# Lipschitz regularity for solutions of a general class of elliptic equations

Greta Marino  and  Sunra Mosconi  Institut für Mathematik
Universität Augsburg
Universitätsstraße 12a, 86159 Augsburg, Germany<EMAIL_ADDRESS>Department of Mathematics and Computer Science
University of Catania
Viale A. Doria 6, I-95125 Catania, Italy<EMAIL_ADDRESS>
###### Abstract.

We prove local Lipschitz regularity for local minimisers of

$W^{1,1}(\Omega)\ni v\mapsto\int_{\Omega}F(Dv)\,dx$

where $\Omega\subseteq\mathbb{R}^{N}$, $N\geq 2$ and
$F:\mathbb{R}^{N}\to\mathbb{R}$ is a quasiuniformly convex integrand in the
sense of Kovalev and Maldonado [30], i. e. a convex $C^{1}$-function such that
the ratio between the maximum and minimum eigenvalues of $D^{2}F$ is
essentially bounded. This class of integrands includes the standard
singular/degenerate functions $F(z)=|z|^{p}$ for any $p>1$ and arises
naturally as the closure, with respect to a natural convergence, of the
strongly elliptic integrands of the Calculus of Variations.

###### Key words and phrases:

Elliptic regularity, Local minimisers, Quasiconformal maps, Calculus of
Variations

###### 2010 Mathematics Subject Classification:

30C65, 35B65, 35J60

###### Contents

  1. 1 Introduction
    1. 1.1 Overview and main result
    2. 1.2 The rôle of the Uhlenbeck structure
    3. 1.3 Outline of the proof
    4. 1.4 Related results
    5. 1.5 Structure of the paper
  2. 2 Quasiuniformly convex integrands
  3. 3 Local minimisers and Sobolev regularity of their stress field
  4. 4 A regularising procedure
  5. 5 Caccioppoli inequality
  6. 6 A family of $1$-homogeneous functions
  7. 7 Proof of the main result

##  1\. Introduction

###  1.1. Overview and main result

Consider a convex and coercive $F:\mathbb{R}^{N}\to\mathbb{R}$. The aim of
this paper is to prove Lipschitz regularity of the local minimisers of the
standard integral of Calculus of Variations

$J(u,\Omega)=\int_{\Omega}F(Du)\,dx$ (1.1)

i. e. of those $u\in W^{1,1}_{\rm loc}(\Omega)$ such that $F(Du)\in L^{1}_{\rm
loc}(\Omega)$ and for any open ${\mathcal{O}}\Subset\Omega$ it holds

$J(u,{\mathcal{O}})\leq J(u+w,{\mathcal{O}})\qquad\forall\,w\in
W^{1,1}_{0}({\mathcal{O}}).$

The integrands $F$ we are interested in satisfy for some $H<\infty$

$\lambda_{\rm max}(D^{2}F(z))\leq H\,\lambda_{\rm
min}(D^{2}F(z))\qquad\text{for a.\,e.\,$z\in\mathbb{R}^{N}$},$ (1.2)

where here and in the following $\lambda_{\rm max}(M)$ and $\lambda_{\rm
min}(M)$ denote respectively the minimum and maximum eigenvalues of the
symmetric matrix $M$. In this introductory paragraph, the integrands obeying
(1.2) will be called uniformly elliptic (even if the term is ubiquitous and
thus may cause some confusion). As will be clarified later, most of the
available Lipschitz regularity results for local minimisers prescribe that the
singular set, where $\lambda_{\rm max}(D^{2}F(z))$ blows up, and the
degeneracy set, where $\lambda_{\rm min}(D^{2}F(z))$ vanishes, are both
bounded. As will be clarified in the following discussion, condition (1.2)
alone instead allows for both sets to be simultaneously unbounded (even dense)
and sizeable (in the sense of Hausdorff dimension).

In order to motivate the uniform ellipticity condition (1.2), let us review
some well known facts regarding local minimisers of (1.1). If $F$ fulfils a
polynomial upper bound, these are finite energy solutions of the corresponding
Euler-Lagrange equation

${\rm div}\,(DF(Du))=0$ (1.3)

in $\Omega$. Existence of minimisers is ensured by standard methods once a
superlinearity condition on $F$ at $\infty$ is imposed, however well known
examples (see [21, 33]) show that under these sôle conditions (sub-polynomial
growth and superlinearity) they may fail to be locally bounded. Regularity of
local minimisers is a classical topic dating back to Hilbert’s XIX problem,
solved through Schauder and DeGiorgi-Nash-Moser theories under the strong
ellipticity assumption

$0<\inf_{z\in\mathbb{R}^{N}}\lambda_{\rm
min}(D^{2}F(z))\leq\sup_{z\in\mathbb{R}^{N}}\lambda_{\rm
max}(D^{2}F(z))<\infty.$

Since the solution of Hilbert’s problem, much effort has been dedicated to
weaken this assumption. A natural quantity arising in the aforementioned
regularity theories is the ellipticity ratio (also called the linear
dilatation of $D^{2}F(z)$), namely

${\rm e}(z)=\frac{\lambda_{\rm max}(D^{2}F(z))}{\lambda_{\rm
min}(D^{2}F(z))}.$

The closure of smooth, strongly elliptic integrands with respect to pointwise
(or, equivalently, $C^{1}_{\rm loc}(\mathbb{R}^{N})$) convergence, when
coupled with a uniform bound on the ellipticity ratio, turns out to be a cone
consisting of

  * •

Affine functions, which are the extremals of the cone

  * •

Convex superlinear $F\in C^{1}_{\rm loc}(\mathbb{R}^{N})\cap W^{2,N}_{\rm
loc}(\mathbb{R}^{N})$ obeying (1.2) a. e. for some $H<\infty$

