{\abs{\frac{\partial u_\varepsilon}{\partial x_l}}}(\alpha)
   &\le \abs{\{ x\in \Omega: \abs{\frac{\partial u_\varepsilon}{\partial x_l}}>\alpha,\,
   |u_\varepsilon|\leq \gamma\}} \\
   & \quad + \abs{\{ x\in \Omega: \abs{\frac{\partial u_\varepsilon}{\partial x_l}}>\alpha,\,
   |u_\varepsilon|> \gamma\}}    \\
   &\leq \frac{1}{\alpha^{p_l}}\int_{\{|u_\varepsilon|\le \gamma\}}
   \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l}\,dx
   + \lambda_{|u_\varepsilon|}(\gamma)    \\
 & \le C\big( \frac{\gamma}{\alpha^{p_l}}+\gamma^{-a}\big).
\end{align*}
Optimizing with respect
to $\gamma$ gives $\gamma=\frac{1}{a}\alpha^{\frac{p_l}{a+1}}$,
which in turn yields the bound
$\lambda_{\abs{\frac{\partial u_\varepsilon}{\partial x_l}}}(\alpha)
\le C\alpha^{-\frac{a p_l}{a+1}}$.
With the choice $a=q^\star$, see \eqref{eq:def:q-qs},
$$
\lambda_{\abs{\frac{\partial u_\varepsilon}{\partial x_l}}}(\alpha)\le
C\alpha^{-\frac{p_l}{\overline{p}}\frac{N(\overline{p}-1)}{N-1}}, \quad \alpha\ge 1.
$$
For $\alpha<1$, $\lambda_{\abs{\frac{\partial u_\varepsilon}{\partial x_l}}}(\alpha)
\le |\Omega|\le |\Omega| 
\alpha^{-\frac{p_l}{\overline{p}}\frac{N(\overline{p}-1)}{N-1}}$. This
proves \eqref{eq:weak_estII}.
\end{proof}

\subsection{Strong $L^1$ convergence of nonlinear vector fields}
\label{strong}
In view of Lemma \ref{lem:weak-apriori}, $u_\varepsilon$
is uniformly bounded in $L^{s_0}(\Omega;\mathbb{R}^m)$
for some $s_0<q^\star$ with $s_0>p_lq/\overline{p}$ for all $l$, and $\frac{\partial u_\varepsilon}{\partial x_l}$
is uniformly bounded in $L^{s_l}(\Omega;\mathbb{R}^m)$
for some $s_l>1$ with $p_l-1< s_l <p_lq/\overline{p}$, $l=1,\dots,N$.
{}From this we get that $u_\varepsilon$ is
uniformly bounded in the isotropic Sobolev space
$$
W^{1,s_{\rm min}}_0(\Omega;\mathbb{R}^m), \quad s_{\rm min}=\min(s_1,\dots,s_N).
$$
Consequently, we can assume without
loss of generality that as $\varepsilon\to 0$
\begin{equation}
   \label{limit:passing1}
    \begin{aligned}
      &u_\varepsilon \to u \quad \text{a.e.~in $\Omega$ and in $L^{s_{\rm min}}(\Omega;\mathbb{R}^m)$},\\
      &u_\varepsilon \rightharpoonup  u \quad \text{in $W^{1,s_{\rm min}}_0(\Omega;\mathbb{R}^m)$},\\
       &\abs{\frac{\partial u_\varepsilon}{\partial x_l}-
      \frac{\partial u}{\partial x_l}} \rightharpoonup h_l
      \quad \text{in $L^{s_l}(\Omega)$}, \quad l=1,\dots,N,\\
      &\sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l} \big)
      \rightharpoonup \beta_l\quad \text{in $L^{s_l'}(\Omega;\mathbb{R}^m)$},
      \quad l=1,\dots,N,\\
      & f_\varepsilon \overset{\star}\rightharpoonup  \mu \quad
      \text{in the sense of measures on $\Omega$}.
   \end{aligned}
\end{equation}

Of course, the convergences provided by \eqref{limit:passing1}
are not strong enough if we want to pass to the limit $\varepsilon\to 0$
in the nonlinear system \eqref{E1_approx:weak},
and the proof of Theorem \ref{thm:theo1} will be completed
by Lemma \ref{lem:A_conv} below. To prove this lemma we follow closely the argument
used in \cite{Dolz-Hung-Mull} for the isotropic $p$-harmonic
system \eqref{p-harmonic} (see also \cite{Evans:LN}),
which is based on using a regularized test function and
a localization procedure to handle the problem
that $u$ does not in general belong
to the anisotropic Sobolev space $W^{1,(p_1,\dots,p_N)}_0$.

\begin{lem}\label{lem:A_conv}
For $l=1,\dots,N$, as $\varepsilon\to 0$ we have
\begin{equation}
   \label{eq:sigma_conv}
   \sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l} \big)
   \to \sigma_l\big(x,\frac{\partial u}{\partial x_l} \big)
   \quad
   \text{a.e.~in $\Omega$ and in $L^1(\Omega;\mathbb{R}^m)$.}
\end{equation}
\end{lem}

\begin{proof}
The main part of the proof consists in showing that
\begin{equation}
   \label{eq:hl_null}
   h_l(x)=0\quad \text{for a.e.~$x\in\Omega$},
   \quad l=1,\dots,N,
\end{equation}
where $h_l$ is defined in \eqref{limit:passing1}.
Suppose for the moment the validity of \eqref{eq:hl_null},
and fix any one of the directions $l=1,\dots,N$.
Then, by Vitali's theorem,
$$
\frac{\partial u_\varepsilon}{\partial x_l} \to \frac{\partial u}{\partial x_l} \quad \text{in $L^1(\Omega;\mathbb{R}^m)$},
$$
and, after extracting a subsequence if necessary,
$\frac{\partial u_\varepsilon}{\partial x_l} \to \frac{\partial u}{\partial x_l}$ a.e.~in $\Omega$. From this
we also have  $\sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)\to
\sigma_l\big(x,\frac{\partial u}{\partial x_l}\big)$
a.e.~in $\Omega$. As $\sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)$ is
uniformly bounded in $L^{s_l'}(\Omega;\mathbb{R}^m)$, Vitali's
theorem gives
$$
\sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l} \big)
\to \sigma_l\big(x,\frac{\partial u}{\partial x_l} \big)
\quad \text{in $L^{t_l}(\Omega;\mathbb{R}^m)$},
$$
for any $1\le t_l<s_l'$, which proves
\eqref{eq:sigma_conv}.