(see Corollary 2.7). The second type of integrands have been studied in the
seminal papers [29, 30] (to which we refer for further details) and are called
$H$-quasiuniformly convex,111Actually, in [29, 30] emphasis is given to the
geometric quasiconformality constant of the gradient map, rather than on its
linear dilatation $H$ defined in (1.2) henceforth abbreviated by $H$-q. u. c..
They are related to the well developed theory of quasiconformal maps, as their
gradient mapping is indeed quasiconformal. It turns out, quite conveniently we
may say, that in dimension $N\geq 2$ condition (1.2) implies not only
superlinearity, but also sub-polynomial growth and strict convexity of the
integrand $F$, granting existence of local minimisers (or actually, minimisers
under various boundary conditions) in $W^{1,1+1/H}_{\rm loc}(\Omega)$.
Moreover, local minimisers and finite energy weak solutions of (1.3) coincide
(see Proposition 3.2). Equation (1.3) can be formally derived with respect to
each variable $x_{\alpha}$, giving for each partial derivative
$\partial_{\alpha}u$

${\rm div}\,(D^{2}F(Du)\,D\partial_{\alpha}u)=0,\qquad\alpha=1,\dots,N,$ (1.4)

which fails to be strongly elliptic for $Du$ belonging to ${\rm
Sing}_{F}\cup{\rm Deg}_{F}$, where

$\begin{split}{\rm
Sing}_{F}&=\left\\{\bar{z}\in\mathbb{R}^{N}:\operatorname*{ess\,lim\,sup}_{z\to\bar{z}}\lambda_{\rm
max}(D^{2}F(z))=\infty\right\\},\\\ {\rm
Deg}_{F}&=\left\\{\bar{z}\in\mathbb{R}^{N}:\operatorname*{ess\,lim\,inf}_{z\to\bar{z}}\lambda_{\rm
min}(D^{2}F(z))=0\right\\}.\end{split}$ (1.5)

If $F$ is q. u. c., the previous sets are related to the so-called
quasiconformal $\infty$\- and $0$\- sets respectively, appearing in the
quasiconformal Jacobian problem [7]. These can be quite large, as shown in
[30, Section 4 and 5]: given an arbitrary $E\subseteq\mathbb{R}^{N}$ of
Hausdorff dimension less than $1$, there exist a q. u. c. $F_{1}$ such that
$E\subseteq{\rm Sing}_{F_{1}}$ and a q. u. c. $F_{2}$ such that
$E\subseteq{\rm Deg}_{F_{2}}$. In particular, both ${\rm Sing}_{F}$ and ${\rm
Deg}_{F}$ can be dense in $\mathbb{R}^{N}$ for $N\geq 2$. On the other hand,
for $F$ q. u. c., neither ${\rm Sing}_{F}$ nor ${\rm Deg}_{F}$ can contain
rectifiable curves (see [7, 30]).

Despite the natural appearance of q. u. c integrands as limits (in the sense
described above) of strongly elliptic ones, the corresponding regularity
theory for (1.4) seems more delicate. Indeed, the $L^{\infty}$ and
$C^{\alpha}$ bounds for $Du$ when $F$ is strongly elliptic essentially depend
on the quantity

$\frac{\sup_{\Omega}\lambda_{\rm max}(D^{2}F(Du))}{\inf_{\Omega}\lambda_{\rm
min}(D^{2}F(Du))}$

rather than on the actually controlled quantity

$\sup_{\Omega}\frac{\lambda_{\rm max}(D^{2}F(Du))}{\lambda_{\rm
min}(D^{2}F(Du))}$

and a naive limiting argument is bound to fail.

Nevertheless, some classes of q. u. c. where the standard regularity theory
can still produce results are already well known. An important type of q. u.
c. integrands are for instance those with Uhlenbeck structure, i. e. depending
only on the modulus of the gradient. Given a non-decreasing
$G:[0,\infty[\to[0,\infty[$, the function $F(z)=G(|z|)$ is q. u. c. if and
only if $G\in C^{1}([0,\infty))\cap W^{2,\infty}_{\rm loc}(\mathbb{R}_{+})$
and there exists a constant $C>0$ such that

$\frac{1}{C}\leq\frac{t\,G^{\prime\prime}(t)}{G^{\prime}(t)}\leq
C\qquad\text{for a.\,e.\,$t>0$}.$ (1.6)

In particular, for a q. u. c. function with Uhlenbeck structure, it always
holds

${\rm Sing}_{F}\subseteq\\{0\\}\quad\text{as well as}\quad{\rm
Deg}_{F}\subseteq\\{0\\}$ (1.7)

which, since obviously ${\rm Sing}_{F}\cap{\rm Deg}_{F}=\emptyset$ for a q. u.
c. $F$, justifies the traditional dichotomy between the singular or degenerate
case of (1.3).

The regularity theory for solutions of (1.3) when $F$ is a Uhlenbeck q. u. c.
integrand is well developed, even when non-homogenous terms with low
summability are included and no variational setting it available. See e. g.
[12] and the literature therein for a survey on the available results for
solutions of

${\rm div}\,\left(\frac{G^{\prime}(|Du|)}{|Du|}\,Du\right)=f$ (1.8)

up to the Lipschitz scale, depending on the summability properties of $f$.
Regarding higher regularity, in [32] minimisers have been proved to be
$C^{1,\alpha}$ for integrands which are even more general than the Uhlenbeck
ones, but still fulfil a uniform radial bound on the ellipticity ratio. By
this we mean that the main assumption for the $C^{1,\alpha}$ regularity is the
existence of two radial functions $\lambda_{\min},\lambda_{\rm
max}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ obeying

$\lambda_{\rm min}(|z|)\leq\lambda_{\rm min}(D^{2}F(z)),\qquad\lambda_{\rm
max}(|z|)\geq\lambda_{\rm max}(D^{2}F(z))$

and

$\sup_{t\in\mathbb{R}_{+}}\frac{\lambda_{\rm max}(t)}{\lambda_{\rm
min}(t)}<\infty$

so that these integrands are q. u. c.. In this framework both ${\rm Sing}_{F}$
and ${\rm Deg}_{F}$ are anyway restricted to have radial symmetry, but since
neither of those can contain circles, (1.7) holds again.

Notice that when $F$ is a q. u. c., normalised in such a way that
$\min_{\mathbb{R}^{N}}F=F(0)=0$ (which can always be safely assumed), both $F$
and $DF$ actually fulfil a two-sided isotropic control of the form

$\frac{1}{C}\,A(|z|)\leq F(z)\leq
C\,A(|z|),\qquad\frac{1}{C}\,A^{\prime}(|z|)\leq|DF(z)|\leq
C\,A^{\prime}(|z|)$

for a $C^{1}$ Young function $A:[0,\infty[\to[0,\infty[$ (see Section 3) but,
as mentioned before, no such isotropic control is available at the level of
the ellipticity ratio for a general q. u. c. integrand.