We now set out to prove \eqref{eq:hl_null}.
Choose a nonnegative function $\alpha\in
C^\infty([0,\infty)\cap L^\infty([0,\infty))$
such that $\alpha(t)=t$ for $t\in [0,\delta]$
for some $\delta>0$, $\alpha'\ge 0$, and
$\alpha'(t)t\le \alpha(t)$ for all $t\ge 0$
(see \cite{Dolz-Hung-Mull:1} for
an explicit example of such a function).
Then define the function $\psi:\mathbb{R}^m\to \mathbb{R}^m$ by
$$
\psi(r)=\frac{r}{|r|}\alpha(|r|),
$$
and note that $\psi(r)=r$ when $|r|\le \delta$.
We also need two scalar functions $\eta$, $\phi$ of
the following type:
\begin{gather*}
   \eta \in C_c^\infty(\mathbb{R}^m), \quad
   0\le \eta \le 1, \quad
   \text{$\mathrm{supp}(\eta)\subset [0,\delta)$},\\
   \phi \in C_c^\infty(\mathbb{R}^n), \quad 0\le \phi \le 1,
   \quad \int \phi\,dx=1.
\end{gather*}
In what follows, let us fix any one of
the directions $l=1,\dots,N$. Denoting by $v$ a comparison function
in $C^1(\Omega;\mathbb{R}^m)$ (to be chosen later), we proceed by using
the triangle and H\"older inequalities:
\begin{equation*}
   \begin{split}
      & \int_{\Omega}\sum_{l=1}^N  \abs{\frac{\partial u_\varepsilon}{\partial x_l}-\frac{\partial u}{\partial x_l}}\eta(u_\varepsilon-v)\phi\,dx
      \\& \le \int_{\Omega}\sum_{l=1}^N  \abs{\frac{\partial u_\varepsilon}{\partial x_l}-\frac{\partial v}{\partial x_l}}\eta(u_\varepsilon-v)\phi\,dx
      + \int_{\Omega} \sum_{l=1}^N  \abs{\frac{\partial v}{\partial x_l}-\frac{\partial u}{\partial x_l}}\eta(u_\varepsilon-v)\phi\,dx
      \\ &\le\sum_{l=1}^N  \Big(\int_{\Omega}
      \abs{\frac{\partial u_\varepsilon}{\partial x_l}-\frac{\partial v}{\partial x_l}}^{p_l}
      \eta(u_\varepsilon-v)\phi\,dx\Big)^{\frac{1}{p_l}}\Big( \int_\Omega
      \eta(u_\varepsilon-v)\phi \,dx \Big)^{\frac{p_l-1}{p_l}}\\
      &\quad +\int_{\Omega} \sum_{l=1}^N  \abs{\frac{\partial v}{\partial x_l}-\frac{\partial u}{\partial x_l}}\eta(u_\varepsilon-v)\phi\,dx.
   \end{split}
\end{equation*}
Equipped with this and \eqref{limit:passing1}, using in particular
that $u_\varepsilon \to u$ a.e.~and the fact that $\eta$, $\psi$, $D\psi$
are continuous and bounded functions, we deduce
\begin{equation}
   \label{strong:7}
   \begin{split}
      &\int_{\Omega} \sum_{l=1}^N  h_l(x) \eta(u-v)\phi\,dx
      \\ &\le \sum_{l=1}^N  L^{\frac{1}{p_l}}_l
      \Big(\int_\Omega \eta(u-v)\phi\,dx\Big)^{\frac{p_l-1}{p_l}}
      +\int_{\Omega} \sum_{l=1}^N  \abs{\frac{\partial v}{\partial x_l}-\frac{\partial u}{\partial x_l}}
      \eta(u-v)\phi\,dx,
   \end{split}
\end{equation}
where
$$
L_l=L_l(\eta,\phi,\psi):=\underset{\varepsilon\to 0}{\operatorname{lim\, sup}}
\int_{\Omega} \abs{\frac{\partial u_\varepsilon}{\partial x_l}-\frac{\partial v}{\partial x_l}}^{p_l}\eta(u_\varepsilon-v)\phi\,dx.
$$
We must analyze $L_l$, and start with the case $p_l\ge 2$.
By \eqref{monotonicity:1},
\begin{equation}
   \label{strong:2}
   \begin{split}
      &\int_{\Omega} c_3\abs{\frac{\partial u_\varepsilon}{\partial x_l}-\frac{\partial v}{\partial x_l}}^{p_l}\eta(u_\varepsilon-v)\phi\,dx
      \\ &
      \le \int_{\Omega} \sum_{l=1}^N  \Big(\sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)-
      \sigma_l\big(x,\frac{\partial v}{\partial x_l}\big)\Big)\cdot\Big(\frac{\partial u_\varepsilon}{\partial x_l}-\frac{\partial v}{\partial x_l}\Big)
      \eta(u_\varepsilon-v)\phi\,dx\\
      &=\int_{\Omega} \sum_{l=1}^N  \Big(\sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)-
      \sigma_l\big(x,\frac{\partial v}{\partial x_l}\big)\Big)
      \cdot \frac{\partial \psi(u_\varepsilon-v) }{\partial x_l}
      \eta(u_\varepsilon-v)\phi\,dx\\
      &= \int_{\Omega} \sum_{l=1}^N  \sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)
      \cdot \frac{\partial\psi(u_\varepsilon-v)}{\partial x_l}\phi\,dx\\
      &\quad - \int_{\Omega} \sum_{l=1}^N  \sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)
      \cdot \frac{\partial\psi(u_\varepsilon-v)}{\partial x_l}
      (1-\eta(u_\varepsilon-v))\phi\,dx\\
      & \quad - \int_{\Omega} \sum_{l=1}^N  \sigma_l\Big(x,\frac{\partial v}{\partial x_l}\big)
      \cdot \big(\frac{\partial u_\varepsilon}{\partial x_l}-\frac{\partial v}{\partial x_l}\Big)\eta(u_\varepsilon-v)\phi\,dx\\
      &=: E_1+E_2+E_3,
   \end{split}
\end{equation}
In the case $p_l<2$, we employ
\eqref{monotonicity:1} instead as follows:
\begin{equation}
   \label{strong:2:case:2}
   \begin{split}
      &\int_{\Omega}\abs{\frac{\partial u_\varepsilon}{\partial x_l}-\frac{\partial v}{\partial x_l}}^{p_l}\eta(u_\varepsilon-v)\phi\,dx
      \\ & \le \bigg(\int_{\Omega}
      \frac{\abs{\frac{\partial u_\varepsilon}{\partial x_l}-\frac{\partial v}{\partial x_l}}^{2}}
      {\Big(\abs{\frac{\partial u_\varepsilon}{\partial x_l}}+\abs{\frac{\partial v}{\partial x_l}}\Big)^{2-p_l}}
      \eta(u_\varepsilon-v)\phi\,dx\bigg)^{\frac{p_l}{2}}
      \\ & \quad\times
      \Big(\int_\Omega
      \Big(\abs{\frac{\partial u_\varepsilon}{\partial x_l}}+\abs{\frac{\partial v}{\partial x_l}}\Big)^{p_l}
      \eta(u_\varepsilon-v)\phi \,dx \Big)^{\frac{2-p_l}{2}}
      \\ & \le c_4^{-p_l/2}
      \left(E_1+E_2+E_3\right)^{\frac{p_l}{2}}\left(\int_\Omega
      \Big(\abs{\frac{\partial u_\varepsilon}{\partial x_l}}+\abs{\frac{\partial v}{\partial x_l}}\Big)^{p_l}
      \eta(u_\varepsilon-v)\phi \,dx \right)^{\frac{2-p_l}{2}}.
   \end{split}
\end{equation}
Thanks to \eqref{E1_tmp},
$$
E_1=\int_\Omega f_\varepsilon \cdot \psi(u_\varepsilon-v)\phi \,dx-
\int_{\Omega}\sum_{l=1}^N  \sigma_l\Bigl(x,\frac{\partial u_\varepsilon}{\partial x_l}\Bigl) \cdot \psi(u_\varepsilon-v)
\frac{\partial \phi}{\partial x_l}\,dx.
$$
Since
\begin{equation*}
   D \psi(r)=\alpha'(|r|)\frac{r\otimes r}{|r|^2}
   +\frac{\alpha(|r|)}{|r|}
   \big(I-\frac{r\otimes r}{|r|^2}\big),
\end{equation*}
there holds
\begin{equation*}
   \sigma_l(x,\xi)\cdot D \psi(r)\xi\ge 0,
   \quad \forall \xi,r\in \mathbb{R}^m.
\end{equation*}
This follows from \eqref{estimation:prob:5}, since
\begin{align*}
   \sigma_l(x,\xi)\cdot D \psi(r)\xi
   =  \frac{\alpha{\abs{r}}}{\abs{r}}\,
   \sigma_l(x,\xi)\cdot \Big(I-
   \Big[\big(1-\frac{\alpha'(\abs{r})\abs{r}}{\alpha(\abs{r})}\big)
   \, \frac{r\otimes r}{\abs{r}^2}\Big]\Big)\xi,
\end{align*}
where the term inside the square brackets can be written as
$a\otimes a$ for some $a\in \mathbb{R}^m$ with $\abs{a}\le 1$ (recall
that $\alpha'(t) t\le \alpha(t)$). Hence
\begin{equation}
   \label{strong:5}
   E_2\le
   \int_{\Omega} \sum_{l=1}^N \sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)\cdot
   D\psi(u_\varepsilon-v)\frac{\partial v}{\partial x_l} (1-\eta(u_\varepsilon-v))\phi\,dx.
\end{equation}
Since $u_\varepsilon \to u$ a.e.~and $\eta$, $\psi$, $D\psi$
are continuous and bounded functions,
we deduce from \eqref{strong:2} that
\begin{equation}
   \label{strong:6}
   \begin{split}
      L_l&\le \sup|\psi|\int_\Omega \phi\, d\mu
      -\int_{\Omega} \sum_{l=1}^N  \beta_l\cdot
      \psi(u-v)\frac{\partial\phi}{\partial x_l}\,dx\\
      &\quad +\int_{\Omega} \sum_{l=1}^N  {\beta}_l\cdot
      D\psi(u-v)\frac{\partial v}{\partial x_l}\left(1-\eta(u-v)\right)\phi\,dx\\
      &\quad  -\int_{\Omega} \sum_{l=1}^N  \sigma_l\big(x,\frac{\partial v}{\partial x_l}\big)\cdot
      \big(\frac{\partial u}{\partial x_l}-\frac{\partial v}{\partial x_l} \big)
      \eta(u-v)\phi\,dx.
   \end{split}
\end{equation}

At this stage we specify the functions $v$, $\eta$, $\psi$, $\phi$.
Fix any point $x=a\in \Omega$ that is simultaneously
a Lebesgue point of $\frac{\partial u}{\partial x_l},h_l,\beta_l$, $l=1,\dots,N$, and
the measure $\mu$. Choose $v$ as the first order Taylor
polynomial of $u$ around $x=a$:
$$
v(x)=u(a)+Du(a)(x-a),
$$
and replace $\phi, \eta,\psi$ in the above
calculations by the following functions:
\begin{gather*}
   \eta_\rho(r)=\tilde{\eta}\big(\frac{r}{\rho}\big),
   \quad \tilde{\eta}\in C_c^\infty(B(0,1)),
   \quad \tilde{\eta}|_{B(0,\frac{1}{2})}\equiv 1,\\
 \phi_\rho(x)
   =\frac{1}{\rho^n}\tilde{\phi}\big(\frac{x-a}{\rho} \big),
   \quad \tilde{\phi}\in C_c^\infty(B(0,1)),
   \quad \int \tilde{\phi}=1,
\end{gather*}
and $\psi_{\rho}(r)=\rho\psi\big(\frac{r}{\rho}\big)$.
%
Denote by $L_l(\rho)$ the corresponding $L_l$, that is,
$L_l(\rho):=L_l(\eta_\rho,\phi_\rho,\psi_\rho)$.
We deduce $\limsup_{\rho\to 0} L_l(\rho)=0$, since
as $\rho \to 0$,
\begin{gather*}
    \frac{1}{\abs{B(a,\rho)}}
    \int_{B(a,\rho)}\abs{\frac{u-v}{\rho}}\,dx \to 0,
    \\ \frac{1}{\abs{B(a,\rho)}}
    \int_{B(a,\rho)}\sum_{l=1}^N \abs{\frac{\partial u}{\partial x_l}-\frac{\partial v}{\partial x_l}}\,dx\to 0,
    \\ \frac{1}{\abs{B(a,\rho)}}
    \int_{B(a,\rho)}\sum_{l=1}^N  \abs{\beta_l(x)-\beta_l(a)}\,dx\to 0,
\end{gather*}
where the second and third terms in \eqref{strong:6} tend
to zero as we have
$$
\psi_\rho(u-v)\frac{\partial\phi}{\partial x_l}
=\mathcal{O}\big(\frac{u-v}{\rho}\big), \quad 1-
\eta_\rho(u-v)= \mathcal{O}\big(\frac{u-v}{\rho}\big).
$$
The first term tends to zero since
$$
\limsup_{\rho \to 0} \mu(B(a,\rho))/\rho^n <\infty,
$$
and thus $\sup|\psi_\rho|\int_\Omega \phi_\rho\, d\mu \le
C\rho \mu(B(a,\rho))/\rho^n$. In the case $p_l<2$,
we also use that the term $(\cdots)^{\frac{2-p_l}{2}}$ in
\eqref{strong:2:case:2} stays finite in the above
localization procedure (since $N\ge 2$).
Since
$$
\frac{1}{\abs{B(a,\rho)}}
\int_{B(a,\rho)}\sum_{l=1}^N  \abs{h_l(x)-h_l(a)}\,dx\to 0
\quad \text{as $\rho\to 0$},
$$
it follows, via \eqref{strong:7}, that $h(a)=0$.
This completes the proof of \eqref{eq:hl_null},
and hence the lemma.
\end{proof}


\section{An extension}
\label{nonlinear:case}


In this section we show that
the results obtained for \eqref{E1_tmp}
can be extended to more general anisotropic
elliptic systems of the form
\begin{equation}
   \label{E1_tmp:bis}
   \begin{aligned}
       -\sum_{l=1}^N \frac{\partial}{\partial x_l}
      \sigma_l\big(x,\frac{\partial u}{\partial x_l} \big)
      +g(x,u) &=\mu,\quad
      \text{in $\Omega$},\\
      \quad u & =0,\quad
      \text{in $ \partial \Omega$},
   \end{aligned}
\end{equation}
where the vector fields $\sigma_1,\dots,\sigma_N$ are as before.
We assume that the nonlinearity $g(x,r):\Omega\times\mathbb{R}^m\to \mathbb{R}^m$
is measurable in $x\in\Omega$
for all $r\in\mathbb{R}^m$, continuous in $r$ for a.e.~$x\in\Omega$,
and satisfies the following conditions:
\begin{gather}
   g(x,r)\cdot (r-r')\ge 0,  \quad
   \text{ $\forall r,r' \in \mathbb{R}^m$ with $|r'|\le |r|$},
   \label{estimate:nonlinear:1}\\
   \sup\left\{\abs{g(x,r)}:
   \abs{r}\leq \tau \right\} \in L^1(\Omega;\mathbb{R}^m),
   \quad \forall r\in \mathbb{R}^m \text{ and } \forall \tau \in \mathbb{R}.
   \label{estimate:nonlinear:3:bis}
\end{gather}