Our interest in this framework is motivated by the results in [4, Chapter 16]
where regularity theory for the corresponding minimisers has been successfully
developed in two space dimensions. More precisely, in [4, Theorem 16.4.5], the
authors prove $C^{1,\alpha}$ regularity of finite energy222In this possibly
non-variational setting, the energy considered is given by the integral of
$({\mathcal{A}}(Du),Du)$. solutions of

${\rm div}\,{\mathcal{A}}(Du)=0$ (1.9)

in the plane, under the even more general assumption that the mapping
${\mathcal{A}}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ fulfils

$\left({\mathcal{A}}(z)-{\mathcal{A}}(w),z-w\right)\geq\delta\,|{\mathcal{A}}(z)-{\mathcal{A}}(w)|\,|z-w|,\qquad\forall\,z,w\in\mathbb{R}^{2}$

for some $\delta>0$. Such mappings are called $\delta$-monotone and have been
introduced in [29], where it is shown that any gradient map of a
quasiuniformly convex integrand is $\delta$-monotone, but notice that in
general equation (1.9) may be non-variational. Hence the results of [4] in the
plane are more general and stronger than ours, but the method developed
therein is based on the reduction of (1.9) to a first-order system of Beltrami
type for the complex gradient of $u$, and thus are constrained to the two-
dimensional setting.

We can now state our main result.

######  Theorem 1.1.

For $N\geq 2$, let $F\in C^{1}_{\rm loc}(\mathbb{R}^{N})\cap W^{2,1}_{\rm
loc}(\mathbb{R}^{N})$ be convex, obey (1.2) a. e. and fulfil the normalisation

$F(z)\geq F(0)=0\qquad\forall\,z\in\mathbb{R}^{N}.$ (1.10)

Then any local minimiser $u$ for $J$ in $\Omega$ is locally Lipschitz. In
particular, there exists a constant $C=C(H,N)>0$ such that if
$B_{2R}\subseteq\Omega$, then

$\sup_{B_{R}}|F(Du)|\leq C\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx.$ (1.11)

######  Remark 1.2 (Comments on the statement).

  * •

Except for the normalisation condition (1.10), the assumptions on $F$ can be
equivalently stated as $F$ being $H$-quasiuniformly convex, or $DF$ being a
quasiconformal map with maximal linear dilatation bounded by $H$. Notice that,
thanks to the quasiconformality of $DF$, the $W^{2,2}_{\rm
loc}(\mathbb{R}^{N})$ regularity of $F$ together with condition (1.2)
automatically improves to $F\in W^{2,N+\varepsilon}_{\rm loc}(\mathbb{R}^{N})$
for some $\varepsilon>0$.

  * •

The normalisation condition (1.10) is made only to have the cleaner estimate
(1.11), as any $H$-q. u. c. integrand $F$ has a unique minimum point
$\bar{z}$. Then, the integrand

$\tilde{F}(z)=F(z+\bar{z})-F(\bar{z})$

is still $H$-q. u. c. and the function $\tilde{u}(x)=u(x)-(x,\bar{z})$ is a
local minimiser for the corresponding integral functional. Notice that (1.11),
having no additional term on the right, prescribes point-wise smallness of
$Du$ for small values of its energy.

  * •

Condition (1.2) giving $H$-q. u. convexity does not distinguish between
singular or degenerate equations, since both $z\mapsto|z|^{1+1/H}$ and
$z\mapsto|z|^{1+H}$ are $H$-q. u. c.. More substantially, a simple
construction (see Example 2.4) shows that for a q. u. c. integrand $F$, both
${\rm Sing}_{F}$ and ${\rm Deg}_{F}$ can be simultaneously non-empty, so that
the traditional dichotomy between singular/degenerate equation cannot possibly
hold in this setting.

  * •

We could also have treated local minimisers of

$J(u,{\mathcal{O}})=\int_{\mathcal{O}}F(Du)+f\,u\,dx,\qquad{\mathcal{O}}\Subset\Omega,$

for sufficiently smooth $f$. However, our method seems to provide Lipschitz
regularity under non-optimal regularity conditions on $f$. The natural
condition ensuring boundedness of $Du$ should be $f\in L^{q}_{\rm
loc}(\Omega)$ for some $q>N$, but it appears that substantial modifications to
the proof are needed to obtain such a result.

###  1.2. The rôle of the Uhlenbeck structure

Let us discuss briefly the rôle of a radial control on the ellipticity ratio
in proving a-priori gradient bounds for finite energy solutions of (1.3) with
a convex $F$. The standard strategy to this end, which goes back to Bernstein,
is to find a suitable coercive function $G:\mathbb{R}^{N}\to\mathbb{R}$ and
coefficients $a_{ij}:\mathbb{R}^{N}\to\mathbb{R}$ such that

  1. (1)

The matrix $A(z)=(a_{ij}(z))$ has controlled ellipticity for every
$z\in\mathbb{R}^{N}$;

  2. (2)

The function $G(Du)$ solves

${\rm div}\,(A(Du)\,D(G(Du)))\geq 0.$ (1.12)

Once these tasks are achieved, linear regularity theory can be applied to
$G(Du)$. If, for instance, $A$ turns out to be strongly elliptic, then
solutions of (1.12) fulfil

$\|G(Du)\|_{L^{\infty}(B_{R})}\leq C\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}G(Du)\,dx$

and we are reduced to control the integrand on the right (or variants of it)
by $F(Du)$. More generally, once suitable refinements of the linear elliptic
regularity step are developed, this scheme is flexible enough to deal with
choices of $A$ which may not be strongly elliptic, as long as their
ellipticity ratio does not blow up too fast for $z\to\infty$ (see e. g. [6]
and the literature therein for the non-uniformly elliptic setting).

For the sake of this discussion, we will suppose that $F$ and $u$ are smooth
and that we have chosen $\lambda_{\rm min},\lambda_{\rm
max}:\mathbb{R}^{N}\to\mathbb{R}_{+}$ in such a way that for all
$z\in\mathbb{R}^{N}$ it holds

$0\leq\lambda_{\rm min}(z)\leq\lambda_{\rm min}(D^{2}F(z)),\qquad\lambda_{\rm
max}(z)\geq\lambda_{\rm max}(D^{2}F(z)),\qquad\frac{\lambda_{\rm
max}(z)}{\lambda_{\rm min}(z)}\leq H.$ (1.13)

The standard approach to construct the couple $(A,G)$ is to multiply each
equation in (1.4) by $\partial_{\alpha}u$ and sum, to obtain

${\rm
div}\,\Big{(}D^{2}F(Du)\,\sum_{\alpha=1}^{N}(D\partial_{\alpha}u)\,\partial_{\alpha}u\Big{)}=\sum_{\alpha=1}^{N}D^{2}F(Du)\,D\partial_{\alpha}u\,D\partial_{\alpha}u\geq
0$ (1.14)

where the last inequality follows from the convexity of $F$. With this choice,
however, we are committing to a radial control on the ellipticity ratio, since

$\sum_{\alpha=1}^{N}(D\partial_{\alpha}u)\,\partial_{\alpha}u=D\frac{|Du|^{2}}{2}$

and a natural choice for $A$ is

$A(z)=\frac{D^{2}F(z)}{\lambda_{\rm min}(z)},$

which is strongly elliptic if the ellipticity ratio is uniformly bounded,
while in the meantime (1.14) reads

${\rm div}\,\left(A(Du)\,\lambda_{\rm min}(Du)\,D\frac{|Du|^{2}}{2}\right)\geq
0.$

In order to construct $G$ we must impose

$DG(z)=\lambda_{\rm min}(z)\,D\frac{|z|^{2}}{2}=\lambda_{\rm min}(z)\,z$
(1.15)

but this relation forces $G$, and thus $\lambda_{\rm min}$, to be radial
functions. If this is so, i. e. $\lambda_{\rm min}(z)=\lambda_{\rm min}(|z|)$,
we can indeed set

$G(z)=\int_{0}^{|z|}\lambda_{\rm min}(s)\,s\,ds$

which fulfils (1.15). Notice finally that, since $F(0)=0$ and $DF(0)=0$, for
$|\omega|=1$ and $t>0$ it holds

$F(t\,\omega)=\int_{0}^{t}\left(D^{2}F(s\,\omega)\,\omega,\omega\right)(t-s)\,ds\geq\int_{0}^{t}\lambda_{\rm
min}(s)\,(t-s)\,ds\geq\int_{0}^{t/2}\lambda_{\rm min}(s)\,s\,ds=G(t/2)$

which can be used to bound $G(Du)$ in terms of the natural integrand
$F(Du)$.333As we will see, if ${\rm e}$ is uniformly bounded it always holds
$F(2\,z)\leq C\,F(z)$. Without a radial control on $\lambda_{\rm min}$ or
$\lambda_{\rm max}$, however, the previous approach fails.

######  Remark 1.3.

It may be worth noting that multiplying (1.4) by $\partial_{\alpha}F(Du)$ and
summing up, one finds

${\rm div}\,\big{(}D^{2}F(Du)D(F(Du))\big{)}={\rm
Tr}\,\big{(}D^{2}F(Du)\,D^{2}u\,D^{2}F(Du)\,D^{2}u).$

As will be quantified in the following paragraph, the convexity of $F$ ensures
that the right-hand side above is still nonnegative, so that we have found
another couple $(A,G)$ fitting the previous scheme, namely,

$A=D^{2}F(Du),\qquad G=F.$

The issue, however, is that $A$ fails to be strongly elliptic and is only
uniformly elliptic. Needless to say, the linear theory for solutions of

${\rm div}\,(A(x)\,Dv)=0$

under the uniform ellipticity condition

$0\leq\lambda_{\rm max}(A(x))\leq H\,\lambda_{\rm
min}(A(x))\qquad\text{a.\,e.}$

is very poor (as can be seen by simple one-dimensional examples) and does not
provide boundedness of solutions. Nevertheless, estimate (1.11) says that the
function $v=F(Du)$ behaves “as if” the operator ${\rm
div}\,\big{(}D^{2}F(Du)\,Dv)$ is strongly elliptic on $v$.

###  1.3. Outline of the proof

The possibility of tackling the regularity problem for finite energy solutions
of (1.3) in dimension $N\geq 3$ under the general uniform ellipticity
condition (1.2) has been considered for the first time in [23]. The approach
adopted therein can be seen in the framework of nonlinear differential
inclusions, originally rooted in the works [5, 37, 18] and well developed
nowadays, thanks to the significant applications investigated in [34, 28, 39,
15], to name a few. Specifically, one can look at (1.3) focusing only on the
stress field $DF(Du)=V$. Formally deriving the equation, we find the system

${\rm Div}\,(DV^{t})=0$ (1.16)

where ${\rm Div}$ is the row-wise divergence operator acting on matrix-valued
functions and $M^{t}$ denotes the transpose of the matrix $M$. Notice that,
contrary to the Laplacian operator

$\Delta V={\rm Div}\,(DV),$

the operator in (1.16) is far from being elliptic: its kernel contains all
compactly supported solenoidal vector fields and there is no hope to prove
regularity of solutions to system (1.16). Ellipticity, however, can be
restored by constraining $DV$ to pointwise belong to a suitable cone. The
natural and fruitful one in our framework, as found in [23], is given by

${\mathcal{K}}_{H}=\left\\{M\in\mathbb{R}^{N\times N}:{\rm
Tr}\,(M\,M)\geq\frac{1}{H}\,{\rm Tr}\,(M\,M^{t})\right\\},$

for $H$ given in (1.2). The fact that $DV$ must belong to such a cone depends
only on the structural condition

$DV=D^{2}F(Du)\,D^{2}u,$

i. e. $DV$ must be the product of a positive definite symmetric matrix with
linear dilatation bounded by $H$ with a symmetric matrix (see Lemma 5.1
below). Note that, as $H\searrow 1$, we have
${\mathcal{K}}_{H}\searrow{\mathcal{K}}_{1}={\rm Sym}_{N}$, that is the linear
space of symmetric $N\times N$ matrices, hence the solutions of the
differential inclusion

$\begin{cases}{\rm Div}\,(DV^{t})=0\\\ DV\in{\mathcal{K}}_{1}\end{cases}$
(1.17)

are just the gradients of harmonic functions.