Condition \eqref{estimate:nonlinear:1}, often called the
angle condition, is also assumed in the recent work
\cite{Bensoussan-Boccardo:02}.
A prototype example of \eqref{E1_tmp:bis} is provided by
the equation
\[
-\sum_{l=1}^N \frac{\partial}{\partial x_l}
\big(\abs{\frac{\partial u}{\partial x_l}}^{p_l-2}\frac{\partial u}{\partial x_l} \big)
+|u|^{\theta-1}u=\mu\,,
\]
for some $\theta>1$.
We look for distributional solutions
to \eqref{E1_tmp:bis} in the following sense:
\begin{defi}\label{definition:nonlinear:1} \rm
A distributional solution of \eqref{E1_tmp:bis}
is a function $u:\Omega\to \mathbb{R}^m$ such that
\eqref{eq:weakreg} and $g(x,u) \in L^1(\Omega;\mathbb{R}^m)$
hold, and $\forall \varphi \in
C^\infty_c(\Omega;\mathbb{R}^m)$
\begin{align*}
   \int_{\Omega}\sum_{l=1}^N 
   \sigma_l\big(x,\frac{\partial u}{\partial x_l}\big) \cdot
   \frac{\partial \varphi}{\partial x_l} \,dx
   +\int_{\Omega}g(x,u)\varphi\,dx
   =\int_{\Omega}\varphi\, d\mu.
\end{align*}
\end{defi}

Our main results are collected in the
following theorem.

\begin{thm} \label{thm:theo1:nonlinear}
Let $\mu =(\mu_1,\dots,\mu_m)^\top$ be a vector-valued Radon measure
on $\Omega$ of finite mass. Then, under
the assumptions stated above and in Section \ref{intro}, \eqref{E1_tmp:bis}
has at least one distributional solution $u$. Moreover, $u$
has regularity as stated in \eqref{eq:weak_est}.
\end{thm}

\begin{proof}
Let $f_\varepsilon$ be as in Section \ref{sec:results}.
Then, by classical arguments, there exists a
sequence of approximate solutions $(u_\varepsilon)_{0<\varepsilon\leq 1} $ 
satisfying the weak formulation
\begin{equation}
   \label{E1_approx:weak:nonlinear}
      \int_\Omega \sum_{l=1}^N
       \sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)\cdot
       \frac{\partial \varphi}{\partial x_l}\,dx
       +\int_\Omega g(x,u_\varepsilon)\cdot \varphi\,dx =
       \int_{\Omega}f_\varepsilon \cdot \varphi\,dx,
\end{equation}
for all $\varphi \in W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m)$.
Substituting $\varphi=T_\gamma(u_\varepsilon)$
in \eqref{E1_approx:weak:nonlinear}, we get
\begin{equation}
   \label{amand1:nonlinear}
   \begin{split}
      & \int_{\Omega} \sum_{l=1}^N
       \sigma\big(x,\frac{\partial u_\varepsilon}{\partial x_l} \big)\cdot \frac{\partial T_\gamma(u_\varepsilon)}
       {\partial x_l}\,dx
       +\int_\Omega g(x,u_\varepsilon)\cdot T_\gamma(u_\varepsilon)\,dx
      =\int_{\Omega} f_\varepsilon T_\gamma(u_\varepsilon)\,dx.
   \end{split}
\end{equation}
By \eqref{estimate:nonlinear:1}, $\int_{\{|u_\varepsilon|\le \gamma\}} g(x,u_\varepsilon)
\cdot T_\gamma(u_\varepsilon)\,dx\ge 0$, and thus
we deduce
\begin{equation}
   \label{C4:nonlinear}
   c_1\sum_{l=1}^N \int_{\{|u|\leq \gamma\}}
      \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l}\,dx
      + \gamma \int_{\{|u_\varepsilon|> \gamma\}} 
      \abs{g(x,u_\varepsilon)}\,dx \le C.
\end{equation}
We obtain from \eqref{C4:nonlinear} and Lemma \ref{lem:weak-apriori}
the weak Lebesgue space estimates
$$
\norm{u_\varepsilon}_{\mathcal{M}^{q^\star}(\Omega;\mathbb{R}^m)}\leq C, \quad
\norm{\frac{\partial u_\varepsilon}{\partial x_l}}_{\mathcal{M}^{p_lq/\overline{p}}(\Omega;\mathbb{R}^m)}\leq C, \quad
l=1,\dots,N,
$$
where the exponents $q$ and $q^\star$ are defined in \eqref{eq:def:q-qs},
and $C$ is a constant independent of $\varepsilon$.
Consequently, we can assume without loss of generality that
the convergence in \eqref{limit:passing1} hold
for our sequence $(u_\varepsilon)_{0<\varepsilon\leq 1} $.

Taking $\gamma=1$ in \eqref{C4:nonlinear} and using
\eqref{estimate:nonlinear:3:bis}, we deduce
\begin{equation}
   \label{estimate:norm:L1}
   \int_{\Omega} \abs{g(x,u_\varepsilon )}\,dx \le C,
\end{equation}
where $C$ is a constant independent of $\varepsilon$. We also have
$g(x,u_\varepsilon )\to g(x,u)$ a.e.~in $\Omega$. In view of Vitali's
theorem, to show that $g(x,u_\varepsilon)$
converges strongly in $L^1(\Omega)$ it remains to prove that
$g(x,u_\varepsilon)$ is equi-integrable.  To this end, let $B$ be a
measurable set in $\Omega$. As usual, we split the integral into two parts
$$
\int_{B} \abs{g(x,u_\varepsilon )}\,dx
= \int_{B\cap \{|u_\varepsilon|\le \gamma\}}  \abs{g(x,u_\varepsilon)}\,dx
+\int_{B\cap \{|u_\varepsilon|>\gamma\}}
\abs{g(x,u_\varepsilon)}\,dx.
$$
Let us call the first and second integrals on the right-hand
side for $I_1$ and $I_2$, respectively.
In view of \eqref{estimate:nonlinear:3:bis},
$\lim\limits_{|B|\to 0} I_1=0$.
Let $0<M<\gamma$, and observe that
$$
\abs{T_\gamma(u_\varepsilon)}\le 
\abs{T_\gamma(u_\varepsilon)} {\bf 1}_{\{|u_\varepsilon|\le M\}}
+\abs{T_\gamma(u_\varepsilon)}{\bf 1}_{\{|u_\varepsilon|>M\}}
\le M + \gamma {\bf 1}_{\{|u_\varepsilon|>M\}},
$$
Using this decomposition in \eqref{amand1:nonlinear} yields
$$
\gamma \int_{\{|u_\varepsilon|> \gamma\}} 
\abs{g(x,u_\varepsilon)}\,dx \le
M \int_\Omega \abs{f_\varepsilon} \,dx
+ \gamma \int_{\{|u_\varepsilon|>M\}} \abs{f_\varepsilon} \,dx.
$$
{}From this inequality we obtain
$$
\lim_{\gamma\to\infty}
\Big(\sup_{0<\varepsilon\le 1}
\int_{\{|u_\varepsilon|> \gamma\}}
\abs{g(x,u_\varepsilon)}\,dx\Big)
=o\big(\frac{1}{M}\big),
$$
and, by sending $M\to \infty$, we conclude the
equi-integrability of $g(x,u_\varepsilon)$.

The proof of Lemma \ref{lem:A_conv} remains more or less
unchanged, except that the term $E_1$ rewrites
in our problem \eqref{E1_tmp:bis} as
\begin{equation}\label{estimate:E1:bis}
   \begin{split}
      E_1=&\int_\Omega f_\varepsilon \psi(u_\varepsilon-v)\phi \,dx
      -\int_\Omega g(x,u_\varepsilon) \psi(u_\varepsilon-v)\phi \,dx\\
      &- \int_{\Omega}\sum_{l=1}^N  \sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)
      \psi(u_\varepsilon-v)
      \frac{\partial \phi}{\partial x_l}\,dx,
      \end{split}
\end{equation}
and estimate \eqref{strong:6} rewrites as
\begin{equation}
   \label{strong:6:bis}
   \begin{split}
       L_l
       &\le \sup \abs{\psi}\Big(\int_\Omega \phi\, d\mu+
       \int_\Omega \abs{g(x,u)}\phi \,dx\Big)
       \\& \quad  -\int_{\Omega} \sum_{l=1}^N  \beta_l\cdot
       \psi(u-v)\frac{\partial\phi}{\partial x_l}\,dx\\
       &\quad +\int_{\Omega} \sum_{l=1}^N  {\beta}_l\cdot
       D\psi(u-v)\frac{\partial v}{\partial x_l}\left(1-\eta(u-v)\right)\phi\,dx\\
       &\quad  -\int_{\Omega} \sum_{l=1}^N  \sigma_l\big(x,\frac{\partial v}{\partial x_l}\big)\cdot
       \big(\frac{\partial u}{\partial x_l}-\frac{\partial v}{\partial x_l} \big)
       \eta(u-v)\phi\,dx.
   \end{split}
\end{equation}
Letting $x=a$ be a Lebesgue point
simultaneously  of $\mu$, $g(x,u)$, $h$, $u$, $Du$,
and $\beta=(\beta_1,\dots\beta_N)$, we
can proceed as in the proof of Lemma \ref{lem:A_conv}.
\end{proof}


\section{A different structure condition}
\label{sec:newstruct}

Zhou \cite{Zhou:00} proved that
the results of Dolzmann, Hungerb\"uhler, and M\"uller
\cite{Dolz-Hung-Mull,Dolz-Hung-Mull:1,Dolz-Hung-Mull:2}
continue to hold under the so-called (isotropic) sign condition. Moreover,
he gave an example of an isotropic elliptic system
that satisfies the sign condition but not the
the angle condition.

In this section we return to problem \eqref{E1_tmp} under assumptions
\eqref{estimation:prob:1}-\eqref{estimation:prob:5}, but
we want to replace the anisotropic angle condition
\eqref{estimation:prob:5} by the following
anisotropic sign condition:
\begin{equation}
   \label{estimation:prob:6:bis}
   \sigma_{i,l}(x,\xi)\xi_i \ge 0, \quad
   \forall (x,\xi) \in\Omega \times\mathbb{R}^N,
\end{equation}
for $i=1,\dots,m$, $l=1,\dots,N$. Here $\sigma_{i,l}$
and $\xi_i$ are the $i$th components of
vectors $\sigma_l$ and $\xi$, respectively.
When $m=2$, \eqref{estimation:prob:5}
implies \eqref{estimation:prob:6:bis}. To see this,  recall
that $(I-a\otimes a)$
projects orthogonally onto the space orthogonal
to $a$, and then choose $a=(1,0)^\top$, $a=(0,1)^\top$
in \eqref{estimation:prob:5}.

It is easy to give an example of an elliptic system which
satisfies \eqref{estimation:prob:1}, \eqref{monotonicity:1},
and \eqref{estimation:prob:6:bis},
but does not satisfy \eqref{estimation:prob:5}.
For example, take $m=2$, $N=2$, and
$$
\sigma_l(x,\xi)=\abs{\xi}^{p_l-2}(\alpha \xi_1, \xi_2)^\top, \quad
l=1,2, \quad \xi=(\xi_1,\xi_2)^\top,
$$
where $0<\alpha \le 0.2$. It is clear that assumptions
\eqref{estimation:prob:1}, \eqref{monotonicity:1}, and the
anisotropic sign condition
\eqref{estimation:prob:6:bis} hold.