For $H>1$, system (1.17) is elliptic at the $L^{2}$ level, but this kind of
restored ellipticity has a limited effect at finer regularity scales. Indeed,
stress fields of $p$-harmonic equation certainly solve the previous nonlinear
differential inclusion for $p=1+1/H$, and for $p\neq 2$ there exists (see
[27]) in the plane a $p$-harmonic function whose stress field vanishes only at
the origin and is homogenous of degree

$d=\frac{1}{6}\left(p+\sqrt{p^{2}+12\,p-12}\right)=\frac{1}{6\,H}\left(H+1+\sqrt{H^{2}+14\,H+1}\right).$

Since the last expression is always less than $1$ for $H>1$, the best
regularity one can expect from solutions of (1.17) is at most $C^{\alpha}$,
for $\alpha=\alpha(H)\in\ [1/3,1[$ and for any $H>1$. We stress here that we
were not able to prove that solutions of (1.17) are bounded, which would imply
the qualitative part of Theorem 1.1 and would be meaningful in light of
current lines of research on differential inclusions (see e. g. [2, 24]). In
fact, we had to resort to additional structure possessed by the original
equation (1.3). More precisely, by still denoting $V=DF(Du)$ for a solution of
(1.3) and by using the differential inclusion $DV\in{\mathcal{K}}_{H}$, we
prove a family of Caccioppoli inequalities of the form

$\int_{B_{r}\cap\\{F(Du)>k\\}}|DV|^{2}\,dx\leq\frac{C_{H}}{(R-r)^{2}}\int_{B_{R}\cap\\{F(Du)>k\\}}|V|^{2}\,dx$
(1.18)

for arbitrary $k\in\mathbb{R}$ and for $R>r>0$. Then we translate this
vectorial Caccioppoli inequality into a family of Caccioppoli inequalities on
suitable scalar functions intrinsically defined as suitable Minkowski
functionals associated to

$G(z)=F(DF^{-1}(z))$

(which is well defined since $DF$ is invertible), namely,

$g_{k}(z)=\inf\left\\{t>0:G(z/t)>k\right\\}$ (1.19)

for any $k\in\mathbb{R}$. Notice that, since $V=DF(Du)$, then

$\\{F(Du)>k\\}=\\{G(V)>k\\}=\\{g_{k}(V)>1\\}$

and the quasiuniform convexity of $F$ is pivotal to prove that (1.18)
translates to

$\int_{B_{r}}|D(g_{k}(V)-1)_{+}|^{2}\,dx\leq\frac{C_{H}}{(R-r)^{2}}\,\int_{B_{R}}|g_{k}(V)|^{2}\,dx.$
(1.20)

Additional fine properties of the family of $1$-homogeneous functions
$\\{g_{k}\\}$ allow to adapt the classical De Giorgi method for proving
boundedness of subsolutions. Notice that (1.20) exhibits two main differences
with respect to a standard Caccioppoli inequality. On the one hand, due to the
vectorial nature of (1.18), its dependence on the level $k$ is encapsulated
rather implicitly in the family of $1$-homogeneous functions $\\{g_{k}\\}$
rather than directly on $V$. More substantially, its right-hand side is quite
bigger than the one usually found for scalar problems, which would provide an
integrand of the form $(g_{k}(V)-1)_{+}$ on the right instead of $g_{k}(V)$.
Nevertheless, the De Giorgi method can still be adapted to this weaker
setting, providing an $L^{\infty}$ bound on $F(Du)$ in terms of the $L^{2}$
norm of $V$. It is quite fortunate that this estimate implies, through a
refinement of the results in [23], the natural bound (1.11).

###  1.4. Related results

In the uniformly elliptic setting the result in Theorem 1.1 has, as already
remarked, been previously obtained in [32] under an Uhlenbeck type control of
the form (1.13), with $\lambda_{\rm min}(z)$ and $\lambda_{\rm max}(z)$
depending only on the modulus of $z$. In [32], actually, $C^{1,\alpha}$
regularity is proved under these assumptions. Lipschitz regularity has been
proved in [13] for solutions of (1.8) coupled with Dirichlet or Neumann
boundary conditions, with optimal regularity assumptions both on $f$ and on
the domain. The corresponding regularity for solutions of systems with
Uhlenbeck structure is treated in [14].

Notice that in [32, 12, 14] the authors assume that the function $G^{\prime}$
appearing in (1.8) (or in the lower/upper controls on $D^{2}F(z)$) belongs to
$C^{1}(\mathbb{R}_{+})$, while we admit $G^{\prime}\in{\rm Lip}_{\rm
loc}(\mathbb{R}_{+})$ (this regularity seems optimal due to [23, Example
3.5]). However, inspecting the proofs in [12, 14] shows that the results
therein hold under this more general assumption.

Our result covers in particular the Finslerian anisotropic setting, which we
will now briefly describe due to its relevance in recent research trends. The
integrand considered in this framework are of the form

$F(z)=G(h(z))$

where $G\in C^{1}(\mathbb{R}_{+})\cap W^{2,\infty}_{\rm loc}(\mathbb{R}_{+})$
is increasing, convex and fulfils (1.6) (see [23, Example 3.7]), while $h$ is
a $C^{2}(\mathbb{R}^{N}\setminus\\{0\\})$ convex, positive, $1$-homogeneous
function (not necessarily symmetric) such that the principal curvatures of
$\partial\\{h<1\\}$ are bounded from below by a positive constant, see [23,
Example 3.7] for more details. The arguments in [11, Section 3] show that
$C^{1,\alpha}$ regularity holds true for the corresponding minimisers when $G$
has more stringent controls of $p$-growth type and $h\in
C^{3,\alpha}(\mathbb{R}^{N}\setminus\\{0\\})$.