Let us verify that the anisotropic angle
condition \eqref{estimation:prob:5} does not hold.
To this end, take $a=\left(\alpha^{1/2},(1-\alpha)^{1/2}\right)^\top$
and $\xi=(1,1)^\top$. Then $|a|=1$ and
$$
\left(I-a\otimes a\right)\xi=
\left(1-\alpha-\alpha^{1/2}(1-\alpha)^{1/2},
\alpha-\alpha^{1/2}(1-\alpha)^{1/2}\right)^\top,
$$
so that
\begin{align*}
   &\sigma_l(x,\xi)\cdot \left[\left(I-a\otimes a\right)\xi\right]
   \\ &
   = 2^{\frac{p_l-2}{2}}
   \left[\alpha \left(1-\alpha-\alpha^{1/2}\left(1-\alpha\right)^{1/2}\right)+
   \alpha-\alpha^{1/2}\left(1-\alpha\right)^{1/2}\right]
   \\ &
   < 2^{\frac{p_l-2}{2}}
   \left[2\alpha-\alpha^{1/2}(1-\alpha)^{1/2}\right]\le 0,
   \quad l=1,2,
\end{align*}
which implies that \eqref{estimation:prob:5} does not hold.

The purpose of this section is to prove
the following theorem.

\begin{thm}
\label{thm:theo1:structured}
Theorem \ref{thm:theo1} continues to hold when
the anisotropic angle condition \eqref{estimation:prob:5}
is replaced by the anisotropic
sign condition \eqref{estimation:prob:6:bis}.
\end{thm}

\begin{proof}
Compared to the proof of Theorem \ref{thm:theo1}, the
main new idea is to use, instead
of \eqref{definition:trancation1}, the
following cubic truncation function
$$
\Theta_{\gamma}(r)=
\Bigl(\max(-\gamma,\min(\gamma,r_1)),\dots,
\max(-\gamma,\min(\gamma,r_N))\Bigl)^\top,
$$
where $r=(r_1,\dots,r_N)^\top\in\mathbb{R}^N$.
Substituting $\varphi=\Theta_{\gamma}(u_\varepsilon)$
in \eqref{E1_approx:weak} yields
\begin{equation}
   \label{estimate:structured:1}
   \sum_{i=1}^m  \int_{\{|u_{\varepsilon,i}|\le \gamma\}}
   \sum_{l=1}^N  \sigma_{i,l}\Bigl(x,\frac{\partial u_\varepsilon}{\partial x_l}\Bigl)
   \frac{\partial u_{\varepsilon,i}}{\partial x_l} \,dx \leq C.
\end{equation}
Using assumptions \eqref{estimation:prob:1},
we deduce from \eqref{estimate:structured:1} that
\begin{equation}
   \label{estimate:structured:2}
   \begin{split}
      &\int_{\{|u_\varepsilon|\le \gamma\}}\sum_{l=1}^N 
      \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l}\,dx
      \\ &
      \le \frac{1}{c_1}
      \int_{\{\max\left(|u_{\varepsilon,1}|,\dots,|u_{\varepsilon,N}|
      \right)\le \gamma\}}
      \sum_{l=1}^N  \sigma_l\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)
      \cdot \frac{\partial u_\varepsilon}{\partial x_l} \,dx
      +\frac{c_2}{c_1}|\Omega|
      \\ &
      \le \frac{1}{c_1} \sum_{i=1}^m  \int_{\{|u_{\varepsilon,i}|\le \gamma\}}
      \sum_{l=1}^N  \sigma_{i,l}\big(x,\frac{\partial u_\varepsilon}{\partial x_l}\big)
      \frac{\partial u_{\varepsilon,i}}{\partial x_l} \,dx 
      +\frac{c_2}{c_1}|\Omega| \leq C.
   \end{split}
\end{equation}

Making similar changes due to the new truncation function
in the rest of the proof of Theorem \ref{thm:theo1}, we
conclude eventually that Theorem \ref{thm:theo1:structured} holds.
\end{proof}


\section{Maximal regularity and a uniqueness result}
\label{uniq-results}


We collect our results in Theorem \ref{thm:exist+regur}
(existence/regularity of solutions)
and Theorem \ref{thm:theo-uniq} (uniqueness of solutions) below.

Before stating the theorems, let us introduce
some notation. First of all, we say that a set
$E\subset\mathbb{R}^N$ is of type A if there exists
a constant $K$ such that
for all $x\in \overline{E}$ and for all
$0<\rho<\textrm{diam}(E)$ there holds
$\abs{Q(x,\rho)\cap E}\ge K \rho^N$, where
$Q(x,\rho)$ denotes the cube
$\left\{y\in\mathbb{R}^N\,:\, |x_l-y_l|<\frac{\rho}{2},\, l=1,\dots,N\right\}$.

In what follows we regard all relevant functions
as defined in $\mathbb{R}^N$ by setting them to zero outside $\Omega$.
A function $g$ belongs to the space ${\rm BMO}(\mathbb{R}^N)$ of functions
of bounded mean oscillation if $g\in L^N(\mathbb{R}^N)$ and
$$
\abs{g}_{{\rm BMO}(\mathbb{R}^N)} =\Big(\sup_{y\in \mathbb{R}^N}\sup_{\rho>0}
\frac{1}{\rho^{N}}\int_{Q(y,\rho)}
\abs{g-(g)_{y,\rho}}^N\,dx\Big)^{1/N}<\infty,
$$
where $(g)_{y,\rho}$ denotes the mean value of $g$ on the cube
$Q(y,\rho)$. The space ${\rm BMO}(\mathbb{R}^N)$ is a Banach space under
the norm
$$
\norm{g}_{{\rm BMO}(\mathbb{R}^N)} =\norm{g}_{L^N(\mathbb{R}^N)}+\abs{g}_{{\rm
BMO}(\mathbb{R}^N)}.
$$

\begin{thm}
\label{thm:exist+regur} Let $\Omega\subset\mathbb{R}^N$ be a bounded open
set such that $\Omega^c=\mathbb{R}^N\setminus \Omega$ is a domain of type
A. Suppose \eqref{estimation:prob:1}-\eqref{estimation:prob:5}
hold and $p_l=N$ for all $l=1,\dots,N$. Let
$\mu=(\mu_1,\dots,\mu_m)^\top$ be a Radon measure on $\Omega$ of
finite mass. Then problem \eqref{E1_tmp} has a solution $u\in
W^{1,(s_1,\dots,s_N)}_0(\Omega;\mathbb{R}^m) \cap {\rm BMO}(\Omega;\mathbb{R}^m)$
for any set of exponents $1\le s_1,\dots,s_N<N$, and the following
a priori estimate holds:
\begin{equation}
   \label{stern}
   \norm{u}_{{\rm BMO}(\Omega;\mathbb{R}^m)}\leq C_1
   \Big(\norm{\mu}_{\mathcal{M}^{\frac{1}{N-1}}(\Omega;\mathbb{R}^m)}+C_2\Big).
\end{equation}
Moreover, $D u$ belongs to the weak Lebesgue space
$\mathcal{M}^{N}(\Omega;\mathbb{R}^{m\times N})$ and
\begin{equation}
   \label{sternstern}
   \norm{D u}_{\mathcal{M}^{N}(\Omega;\mathbb{R}^{m\times N})}\leq C_3
   \left(\norm{\mu}_{\mathcal{M}^{\frac{1}{N-1}}}+C_4\right).
\end{equation}
The constants $C_i$, $i=1,2,3,4$, depend only on $c_1$, $c_2$,
$c_1'$, $c_2'$, $N$, $|\Omega|$, and the constant $K$ in the
definition of property A.
\end{thm}


\begin{thm}\label{thm:theo-uniq}
Suppose \eqref{estimation:prob:1}, \eqref{monotonicity:1}, and \eqref{estimation:prob:5}
hold and $p_l=N$ for all $l=1,\dots,N$.
Let $\mu=(\mu_1,\dots,\mu_m)^\top$ be a Radon measure on
$\Omega$ of finite mass.
Let $u,v$ be two solutions
of \eqref{E1_tmp} such that $u,~v \in W_0^{1,1}(\Omega;\mathbb{R}^m)$,
$u-v \in W_0^{1,1}(\Omega;\mathbb{R}^m)$, and
$Du,Dv \in \mathcal{M}^{N}(\Omega;\mathbb{R}^{m\times N})$.
Then $u=v$ a.e.~in $\Omega$.
\end{thm}

Let us now embark on the proofs
of Theorems \ref{thm:exist+regur}
and \ref{thm:theo-uniq}.

\subsection*{Proof of Theorem \ref{thm:exist+regur}}
The following lemma contains so-called Caccioppoli estimates,
which are at the heart of the matter of the regularity
theory developed in  \cite{Dolz-Hung-Mull:2}.

\begin{lem}\label{s2t2}
Let $u\in W_0^{1,N}(\Omega;R^m)$ be a solution of \eqref{E1_tmp}
with $p_l=N$ for all $l=1,\dots,N$, $\mu=f$, and
$f\in L^1(\Omega;\mathbb{R}^m)$. Fix two positive numbers
$\rho,R$ such that $0<\rho<R$.
Then there exist constants
$C_1$, $C_2$ such that for all cubes $Q(y,R)\subset\Omega$,
for all $\beta \in \mathbb{R}^m$, and for all $\gamma>0$
\begin{equation}
   \label{s2f4:1}
   \begin{split}
      &\int_{\{|u-\beta|\le\gamma\}\cap Q(y,\rho)}
      \sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^N\,dx
      \\ &
      \leq \frac{C_1}{(R-\rho)^N}\int_{Q(y,R)\setminus Q(y,\rho)}
      \abs{u-\beta}^N\,dx+
      C_2\Big(\gamma \int_{Q(y,R)}\abs{f}\,dx +R^N\Big),
   \end{split}
\end{equation}
and for all cubes $Q(y,R)\subset\mathbb{R}^N$ and for all $\gamma> 0$
\begin{equation}
   \label{s2f4:2:bis}
   \begin{split}
      &\int_{\{|u|<\alpha\}\cap Q(y,\rho)}
      \sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^N dx\\
      & \le
      \frac{C_1}{(R-\rho)^N}
      \int_{Q(y,R)\setminus Q(y,\rho)}
      \abs{u}^N dx+
      C_2\Big(\gamma \int_{Q(y,R)}\abs{f}\,dx +R^N\Big).
   \end{split}
\end{equation}
\end{lem}