While the non-uniformly elliptic case is not treated in this research, it is
worth mentioning that there are many instances where Lipschitz regularity can
be obtained even if the ellipticity ratio is unbounded. Generally speaking, in
order to get Lipschitz continuity of local minimisers, one usually requires a
growth control on the ellipticity ratio outside a bounded set, but in most
cases the resulting Lipschitz bound critically depends on the diameter of the
aforementioned set. The literature in this area is huge and, regarding local
minimisers, we refer to [6] for some recent results and for a rather
comprehensive description of this research topic. Lipschitz regularity can be
obtained for rather wild functionals by the so called Hilbert-Haar method,
adapted to the Calculus of Variations by Stampacchia and Hartman [36, 25].
This allows to obtain Lipschitz regularity of minimisers having a prescribed
boundary value obeying the so-called bounded slope condition, under very loose
conditions on the integrand. We refer to [8] and the literature therein for
more details on this approach. Another class of non-uniformly elliptic
integrands are the so-called orthotropic ones, where both ${\rm Sing}_{F}$ and
${\rm Deg}_{F}$ are unbounded, being union of hyperplanes $\\{z_{i}=0\\}$. In
this setting, assuming a-priori boundedness of the minimiser is the key to
infer Lipschitz regularity. We refer to [9] and the literature therein for
further details on this class of integrands.

###  1.5. Structure of the paper

In Section 2 we collect the properties of quasiuniformly convex functions
which are relevant to the proof. Section 3 is devoted to a refinement of the
Sobolev regularity of the stress field $V=DF(Du)$, originally obtained in
[23]. In Section 4 we construct a sequence of approximating elliptic problems
and corresponding solutions, allowing to reduce the proof of Theorem 1.1 to
the smooth setting. In Section 5 we prove the Caccioppoli inequality (1.18),
while in Section 6 we derive several properties of the family of
$1$-homogeneous functions defined in (1.19). The final Section 7 is devoted to
the proof of Theorem 1.1.

Acknowledgments. We thank prof. L. Kovalev for improving a preliminary version
of the paper. The authors are member of GNAMPA of INdAM. G. M. acknowledges
the support of DFG via grant GZ: MA 10100/1-1 project number 496629752. S. M.
is partially supported by the projects PIACERI linea 2 and linea 3 of the
University of Catania and by the GNAMPA’s projects Equazioni alle derivate
parziali di tipo ellittico o parabolico con termini singolari and Problemi
ellittici e parabolici con termini di reazione singolari e convettivi.

Notations. In the whole paper we restrict to the case $N\geq 2$. By $|v|$ we
denote the Euclidean norm of a vector $v\in\mathbb{R}^{N}$, while $(v,w)$
stands for the scalar product of any $v,w\in\mathbb{R}^{N}$. Given a vector
field, upper and lower indexes stand for its components and its derivatives,
respectively. We sum over repeated indexes. With the symbol $\Omega$ we mean a
bounded, open subset of $\mathbb{R}^{N}$, while $B_{r}(x_{0})$ denotes a ball
with center $x_{0}\in\mathbb{R}^{N}$ and radius $r>0$, and by $B_{r}$ we
indicate a ball of radius $r$, not necessarily centred at the origin. For any
measurable $E\subset\mathbb{R}^{N}$, $|E|$ denotes its $N$-dimensional
Lebesgue measure. We will omit the domain of integration when it is the whole
$\mathbb{R}^{N}$, if this causes no confusion. Furthermore, for the sake of
notational simplicity we set $\|f\|_{m}:=\|f\|_{L^{m}(\mathbb{R}^{N})}$.

Let $M=(m_{ij})$ be an $N\times N$ matrix with real entries, and let $M^{t}$
denote its transpose. We consider the Frobenius norm

$|M|_{2}=\left(\sum_{ij=1}^{N}|m_{ij}|^{2}\right)^{1/2}$

arising from the scalar product
$(M_{1},M_{2})_{2}=\operatorname{Tr}(M_{1}\,M_{2}^{t})$. We further denote by
${\rm Id}$ the identity matrix. Finally, for any matrix $M$, $\sigma_{\rm
max}(M)$ and $\sigma_{\rm min}(M)$ denote its maximum and minimum singular
values (i. e. the square roots of the eigenvalues of $M\,M^{t}$),
respectively. If $M$ is symmetric and non-negative definite, we will use the
notation $\lambda_{\rm max}(M)$ and $\lambda_{\rm min}(M)$ as in this case
eigenvalues and singular values coincide.

##  2\. Quasiuniformly convex integrands

######  Definition 2.1.

A map $\Phi:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is $K$-quasiconformal if it is a
$W^{1,1}_{\rm loc}(\mathbb{R}^{N};\mathbb{R}^{N})$ homeomorphism, it is a. e.
differentiable and the inequality

$\frac{1}{K}\,\sigma_{\rm max}(D\Phi)^{N}\leq|J\,\Phi|\leq K\,\sigma_{\rm
min}(D\Phi)^{N}$

holds a. e. in $\mathbb{R}^{N}$.

Let $\overline{\mathbb{R}}^{N}$ denote the one-point compactification of
$\mathbb{R}^{N}$. By [38, Theorem 17.3] any quasiconformal map
$\Phi:\mathbb{R}^{N}\to\mathbb{R}^{N}$ can be extended to a homeomorphism of
$\overline{\mathbb{R}}^{N}$ by setting $\Phi(\infty)=\infty$, in the meantime
keeping its (geometric) quasiconformality constant $K$ unaltered. Moreover,
$\Phi\in W^{1,N}_{\rm loc}(\mathbb{R}^{N})$ and is quasisymmetric, that is,
there are an increasing homeomorphism $\eta:\mathbb{R}_{+}\to\mathbb{R}_{+}$
and a constant $C>0$ such that

$\frac{\left|\Phi(z)-\Phi(z_{0})\right|}{|\Phi(w)-\Phi(z_{0})|}\leq
C\,\eta\left(\frac{|z-z_{0}|}{|w-z_{0}|}\right)$ (2.1)

for all $z_{0}\in\mathbb{R}^{N}$ and
$z,w\in\mathbb{R}^{N}\setminus\\{z_{0}\\}$. Set, for $\alpha>0$ and $t\geq 0$,

$\eta_{\alpha}(t)=\max\bigl{\\{}t^{\alpha},t^{1/\alpha}\bigr{\\}}.$ (2.2)

By [1, Theorems 3.18 and 5.1], any $K$-quasiconformal map $\Phi$ fulfils (2.1)
with

$C=C(K),\qquad\eta=\eta_{K^{1/(N-1)}}.$

The maximal linear dilatation of $\Phi$ is defined as the (finite) number

$H=\operatorname*{ess\leavevmode\nobreak\
sup}_{z\in\mathbb{R}^{N}}\frac{\sigma_{\rm max}(D\Phi(z))}{\sigma_{\rm
min}(D\Phi(z))}.$

Elementary linear algebra shows that such a quasiconformal $\Phi$ is actually
$H^{N-1}$-quasiconformal, hence it fulfils the distortion estimate

$\frac{\left|\Phi(z)-\Phi(z_{0})\right|}{|\Phi(w)-\Phi(z_{0})|}\leq
C\,\eta_{H}\left(\frac{|z-z_{0}|}{|w-z_{0}|}\right)$ (2.3)

for all $z_{0}\in\mathbb{R}^{N}$ and all
$z,w\in\mathbb{R}^{N}\setminus\\{z_{0}\\}$. Notice that the constant $C$
actually depends on $N$ as well. Finally, recall that if $\Phi$ is
quasiconformal then so is $\Phi^{-1}$ and the maximal linear dilatations of
$\Phi$ and $\Phi^{-1}$ coincide. Hence (2.3) holds true for $\Phi^{-1}$ as
well.

######  Definition 2.2.

A map $\Phi:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is $\delta$-monotone for some
$\delta\in\,]0,1]$ if

$\left(\Phi(z)-\Phi(w),z-w\right)\geq\delta\,|\Phi(z)-\Phi(w)|\,|z-w|.$

Kovalev’s theorem [29] shows that any non-constant $\delta$-monotone map is
quasiconformal. In particular it is a homeomorphism and, directly from the
definition, its inverse is $\delta$-monotone as well. Kovalev theorem has a
quantitative version proved in [3, Theorem 1]. It states that any
$\delta$-monotone map has a maximal linear dilatation obeying

$H\leq\frac{1+\sqrt{1-\delta^{2}}}{1-\sqrt{1-\delta^{2}}}$

and this bound is sharp, i. e. it reduces to an equality for the
$\delta$-monotone linear map $v\mapsto A\,v$, where

$A=\begin{pmatrix}1+\sqrt{1-\delta^{2}}&0\\\
0&1-\sqrt{1-\delta^{2}}\end{pmatrix}.$ (2.4)

The opposite implication is in general not true. To see this, consider the
linear map $v\mapsto B\,v$, with $B$ given by

$B=\begin{pmatrix}1&-2\\\ 2&-1\end{pmatrix}$

which is quasiconformal with $H=3$ but not even monotone. However, full (even
quantitatively) equivalence of the two concepts holds true in the class of
gradient mappings.

######  Definition 2.3.

A differentiable function $F:\mathbb{R}^{N}\to\mathbb{R}$ is
$H$-quasiuniformly convex (briefly, $H$-q. u. c.) if it is convex and its
gradient map is quasiconformal, with maximal linear dilatation bounded by $H$.

More explicitly (and by making use of Aleksandrov’s theorem), $F$ is $H$-q. u.
c. if

  * (i)

$F$ is convex, $C^{1}$ and $W^{2,1}_{\rm loc}(\mathbb{R}^{N})$, and not
affine;

  * (ii)

It holds

$\lambda_{\rm max}(D^{2}F(z))\leq H\,\lambda_{\rm min}(D^{2}F(z))$ (2.5)

for a. e. point $z$ of second order differentiability.

By [30, Theorem 3.1 and Lemma 3.2], any q. u. c. function is strictly convex
and coercive, thus it has a unique minimum point. Since, given two symmetric
matrices $M_{1},M_{2}$, it holds

$\lambda_{\rm min}(M_{1}+M_{2})\geq\lambda_{\rm min}(M_{1})+\lambda_{\rm
min}(M_{2}),\qquad\lambda_{\rm max}(M_{1}+M_{2})\leq\lambda_{\rm
max}(M_{1})+\lambda_{\rm max}(M_{2}),$ (2.6)

the set of $H$-q. u. c. functions turns out to be a convex cone, i. e. it is
closed by sum and positive scalar multiples. Moreover, it is also closed by
isometric change of variables, as well as dilations. Finally, summing an
affine function to a $H$-q. u. c. function still gives a $H$-q. u. c.
function. For these reasons we will frequently normalize q. u. c. integrands
by requiring that

$F(z)\geq F(0)=0$ (2.7)

and that $i_{F}=1$, where

$i_{F}:=\inf_{|z|=1}|DF(z)|,$ (2.8)

simply by considering

$\tilde{F}(z)=\frac{1}{i_{F}}\,\left(F(z+\bar{z})-F(\bar{z})\right),\qquad\text{where
}{\rm Argmin}\,(F)=\\{\bar{z}\\}.$ (2.9)

######  Example 2.4.

Some examples of q. u. c. integrands have already been discussed in the
previous section. Here, given two points $z_{1}\neq z_{2}$ in
$\mathbb{R}^{N}$, we construct a q. u. c. function $F$ such that ${\rm
Sing}_{F}=\\{z_{1}\\}$ and ${\rm Deg}_{F}=\\{z_{2}\\}$, the latter sets being
as in (1.5).

Without loss of generality, assume $z_{1}=0$ and $z_{2}=w$, for fixed
$w\in\mathbb{R}^{N}$, and set $r=|w|$. For $p>2>q>1$ choose

$d(z)=\frac{|z|^{p}}{p},\qquad s(z)=\frac{|z-w|^{q}}{q},$

so that a direct computation shows (cf. [23, Example 3.6])

$\displaystyle\lambda_{\rm min}(D^{2}d(z))$ $\displaystyle=|z|^{p-2},$
$\displaystyle\lambda_{\rm max}(D^{2}d(z))$ $\displaystyle=(p-1)\,|z|^{p-2},$
(2.10) $\displaystyle\lambda_{\rm min}(D^{2}s(z))$
$\displaystyle=(q-1)\,|z|^{q-2},$ $\displaystyle\lambda_{\rm max}(D^{2}s(z))$
$\displaystyle=|z|^{q-2}.$ (2.11)