\begin{proof}
Following \cite{Dolz-Hung-Mull:2}, let $\chi\in C^\infty_c(Q(y,R))$ be a
cut-off function satisfying
\begin{gather*}
\text{$\chi(x)=1$ if $x \in Q(y,\rho)$, $0\leq\chi\leq 1$, and} \\
\abs{\frac{\partial \chi}{\partial x_l}}
      \leq C/(R-\rho),\quad l=1,\dots,N,
\end{gather*}
for some finite constant $C$. Let
$\alpha_\gamma:\mathbb{R}\rightarrow \mathbb{R}$ be any smooth
function with the following properties:
\begin{equation}
   \label{def:galpha}
   \begin{gathered}
   \alpha_\gamma(s)=s \text{ if } s \in [0,\gamma],\;
   0\leq \alpha_\gamma\leq N \gamma,\; \alpha_\gamma' \leq 1,\\
   0<c\big(\frac{\alpha_\gamma(s)}{s}\big)^{N/(N-1)}
   \leq \alpha_\gamma'(s)
   \leq\frac{\alpha_\gamma(s)}{s}\leq 1\quad\mbox{ on }(0,\infty),
   \end{gathered}
\end{equation}
where $c>0$ is a constant. An example of
such a function can be found in \cite{Dolz-Hung-Mull:1}.
Now we define the cut-off function $\psi_\gamma:\mathbb{R}^m\to\mathbb{R}^m$ by
$$
\psi_\gamma(r)=\frac{r}{\abs{r}}\,\alpha_\gamma(\abs{r}).
$$
A calculation reveals that
\begin{align*}
   D \psi_\gamma(r)&=\alpha_\gamma'(\abs{r})\frac{r\otimes r}{\abs{r}^2}
   +\frac{\alpha_\gamma(\abs{r})}{\abs{r}}
   \big(I-\frac{r\otimes r}{\abs{r}^2}\big)
   \\ & = \alpha'(\abs{r}) I +
   \big[\frac{\alpha_\gamma(\abs{r})}{\abs{r}}
   - \alpha'(\abs{r})\big]\big(I-\frac{r\otimes r}{\abs{r}^2}\big).
\end{align*}
Hence, by \eqref{estimation:prob:5} and \eqref{def:galpha},
there holds
$$
\sigma_l(x,\xi)\cdot D \psi_\gamma(r)\xi\ge \sigma_l(x,\xi)\cdot
\xi \alpha_\gamma'(\abs{r}), \quad \forall \xi,r\in \mathbb{R}^m,\,
l=1,\dots,N,
$$
and, by \eqref{estimation:prob:1},
\begin{equation}
   \label{strong:4}
   \begin{split}
      &\sum_{l=1}^N  \sigma_l\big(x,\frac{\partial u}{\partial x_l}\big)
      \cdot \frac{\partial}{\partial x_l}
      \psi_\gamma(u)
      \\ &  \ge \alpha_\gamma'(\abs{u})
      \sum_{l=1}^N 
      \sigma_l\big(x,\frac{\partial u}{\partial x_l}\big)\cdot \frac{\partial u}{\partial x_l}
      \geq \alpha_\gamma'(\abs{u})
      \Big(c_1\sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^N-c_2N\Big).
   \end{split}
\end{equation}
Using $\chi^N\psi_\gamma(u-\beta)$ as a test function in the
weak formulation of \eqref{E1_tmp} yields
\begin{equation}
   \label{est:def:galpha:1}
   \begin{split}
      &\int_{\Omega}\chi^N
      \sum_{l=1}^N  \sigma_l\big(x,\frac{\partial u}{\partial x_l}\big)\cdot
      \frac{\partial}{\partial x_l}\psi_\gamma(u-\beta)\,dx\\
      & \quad =-\int_{\Omega} N\chi^{N-1}\psi_\gamma(u-\beta)\,dx
      \sum_{l=1}^N  \sigma_l\big(x,\frac{\partial u}{\partial x_l}\big)
      \cdot \frac{\partial \chi}{\partial x_l}\,dx
      \\
      & +\int_{\Omega} \chi^N f\psi_\gamma(u-\beta) \,dx.
   \end{split}
\end{equation}
Using \eqref{def:galpha}, \eqref{strong:4}, \eqref{estimation:prob:1}, and
H\"older's inequality, we deduce from \eqref{est:def:galpha:1}
\begin{equation}\label{est:def:galpha:2}
   \begin{split}
      &c_1\int_\Omega\chi^N\sum_{l=1}^N \abs{\frac{\partial u}{\partial x_l}}^N
      \alpha_\gamma'(\abs{u-\beta})\,dx\\
      &\leq
      \frac{C}{(R-\rho)}\Big(\int_{\Omega}\chi^N
      \alpha_\gamma'(\abs{u-\beta})
      \Big(c_1'\sum_{l=1}^N \abs{\frac{\partial u}{\partial x_l}}^{N-1}+Nc_2'\Big)^{\frac{N}{N-1}}\,dx
      \Big)^{\frac{N-1}{N}}
      \\ & \quad
      \times \Big(\int_{Q(y,R)\setminus Q(y,\rho)}
      \abs{u-\beta}^N dx\Big)^{1/N}
      +\tilde{C}\Big(\gamma\int_{Q(y,R)}\abs{f}\,dx+R^N\Big).
   \end{split}
\end{equation}
An application of Young's inequality yields
\begin{align*}
   &\int_\Omega\chi^N \alpha_\gamma'(\abs{u-\beta})
   \sum_{l=1}^N \abs{\frac{\partial u}{\partial x_l}}^N \,dx
   \\ & \leq
   \frac{C_1}{(R-\rho)^N}\int_{Q(y,R)\setminus Q(y,\rho)} \abs{u-\beta}^N\,dx+
   C_2\Big(\gamma\int_{Q(y,\rho)}\abs{f}\,dx+R^N\Big),
\end{align*}
for some constants $C_1,C_2$.
Now \eqref{s2f4:1} follows from
the definition of $\alpha_\gamma$.
Using $\chi^N\psi_\gamma(u)$ as a test function in the
weak formulation of \eqref{E1_tmp} and proceeding as in the proof of
\eqref{s2f4:1}, we deduce easily \eqref{s2f4:2:bis}.
\end{proof}

We quote the following key lemma from \cite{Dolz-Hung-Mull:2}.

\begin{lem}[Dolzmann, Hungerb\"uhler, and
M\"uller \cite{Dolz-Hung-Mull:2}] \label{lem2:regul:1}
 Suppose $u$ belongs to $W_0^{1,N}(\Omega;R^m)$ and there exists
 $f \in L^1(\Omega;\mathbb{R}^m)$ such that the Caccioppoli estimates
\eqref{s2f4:1}, \eqref{s2f4:2:bis} hold. Then $u\in{\rm
BMO}(\Omega;\mathbb{R}^m)$, $Du\in\mathcal{M}^N(\Omega;\mathbb{R}^{m\times N})$,
and
$$
\abs{u}_{{\rm BMO}(\Omega;\mathbb{R}^m)}
+\norm{Du}_{\mathcal{M}^N(\Omega;\mathbb{R}^{m\times N})}\leq C
\big(\norm{f}_{L^1(\Omega;\mathbb{R}^m)}^{1/(N-1)} +1\big),
$$
where $C>0$ is a constant depending only on $N$, $\abs{\Omega}$, and
the constant $K$ in the definition of property A.
\end{lem}

\subsection*{Concluding the proof of Theorem \ref{thm:exist+regur}}
It is possible to construct a sequence of approximate solutions
$u_\varepsilon \in W_0^{1,N}(\Omega;\mathbb{R}^m)$ satisfying \eqref{E1_approx:weak},
with $f_\varepsilon\in L^1(\Omega;\mathbb{R}^m)\cap L^\infty(\Omega;\mathbb{R}^m)$
satisfying \eqref{Zu}.
In view of Lemmas \ref{s2t2} and \ref{lem2:regul:1}, the proof
of Theorem \ref{thm:exist+regur} is obtained by routine arguments.

\subsection*{Proof of Theorem \ref{thm:theo-uniq}}
The main obstacle that one encounters
when attempting to prove uniqueness is that
if $u,v$ are two solutions of \eqref{E1_tmp}, then $w=u-v$ is not
in $L^{\infty}(\Omega;\mathbb{R}^m)$ and therefore
cannot be used as a test function in the weak formulation.
To handle this problem, we implement
the technique developed in Dolzmann, Hungerb\"uhler, and
M\"uller \cite{Dolz-Hung-Mull:2}, which in turn was
motivated by earlier work by Acerbi and Fusco \cite{AcFu:88}.
The idea is to approximate
the function $w$ by a Lipschitz function
$w_\lambda$ that coincides with $w$ on a large set. Moreover, precise
estimates of the measure of the set where these
two functions do not coincide can be provided
if $w$ has ``maximal regularity''.

We start by recalling the
key approximation lemma.

\begin{lem}[Dolzmann, Hungerb\"uhler, and
M\"uller \cite{Dolz-Hung-Mull:2}, see also \cite{AcFu:88}]
\label{lem-uniq}
Let $\Omega\subset\mathbb{R}^N$ be a bounded open set such
that $\Omega^c$ is a domain of type A and fix $1< p<\infty$.
Let $w\in W^{1,1}_0(\Omega;\mathbb{R}^m)$ be such that
$Dw\in \mathcal{M}^p(\Omega;\mathbb{R}^{m\times N})$.
Then there exists for each $\lambda>0$ a function
$w_\lambda\in W^{1,\infty}(\Omega;\mathbb{R}^m)$ such that
$\norm{w_\lambda}_{W^{1,\infty}(\Omega;\mathbb{R}^m)}\le C_1\lambda$ and
$$
\abs{\left\{x\in \Omega: w(x)\neq w_\lambda(x)\right\}}
\le C_2 \lambda^{-p}\norm{D w}_{\mathcal{M}^p(\Omega;\mathbb{R}^{m\times N})}.
$$
The constants $C_1$ and $C_2$ depend only on $\abs{\Omega}$ and $N$.
If $w\in W^{1,p}(\Omega;\mathbb{R}^m)$, then
$$
\abs{\{x\in \Omega: w(x)\neq w_\lambda(x)\}}
= o\big(\lambda^{-p}\big).
$$
\end{lem}

Let $A_\lambda:=\{x\in \Omega: w(x)\neq w_\lambda(x)\}$.
To prove Theorem \ref{thm:theo-uniq}, observe that
$w:=u-v \in W_0^{1,1}(\Omega;\mathbb{R}^m)$ and introduce according to
Lemma \ref{lem-uniq} the function $w_\lambda$.
Since $u$ and $v$ are solutions, we have
\begin{equation}
   \label{Eq:uniq}
   \sum_{l=1}^N  \frac{\partial}{\partial x_l}\Big(
   \sigma_l\big(x,\frac{\partial u}{\partial x_l} \big)
   -\sigma_l\big(x,\frac{\partial v}{\partial x_l} \big)\Big)=0
   \quad \text{in ${\mathcal D}'(\Omega;\mathbb{R}^m)$}.
\end{equation}
Using $w_\lambda$ as a test function in \eqref{Eq:uniq} yields
\begin{equation}
   \label{Eq:uniq:1}
   \sum_{l=1}^N  \int_{\Omega}\Big(\sigma_l\big(x,\frac{\partial u}{\partial x_l} \big)-
   \sigma_l\big(x,\frac{\partial v}{\partial x_l} \big)\Big)\cdot
   \frac{\partial w_\lambda}{\partial x_l}\,dx=0.
\end{equation}

Since $\frac{\partial w_\lambda}{\partial x_l}=\frac{\partial u}{\partial x_l}-\frac{\partial v}{\partial x_l}$ a.e.~on
$\Omega\setminus A_\lambda$, we deduce from \eqref{Eq:uniq:1} and
\eqref{estimation:prob:1}, with $p_l=N$ for all
$l=1,\dots,N$,
\begin{equation}
   \label{Eq:uniq:2}
   \begin{split}
      &c_1 \sum_{l=1}^N  \int_{\Omega\setminus A_\lambda}
      \big|\frac{\partial u}{\partial x_l} -\frac{\partial v}{\partial x_l}\big|^N \,dx
      \\ & \quad\le C\lambda
      \sum_{l=1}^N  \int_{A_\lambda}
      \Big(\abs{\frac{\partial u}{\partial x_l}}^{N-1}+\abs{\frac{\partial v}{\partial x_l}}^{N-1}+1\Big)\,dx
      \\& \quad
      \le C\lambda \sum_{l=1}^N  \abs{A_\lambda}^{1/N}
      \Big(\norm{ \abs{\frac{\partial u}{\partial x_l}}\,}^{N-1}_{\mathcal{M}^{N}(\Omega;\mathbb{R}^m)}+
      \norm{\, \abs{\frac{\partial v}{\partial x_l}}\, }^{N-1}_{\mathcal{M}^{N}(\Omega;\mathbb{R}^m)}\Big)
      +C\lambda N \abs{A_\lambda}
      \\ & \quad \le \tilde{C},
    \end{split}
\end{equation}
where the last bound is a consequence of Lemma \ref{lem-uniq}.
Consequently, sending $\lambda\to \infty$, we have
$Dw=D(u-v)\in L^N(\Omega;\mathbb{R}^{m\times N})$. We can therefore
use the last part of Lemma \ref{lem-uniq} when
sending $\lambda \to \infty$ in \eqref{Eq:uniq:2}.
The result is that $Dw=0$, which concludes the proof
Theorem \ref{thm:theo-uniq}.


\section{Anisotropic harmonic maps into spheres}
\label{sec:harmonic}

Let $\Omega$ be a bounded smooth open connected subset of $\mathbb{R}^N$ ($N\ge 2$)
and $1\le p_1,\dots,p_N<\infty$. In this section we need
to use the anisotropic Sobolev space
$W^{1,(p_1,\dots,p_N)}(\Omega)$, which is defined by
\begin{align*}
   &W^{1,(p_1,\dots,p_N)}(\Omega)=
   \{g \in W^{1,1}(\Omega)\,:\,
   \frac{\partial g}{\partial x_l}\in L^{p_l}(\Omega), \,
   l=1,\dots,N\},
   \\ &
   \norm{g}_{W^{1,(p_1,\dots,p_N)}({\Omega})} =
   \sum_{l=1}^N  \Big(\norm{g}_{L^{p_l}(\Omega)}+
   \norm{\frac{\partial g}{\partial x_l}}_{L^{p_l}(\Omega)}\Big).
\end{align*}
Let $u$ satisfy $I[u]=\min_{w\in \mathcal{A}} I[w]$,
where the anisotropic energy functional $I$
and the set of admissible functions $\mathcal{A}$
are defined in \eqref{eq:I_def} and \eqref{eq:def_Aset}, respectively.
Pick any $\phi\in W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m)\cap L^\infty(\Omega;\mathbb{R}^m)$.
Since $\abs{u}=1$ a.e.~in $\Omega$, $w(\tau)=(u+\tau \phi)/\abs{u+\tau \phi}\in \mathcal{A}$
for small enough $\tau$'s. Hence $J(\tau)=I[w(\tau)]$ has
a minimum at $\tau=0$ and $J'(0)=0$. A calculation
of $J'(0)$ then shows that $u$ solves the
Euler-Lagrange system \eqref{E1_map} in the weak sense,
which motivates the next definition.

\begin{defi} \rm
A vector-valued function
$$
u=(u_1,\dots,u_m)^\top\in W^{1,(p_1,\dots,p_N)}(\Omega;\mathbb{S}^{m-1})
$$
is called a $(p_1,\dots,p_N)$-harmonic
map from $\Omega$ into $\mathbb{S}^{m-1}$ provided
\begin{equation}
   \label{def:weak_harmonic}
   \int_{\Omega} \sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^{p_l-2}\frac{\partial u}{\partial x_l}\cdot \frac{\partial\phi}{\partial x_l}\,dx
   = \int_{\Omega} \sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^{p_l} u\cdot \phi\,dx,
\end{equation}
for all $\phi\in W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m)\cap L^\infty(\Omega;\mathbb{R}^m)$.
We also use the term ``anisotropic harmonic'' for such a map.
\end{defi}

Since we have not been able to find the proof of the
following anisotropic Sobolev-Poincar\'e inequality
in the literature, we have chosen to include a proof
of it by the usual ``contradiction method'', relying
on the following anisotropic Sobolev inequality \cite{Troisi,AcerFus:94}:
Let $Q$ be a cube with faces parallel to the coordinate planes.
Suppose $g\in W^{1,(p_1,\dots,p_N)}(Q)$ and $\overline{p}<N$. Then
\begin{equation}
   \label{Troisi_new}
   \norm{g}_{L^{\overline{p}^\star}(Q)}\leq
   C \prod_{l=1}^N
   \Big(\norm{\frac{\partial g}{\partial x_l}}_{L^{p_{l}}(Q)}
   + \norm{g}_{L^{p_{l}}(Q)}\Big)^{1/N},
\end{equation}
and the inequality between geometric and
arithmetic means implies that the right-hand side can be bounded
by $\frac{C}{N} \sum_{l=1}^N 
   \left(\norm{\frac{\partial g}{\partial x_l}}_{L^{p_{l}}(Q)}
   + \norm{g}_{L^{p_{l}}(Q)}\right)$. Hence, the space
$W^{1,(p_1,\dots,p_N)}(Q)$ is continuously embedded into $L^{\overline{p}^\star}(Q)$.

\begin{lem}
\label{lem:Poincare}
Let $Q(x_0,\rho)=\{x\in\mathbb{R}^N\,:\, \abs{x_l-x_{0,l}}<
\frac{\rho}{2},\, l=1,\dots,N\}$, where $x_0\in\mathbb{R}^N$, $\rho>0$.
Suppose $g\in W^{1,(p_1,\dots,p_N)}(Q(x_0,\rho))$.
Suppose the anisotropy $(p_1,\dots,p_N)$ is such
that \eqref{ass:pl_har_ass} holds. Then for each $1\le p <\overline{p}^\star$
\begin{equation}
   \label{Poincare}
   \begin{split}
      &\Big(\frac{1}{\rho^N}\int_{Q(x_0,\rho)}
      \abs{g-(g)_{x_0,\rho}}^p\,dx\Big)^{1/p}
      \le C \rho \sum_{l=1}^N  \Big(\frac{1}{\rho^N}\int_{Q(x_0,\rho)}
      \abs{\frac{\partial g}{\partial x_l}}^{p_l}\Big)^{1/p_l},
   \end{split}
\end{equation}
for some constant $C=C(N,p_1,\dots,p_N,p)$. Here
$(g)_{x_0,\rho}$ denotes the average
value of $g$ over the cube $Q(x_0,\rho)$.
\end{lem}

\begin{proof} We divide the proof into two steps.

 \textit{Step 1 ($x_0=0$, $\rho=1$).} We argue by
contradiction. Suppose the assertion is not
true. Then for each $n=1,2,\dots$, there
would exist a function $g_n\in W^{1,(p_1,\dots,p_N)}(\Omega)$
such that
\begin{equation}
   \label{Poincare:I}
   \sum_{l=1}^N  \norm{\frac{\partial g_n}{\partial x_l}}_{L^{p_l}(Q(0,1))}
   < \frac{1}{n} \norm{g_n- (g_n)_{0,1}}_{L^p(Q(0,1))},
\end{equation}
where, by the anisotropic Sobolev inequality \eqref{Troisi_new}, the right-hand
side is bounded by a constant (independent of $n$)
times $1/n$.
Define
$$
h_n = \frac{g_n - (g_n)_{0,1}}{\norm{g_n-(g_n)_{0,1}}_{L^p(Q(0,1))}}.
$$
Then $(h_n)_{0,1}=0$ and $\norm{h_n}_{L^p(Q(0,1))}=1$.
By \eqref{Poincare:I}, we have, passing
if necessary to a subsequence, that
$h_n\to h$ a.e.~in $Q(0,1)$ and also in $L^p(Q(0,1))$, where
$h$ is some limit function. It follows that
\begin{equation}
   \label{Poincare:II}
   (h)_{0,1}=0, \quad
   \norm{h}_{L^p(Q(0,1))}=1.
\end{equation}
On the other hand, it follows from
\eqref{Poincare:I} that $\frac{\partial h}{\partial x_l}=0$ for all $i=1,\dots,N$,
and hence $h$ is constant, which contradicts \eqref{Poincare:II}.

\textit{Step 2 (the general case)}. Let $g:Q(x_0,\rho)\to\mathbb{R}$,
and scale this function to the unit cube by setting
$h(x)=g(x_0+\rho x)$ for $x\in Q(0,1)$. By Step 1,
$$
\Big(\int_{Q(0,1)} \abs{h}^p\,dx
\Big)^{\frac{1}{p}}
\le C \sum_{l=1}^N  \Big(\int_{Q(0,1)}\abs{\frac{\partial h}{\partial x_l}}^{p_i}\,dx\Big)^{\frac{1}{p_l}}.
$$
Changing variables in this inequality yields \eqref{Poincare}.
\end{proof}

Before we continue, we need to introduce
some additional notations and function spaces.
A function $g\in L^1(\mathbb{R}^N)$ belongs to
the Hardy space $\mathcal{H}^1(\mathbb{R}^N)$
if the grand maximal function
$g^\star:=\sup_{\rho>0} \abs{g\star \omega_\rho}$
belongs to $L^1(\mathbb{R}^N)$, where $\omega_\rho(x)=\rho^{-N}\omega_1(x/\rho)$,
$\omega_1\in C^{\infty}_c(B(0,1))$, $\int\omega_1=1$.
The definition does not depend on the choice
of $\omega_1$. The Hardy space is a Banach space under
the norm $\norm{g}_{\mathcal{H}^1(\mathbb{R}^N)}=\norm{g}_{L^1(\mathbb{R}^N)}
+\norm{g^\star}_{L^1(\mathbb{R}^N)}$. If $g\in\mathcal{H}^1(\mathbb{R}^N)$,
then necessarily $\int g=0$. The dual
space of $\mathcal{H}^1(\mathbb{R}^N)$ is the
space ${\rm BMO}(\mathbb{R}^N)$ of functions of
bounded mean oscillations. Here a function $h\in L^1_{\rm loc}(\mathbb{R}^N)$
belongs to $BMO(\mathbb{R}^N)$ if
$\abs{h}_{{\rm BMO}(\mathbb{R}^N)}=\sup_{x,\rho}\frac{1}{\rho^N}\int_{Q(x,r)}
\abs{h(y)-(h)_{x,\rho}}\,dy$ is finite.
The space $VMO(\mathbb{R}^N)$ of functions of
vanishing mean oscillations, which is defined
as the closure of $C_0(\mathbb{R}^N)$ in $BMO(\mathbb{R}^N)$, is the
predual of $\mathcal{H}^1(\mathbb{R}^N)$. We shall need the local
Hardy space $\mathcal{H}^1_{\rm loc}(\Omega)$. Let
$K$ be any compact subset of $\Omega$
and set $\epsilon_K=\mathrm{dist}(K,\mathbb{R}^N\setminus\Omega)$.
Then $g\in \mathcal{H}^1_{\rm loc}(\Omega)$ if for any compact
subset $K\subset\Omega$ there holds
$\sup_{0<\rho<\epsilon_K} \abs{g\star \omega_\rho}\in L^1(K)$.
We refer to Stein \cite{Stein-HA:93} for
more information about the spaces just introduced.

Coifman, Lions, Meyer, and Semmes \cite{CLMS:93} proved that if
two vector fields $B$ and $E$ in conjugate Lebesgue spaces
$L^p(\mathbb{R}^N;\mathbb{R}^N)$ and $L^{p'}(\mathbb{R}^N;\mathbb{R}^N)$ satisfy $\mathop{\rm curl} B=0$ and
$\mathop{\rm div} E=0$ in the sense of distributions, then their
scalar product $B\cdot E$, which a priori only belongs to
$L^1(\mathbb{R}^N)$ by H\"older's inequality, belongs to the Hardy space
$\mathcal{H}^1(\mathbb{R}^N)$, which is a strict subspace of $L^1(\mathbb{R}^N)$. Thus the
nonlinear quantity $B\cdot E$ possesses a compensated
integrability property. We shall require an anisotropic version of
(a special case) of this result. The proof follows closely that in
\cite{CLMS:93}, with some minor modifications to account for the
anisotropy of the involved vector fields.

\begin{thm}
\label{thm:CI} Let $1<p_l<\infty$ and $1/p_l+1/p_l'=1$,
$l=1,\dots,N$. Suppose $B=D\pi$ for some function $\pi\in
W^{1,(p_1,\dots,p_N)}(\mathbb{R}^N)$ and $E=(E_1,\dots,E_N)^\top$, $E_l\in
L^{p_l'}(\mathbb{R}^N)\cap L^1(\mathbb{R}^N)$, $\mathop{\rm div} E=0$. Suppose the
anisotropy $(p_1,\dots,p_N)$ is such that \eqref{ass:pl_har_ass}
holds. Then $B\cdot E$ belongs to $\mathcal{H}^1(\mathbb{R}^N)$ and
$$
\norm{B\cdot E}_{\mathcal{H}^1(\mathbb{R}^N)}
\le C \Big(\sum_{l=1}^N  \norm{\frac{\partial \pi}{\partial x_l}}_{L^{p_l}(\mathbb{R}^N)}\Big)
\Big(\sum_{l=1}^N  \norm{E_l}_{L^{p_l'}(\mathbb{R}^N)}\Big),
$$
where the universal constant $C$
depends on $N,p_1,\dots,p_N$.
If the domain of definition $\mathbb{R}^N$
for $B$ and $E$ are replaced by $\Omega$, then
the theorem remains true with $\mathcal{H}^1(\mathbb{R}^N)$ replaced by
$\mathcal{H}^1_{\rm loc}(\Omega)$.
\end{thm}

\begin{rem}\rm
Theorem \ref{thm:CI} shows
that the product $B\cdot E$ has a compensated integrability
property as long as the anisotropy $(p_1,\dots,p_N)$
is not too much spread out, which
is reflected in the condition $\overline{p}^\star>p_{\rm max}$.
\end{rem}

\begin{proof}
It is clear that $D \pi \cdot E\in L^1(\mathbb{R}^N)$ and
that $D\pi \cdot E= \sum_{l=1}^N  \frac{\partial}{\partial x_l} (\pi E_l)$
in the sense of distributions. For any $x\in\mathbb{R}^N$
and any $\rho>0$, we need to estimate the convolution product
$(D\pi\cdot E)\star \omega_\rho(x)$:
\begin{align*}
   \abs{(D\pi\cdot E)\star \omega_\rho(x)}
 &= \big|\int_{\mathbb{R}^N} (D\pi \cdot E)(y)\omega_\rho(x-y)\,dy\big|
   \\ &
   = \big|\int_{\mathbb{R}^N}\sum_{l=1}^N (\pi E_l)(y)\frac{\partial}{\partial y_l}
   \omega_\rho(x-y)\,dy\big|
   \\ &
   = \big|\int_{\mathbb{R}^N}\sum_{l=1}^N  (\pi(y)-(\pi)_{x,\rho})
   E_l(y)\frac{\partial}{\partial y_l}\omega_\rho(y-x)\,dy\big|
   \\ &
   \le C\frac{1}{\rho^{N+1}}\int_{Q(x,\rho)}
   \sum_{l=1}^N  \abs{\pi(y)-(\pi)_{x,\rho}}\,
   \abs{E_l(y)}\,dy.
\end{align*}
Next we choose $(q_1,\dots,q_N)$ such
that $q_l<p_l$ for all $l=1,\dots,N$
and $\overline{q}^\star>p_{\rm max}>q_{\rm max}$. We can do this 
since $\overline{p}^\star>p_{\rm max}$.
To be specific, choose $q_l=\theta p_l$, $l=1,\dots,N$, for some
$\theta \in \big(\frac{\overline{p}^\star}{\overline{p}^\star+N},1\big)$ to be specified later.
One can check that
$$
\theta \overline{p}^\star
=\frac{N\theta}{N\theta-(1-\theta)\overline{q}^\star}\overline{q}^\star
=:e(\theta) \overline{q}^\star.
$$
Since $0<\overline{q}^\star<\overline{p}^\star$ and 
$\theta>\frac{\overline{p}^\star}{\overline{p}^\star+N}$, there holds
$1<e(\theta)<\frac{N\theta}{N\theta-(1-\theta)\overline{p}^\star}<\infty$.
Moreover, $e(\theta)\downarrow 1$ as
we let $\theta\uparrow 1$.
Using $\overline{p}^\star>p_{\rm max}$ to write $\overline{p}^\star=p_{\rm max}+\kappa$
for some $\kappa>0$, we obtain
$$
\overline{q}^\star=\frac{\theta}{e(\theta)}\overline{p}^\star
= p_{\rm max} + \Delta(\theta),
\quad
\Delta(\theta):=\big(\frac{\theta}{e(\theta)}-1\big)p_{\rm max}
+\frac{\theta}{e(\theta)}\kappa.
$$
Clearly, by choosing $\theta$ close enough to $1$, we can
ensure $\Delta(\theta)>0$. Hence, for such a choice of $\theta$, we have
$\overline{q}^\star>p_{\rm max}>q_{\rm max}$. 
Having chosen the $q_l$'s, we choose $(s_1,\dots,s_N)$ such that
$p_l<s_l<\overline{q}^\star$ for all $l$. Indeed, we can take
$\frac{1}{s_l}=\frac{1}{q_l}-\delta_l$,
with $\delta_l\in (0, \frac{1}{q_l}-\frac{1}{\overline{q}^\star})$.

We now continue using
H\"older's inequality to obtain
\begin{align*}
   & \abs{(D\pi\cdot E)\star \omega_\rho(x)}
   \\ &\le C \sum_{l=1}^N  \frac{1}{\rho}
   \Big(\frac{1}{\rho^N}\int_{Q(x,\rho)}
   \abs{\pi(y)-(\pi)_{x,\rho}}^{s_l}\Big)^{1/s_l}
   \Big(\frac{1}{\rho^N}\int_{Q(x,\rho)}
   \abs{E_l(y)}^{s_l'}\Big)^{1/s_l'}.
\end{align*}
Since $s_l<\overline{q}^\star$ and $\pi\in W^{1,(q_1,\dots,q_N)}_0(Q(x,\rho))$, we can use
the anisotropic Sobolev-Poincar\'e inequality (see Lemma \ref{lem:Poincare}) to obtain
\begin{align*}
   & \abs{(D\pi\cdot E)\star \omega_\rho(x)}
   \\ &\le
   C \sum_{l=1}^N \sum_{j=1}^N 
   \Big(\frac{1}{\rho^N}\int_{Q(x,\rho)}
   \abs{\frac{\partial\pi(y)}{\partial y_j}}^{q_j}\Big)^{1/q_j}
   \Big(\frac{1}{\rho^N}\int_{Q(x,\rho)}
   \abs{E_l(y)}^{s_l'}\Big)^{1/s_l'}.
\end{align*}
We need the Hardy-Littlewood maximal function
$$
M[g](x)=\sup_{\rho>0}\frac{1}{\rho^N}\int_{Q(x,\rho)} 
|g(y)|\,dy,
$$
which is bounded on $L^p(\mathbb{R}^N)$, that is,
$$
\norm{M[g]}_{L^p(\mathbb{R}^N)}\le C(p) 
\norm{g}_{L^p(\mathbb{R}^N)},
$$
for $1<p<\infty$. Using the maximal function we find that
\[
\sup_{\rho>0} \abs{(D\pi\cdot E)\star \omega_\rho(x)}
\le  C \sum_{l=1}^N \sum_{j=1}^N  
\Big(M\big[\abs{\frac{\partial\pi}{\partial y_j}}^{q_j}\big](x)\Big)^{1/q_j}
\Big(M\big[\abs{E_l}^{s_l'}\big](x)\Big)^{1/s_l'}.
\]
Integrating over $x\in\mathbb{R}^N$, using
H\"older's inequality, and finally
using the boundedness of the maximal function
(recall that $p_j>q_j$ and $p_l'>s_l'$), we get
\begin{align*}
& \int_{\mathbb{R}^N}\sup_{\rho>0} \abs{(D\pi\cdot E)\star \omega_\rho(x)}\,dx
   \\
&\le   C \sum_{l=1}^N \sum_{j=1}^N  \Big( \int_{\mathbb{R}^N} \Big(
   M\big[\abs{\frac{\partial\pi}{\partial y_j}}^{q_j}\big](x)\Big)^{p_j/q_j}\,dx
   \Big)^{\frac{1}{p_j}}
  \Big(\int_{\mathbb{R}^N}
   \left(M\left[\abs{E_l}^{s_l'}\right](x)\right)^{p_l'/s_l'}
   \,dx\Big)^{1/p_l'}
   \\ &\le
   C \Big(\sum_{j=1}^N  \norm{\frac{\partial\pi}{\partial y_j}}_{L^{p_j}(\mathbb{R}^N)}\Big)
   \Big(\sum_{l=1}^N  \norm{E_l}_{L^{p_l'}(\mathbb{R}^N)}\Big),
\end{align*}
which concludes the proof of the theorem.
\end{proof}

We have come to the main result of this section, namely
a compactness theorem
for $(p_1,\dots,p_N)$-harmonic maps. This result can be viewed
as an anisotropic version of a result of Toro and Wang \cite{ToroWang:95}
for $p$-harmonic maps, and our proof proceeds
along the lines of \cite{ToroWang:95}.

\begin{thm}
Suppose $(u_\varepsilon)_{0<\varepsilon\le 1}
\subset W^{1,(p_1,\dots,p_N)}(\Omega;\mathbb{S}^{m-1})$ is a
sequence of \break $(p_1,\dots,p_N)$-harmonic maps such that
$$
u_\varepsilon \rightharpoonup u \quad 
\text{in $W^{1,(p_1,\dots,p_N)}(\Omega;\mathbb{S}^{m-1})$ as $\varepsilon\to 0$.}
$$
Then $u$ is a $(p_1,\dots,p_N)$-harmonic
map from $\Omega$ into $\mathbb{S}^{m-1}$.
\end{thm}

\begin{proof}
Each $u_\varepsilon$ is a weak solution of
$$
-\sum_{l=1}^N  \frac{\partial}{\partial x_l} 
\Big(\abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2}\frac{\partial u_\varepsilon}{\partial x_l}  \Big)
=f_\varepsilon, \quad
f_\varepsilon:=\sum_{l=1}^N  \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l} u_\varepsilon.
$$
Clearly, as $u_\varepsilon$ is uniformly bounded in
$W^{1,(p_1,\dots,p_N)}(\Omega;\mathbb{R}^m)$ and 
$\abs{u_\varepsilon}=1$ a.e.~in $\Omega$, we have
that $f_\varepsilon$ is uniformly bounded in $L^1(\Omega)$.
Thus the above system fits into the
theory developed previously in this paper.

As in \cite{Helein:2002,ToroWang:95}, the main point of the proof
is exploit that the term $f_\varepsilon$
has a particular structure due to the constraint
$\abs{u_\varepsilon}=1$ a.e.~in $\Omega$, which implies that it in fact belongs
to the Hardy space $\mathcal{H}^1_{\rm loc}(\Omega)$ and not just $L^1(\Omega)$.
Indeed, observe that, for any $i=1,\dots,N$,
\begin{align*}
   f_{\varepsilon,i}&= \sum_{k=1}^m  \sum_{l=1}^N  
   \frac{\partial}{\partial x_l} u_{\varepsilon,k} 
   \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2}
   \big(u_{\varepsilon,i}\frac{\partial}{\partial x_l} u_{\varepsilon,k}
   - u_{\varepsilon,k}\frac{\partial}{\partial x_l} u_{\varepsilon,i}\big)
   \\ & = \sum_{k=1}^m  B_{\varepsilon,k}\cdot E_{\varepsilon,i,k},
\end{align*}
where the vector fields $B_{\varepsilon,k}=(B_{\varepsilon,k})_{l=1}^N$
and $E_{\varepsilon,i,k}=(E_{\varepsilon,i,k})_{l=1}^N$ are defined by
$(B_{\varepsilon,k})_l = \frac{\partial}{\partial x_l} u_{\varepsilon,k}$, $l=1,\dots,N$, and
$$
(E_{\varepsilon,i,k})_l = \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2} 
\big(u_{\varepsilon,i}\frac{\partial}{\partial x_l}
u_{\varepsilon,k} - u_{\varepsilon,k}\frac{\partial}{\partial x_l} 
u_{\varepsilon,i}\big), \quad l=1,\dots,N.
$$
Clearly, $\mathop{\rm curl} B_{\varepsilon,k}=0$. Let us show that $E_{\varepsilon,i,k}$
is divergence free:
\begin{align*}
   \mathop{\rm div} E_{\varepsilon,i,k}
   &= \sum_{l=1}^N  \frac{\partial}{\partial x_l} (E_{\varepsilon,i,k})_l
   \\ &= \sum_{l=1}^N  \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2} \frac{\partial}{\partial x_l} u_{\varepsilon,i} \frac{\partial}{\partial x_l} u_{\varepsilon,k}
  +\sum_{l=1}^N  \frac{\partial}{\partial x_l} \Big(\abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2}\frac{\partial}{\partial x_l} u_{\varepsilon,k}\Big)u_{\varepsilon,i}
   \\ & \quad
   - \sum_{l=1}^N  \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2}\frac{\partial}{\partial x_l} u_{\varepsilon,k} \frac{\partial}{\partial x_l} u_{\varepsilon,i}
   - \sum_{l=1}^N  \frac{\partial}{\partial x_l}\Big(\abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2}\frac{\partial}{\partial x_l} u_{\varepsilon,i}\Big)u_{\varepsilon,k}
   \\ & = \sum_{l=1}^N  \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l}u_{\varepsilon,k}u_{\varepsilon,i}
   -\sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^{p_l} u_{\varepsilon,i} u_{\varepsilon,k} =0.
\end{align*}
According to Theorem \ref{thm:CI},
$E_{\varepsilon,i,k}\cdot B_{\varepsilon,k}$ is then bounded in
$\mathcal{H}^1_{\rm loc}(\Omega)$.

Adapting the methods and results in Subsection \ref{strong},
we can without loss of generality assume in the following
that as $\varepsilon\to 0$,
\begin{equation}
   \label{assume:conv}
   \begin{gathered}
      \text{$u_\varepsilon \to u$ a.e.~in $\Omega$ and $\frac{\partial u_\varepsilon}{\partial x_l} 
      \to \frac{\partial u}{\partial x_l}$ a.e.~in $\Omega$},\\
      \text{$\abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2}
      \frac{\partial u_\varepsilon}{\partial x_l}\rightharpoonup
      \abs{\frac{\partial u}{\partial x_l}}^{p_l-2}
      \frac{\partial u}{\partial x_l}$ in $L^{p_l'}(\Omega;\mathbb{R}^m)$,}
   \end{gathered}
\end{equation}
for $l=1,\dots,N$. Therefore,
$$
f_{\varepsilon,i}\to f_i:= \sum_{k=1}^m \sum_{l=1}^N  \frac{\partial}{\partial x_l} u_k
\abs{\frac{\partial u}{\partial x_l}}^{p_l-2}\big(u_i\frac{\partial}{\partial x_l} u_k
- u_k\frac{\partial}{\partial x_l} u_i\big) \quad \text{a.e.~in $\Omega$},
$$
and $f_i\in L^1(\Omega)$, $i=1,\dots,N$.
Of course, the main difficulty is to
improve this a.e.~convergence
to the convergence
\[
\int_{\Omega} f_\varepsilon \cdot \phi
\,dx \to \int_{\Omega} f\cdot\phi\,dx\quad
\text{for any $\phi\in W^{1,(p_1,\dots,p_N)}_0\cap L^\infty$}.
\]
For each $i=1,\dots,N$, by Theorem \ref{thm:CI}, $f_{\varepsilon,i}$ is
bounded in $\mathcal{H}^1_{\rm loc}(\Omega)$, and for any compact
$K\subset \Omega$ we have the bound
\begin{equation}
   \label{Hardy:tmpI}
   \begin{split}
      &\norm{f_{\varepsilon,i}}_{\mathcal{H}^1(K)}
   \le C \sum_{k=1}^m 
      \Big(\sum_{l=1}^N  \norm{\frac{\partial}{\partial x_l} u_{\varepsilon,k}}_{L^{p_l}(\Omega)}\Big)
      \\ & \quad \times
      \Big(\sum_{l=1}^N  \norm{\abs{\frac{\partial u}{\partial x_l}}^{p_l-2}\big(u_i\frac{\partial}{\partial x_l} u_k
      - u_k\frac{\partial}{\partial x_l} u_i\big)}_{L^{p_l'}(\Omega)} \Big)\le C,
   \end{split}
\end{equation}
where the last constant is independent of $\varepsilon$ 
since $\frac{\partial u_\varepsilon}{\partial x_l}$ is
bounded in $L^{p_l}(\Omega;\mathbb{R}^m)$, $l=1,\dots,N$.

Let $\eta\in C^\infty_c(\Omega)$, $\int_{\Omega} \eta\,dx\neq 0$,
and introduce
$$
\begin{gathered}
   A_{\varepsilon,i}=\int_{\Omega} \eta f_{\varepsilon,i}\,dx/\int_{\Omega}\eta\,dx\in\mathbb{R}, \quad
   i=1,\dots,N,\\
   F_{\varepsilon,i}=\eta \left(f_{\varepsilon,i}-A_{\varepsilon,i}\right),\quad
   i=1,\dots,N\,.
\end{gathered}
$$
Note that $\int_{\Omega} F_{\varepsilon,i}\,dx=0$. Now we extend
all relevant functions defined on $\Omega$ to $\mathbb{R}^N$
by setting them to zero off $\Omega$.

According to  Semmes \cite[Proposition 1.92]{Semmes:92},
$F_{\varepsilon,i}$ is bounded in $\mathcal{H}^1(\mathbb{R}^N)$
and if $K=\mathrm{supp}\, (\eta)$ then
$$
\norm{F_{\varepsilon,i}}_{\mathcal{H}^1(\mathbb{R}^N)} \le C\big(1 +
\norm{F_{\varepsilon,i}}_{L^1(\mathbb{R}^N)} +
\norm{f_{\varepsilon,i}}_{\mathcal{H}^1(K)}\big), \quad i=1,\dots,N,
$$
where the right-hand side is bounded by
a constant independent of $\varepsilon$, thanks to \eqref{Hardy:tmpI}.
Observe that by \eqref{E1_map} and the last
part of \eqref{assume:conv} we have
\begin{align*}
   A_{\varepsilon,i} &= \frac{\int_{\Omega}\sum_{l=1}^N  \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2}\frac{\partial}{\partial x_l} u_{\varepsilon,i}
   \frac{\partial}{\partial x_l} \eta\,dx}{\int_{\Omega} \eta\,dx}
   \\ & 
   \to  \frac{\int_{\Omega}\sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^{p_l-2}
   \frac{\partial}{\partial x_l} u_i
   \frac{\partial}{\partial x_l} \eta\,dx}{\int_{\Omega} \eta\,dx}=:A_i,
\end{align*}
for $i=1,\dots,N$. Hence
$F_{\varepsilon,i}\to F_i:=\eta(f_i-A_i)$ a.e.~in $\mathbb{R}^N$
and, as mentioned before, $F_{\varepsilon,i}$ is
bounded in $\mathcal{H}^1(\mathbb{R}^N)$. Thanks to a theorem
of Jones and Journ{\'e} \cite{JonesJourne},
this implies that $F_{\varepsilon,i}\overset{\star}\rightharpoonup  F_i$
in $\mathcal{H}^1(\mathbb{R}^N)$, that is,
$$
\int_{\mathbb{R}^N} F_{\varepsilon,i}\Psi\,dx \to \int_{\mathbb{R}^N} F_i\Psi\,dx, \quad
\forall \Psi\in VMO(\mathbb{R}^N).
$$

Now we have all the necessary tools at our disposal
for concluding the proof of the theorem.
Let $\phi\in W^{1,(p_1,\dots,p_N)}_0(\Omega;\mathbb{R}^m)\cap L^\infty(\Omega;\mathbb{R}^m)$
and choose $\eta\in C^\infty_c(\Omega)$ such that
$\eta\equiv 1$ on $K=\mathrm{supp}\, (\phi)$. Then
\begin{align*}
   &\int_{\Omega} \sum_{l=1}^N  \abs{\frac{\partial u_\varepsilon}{\partial x_l}}^{p_l-2}\frac{\partial u_\varepsilon}{\partial x_l} \cdot \frac{\partial\phi}{\partial x_l}\,dx
   \\ & = \int_{\Omega} \sum_{i=1}^N  f_{\varepsilon,i}\phi_i\,dx
   \\ & = \int_{\mathbb{R}^N} \sum_{i=1}^N  F_{\varepsilon,i} \phi_i \,dx
   + \int_{\mathbb{R}^N} \sum_{i=1}^N  \eta A_{\varepsilon,i} \phi_i\,dx.
\end{align*}
Sending $\varepsilon\to 0$ yields
\begin{align*}
   \int_{\Omega} \sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^{p_l-2}\frac{\partial u}{\partial x_l} \cdot \frac{\partial\phi}{\partial x_l}\,dx
 & = \int_{\mathbb{R}^N} \sum_{i=1}^N  F_i \phi_i \,dx
   + \int_{\mathbb{R}^N} \sum_{i=1}^N  \eta A_i \phi_i\,dx
  \\ & 
   = \int_{\Omega} \sum_{i=1}^N  f_i\phi_i\,dx =
   \int_{\Omega} \sum_{l=1}^N  \abs{\frac{\partial u}{\partial x_l}}^{p_l}u\cdot \phi\,dx.
\end{align*}
Hence $u$ is a $(p_1,\dots,p_N)$-harmonic map.
\end{proof}

\subsection*{Acknowledgments}
This work was partially supported by the European network  HYKE, 
contract HPRN-CT-2002-00282. M. Bendahmane was supported in part 
by The Gulbenkian Foundation and FCT of Portugal. K. H. Karlsen 
was supported in part by an Outstanding Young Investigators 
Award from the Research Council of Norway.

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\end{thebibliography}


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