Choose $\varphi\in C^{\infty}(\mathbb{R}^{N}\setminus\\{w\\})$ such that

$\varphi(z)=\begin{cases}s(z)&\text{if $|z-w|\leq r/4$}\\\ 1&\text{if
$|z-w|\geq r/2$}\end{cases}$

and consider the (positive and finite) numbers

$\displaystyle\alpha$ $\displaystyle=\inf\left\\{\lambda_{\rm
min}(D^{2}d(z)):|z|\geq r/2\right\\},$ $\displaystyle\beta$
$\displaystyle=\sup\left\\{|D^{2}\varphi(z)|_{2}:|z-w|\geq r/4\right\\},$
$\displaystyle\gamma$ $\displaystyle=\sup\left\\{\lambda_{\rm
max}(D^{2}d(z)):|z-w|\leq r/4\right\\},$ $\displaystyle\delta$
$\displaystyle=\inf\left\\{\lambda_{\rm min}(D^{2}s(z)):|z-w|\geq
r/4\right\\}.$

For any $\varepsilon>0$, the function $F(z)=d(z)+\varepsilon\,\varphi(z)$
belongs to $C^{1}(\mathbb{R}^{N})\cap W^{2,N}_{\rm loc}(\mathbb{R}^{N})$ and
furthermore $F\in C^{2}(\mathbb{R}^{N}\setminus\\{w\\})$, so that ${\rm
Sing}_{F}\subseteq\\{w\\}$.

For $z\in B_{r/2}(0)$ we have

$\lambda_{\rm min}(D^{2}F(z))=\lambda_{\rm min}(D^{2}d(z)),\qquad\lambda_{\rm
max}(D^{2}(F(z))=\lambda_{\rm max}(D^{2}d(z)),$

which shows through (2.10) that

$\frac{\lambda_{\rm max}(D^{2}F(z))}{\lambda_{\rm min}(D^{2}F(z))}\leq
p-1\qquad\text{in $B_{r/2}(0)$}$

and $\lambda_{\rm min}(D^{2}F(z))\to 0$ for $z\to 0$. From (2.6) we infer that
for all $z\in B_{r/4}(w)$

$\lambda_{\rm min}(D^{2}F(z))\geq\alpha+\varepsilon\,\lambda_{{\rm
min}}(D^{2}s(z)),\qquad\lambda_{\rm
max}(D^{2}F(z))\leq\gamma+\varepsilon\,\lambda_{{\rm max}}(D^{2}s(z)),$

so that by (2.11) it follows

$\frac{\lambda_{\rm max}(D^{2}F(z))}{\lambda_{\rm
min}(D^{2}F(z))}\leq\max\left\\{\frac{\gamma}{\alpha},\frac{1}{q-1}\right\\}\qquad\text{in
$B_{r/4}(w)$}$

and $\lambda_{\rm min}(D^{2}F(z))\to\infty$ for $z\to w$. Finally, for
$z\notin B_{r/2}(0)\cup B_{r/4}(w)$ we have

$\lambda_{\rm min}(D^{2}F(z))\geq\lambda_{\rm
min}(D^{2}d(z))-\varepsilon\,\beta,\qquad\lambda_{\rm
max}(D^{2}F(z))\leq\lambda_{\rm max}(D^{2}d(z))+\varepsilon\,\beta,$

so that for $\varepsilon\,\beta<\alpha/2$ it holds $\lambda_{\rm
min}(D^{2}F(z))\geq\alpha/2$, while (2.10) yields

$\frac{\lambda_{\rm max}(D^{2}F(z))}{\lambda_{\rm min}(D^{2}F(z))}\leq
3\,(p-1)\qquad\text{in $\mathbb{R}^{N}\setminus\big{(}B_{r/2}(0)\cup
B_{r/4}(w)\big{)}$}.$

All in all, for $\varepsilon<\alpha/(2\,\beta)$, $F$ is q. u. c. with ${\rm
Sing}_{F}=\\{w\\}$, ${\rm Deg}_{F}=\\{0\\}$.

Many properties of q. u. c. functions have been studied in [29, 30], to which
we refer for further details and characterisations. Here we gather the ones
that are needed in the proof of our main result. We start by a converse of
Kovalev’s theorem relating the $\delta$-monotonicity to the quasiconformality
in a quantitative form.

######  Lemma 2.5.

Let $F:\mathbb{R}^{N}\to\mathbb{R}$ be differentiable and not affine. Then,
$F$ is $H$-q. u. c. if and only if $DF$ is $\delta$-monotone, where

$\delta=\frac{2\,\sqrt{H}}{H+1},\qquad\text{or}\qquad
H=\frac{1+\sqrt{1-\delta^{2}}}{1-\sqrt{1-\delta^{2}}},$

and the bounds are sharp.

###### Proof.

The fact that if $DF$ is $\delta$-monotone and non-constant then $F$ is $H$-q.
u. c. has been proved, as mentioned before, in [3, Theorem 1]. To prove the
opposite implication, suppose $F$ is $H$-q. u. c., so that $DF$ is
quasiconformal. By [29, Section 3], the $\delta$-monotonicity of $DF$ is
equivalent to

$\left(D^{2}F(z)\,v,v\right)\geq\delta\,|D^{2}F(z)\,v|\,|v|$ (2.12)

at a. e. points of second order differentiability where (2.5) holds true. Note
that we can assume that $D^{2}F$ is symmetric, as this follows from
Alexandrov’s theorem (see [35, Corollary 2.9]), and that it is strictly
positive definite, thanks to the quasiconformality of $DF$. Since (2.12) is
invariant by orthogonal change of variables, we can assume that $D^{2}F$ is
diagonal with positive eigenvalues
$\lambda_{1}<\lambda_{2}<\dots<\lambda_{N}$. We are therefore reduced to find,
for $v=(v_{1},\dots,v_{N})$, the value of

$I:=\inf_{v\neq
0}\frac{\sum_{i}\lambda_{i}\,v_{i}^{2}}{\Big{(}{\sum_{i}\lambda_{i}^{2}\,v_{i}^{2}}\Big{)}^{1/2}\Big{(}{\sum_{i}v_{i}^{2}}\Big{)}^{1/2}}.$

In order to determine the latter we invoke Cassels inequality (see [40,
Appendix]), which reads as

$\frac{\sum_{i}w_{i}\,a^{2}_{i}\sum_{i}w_{i}\,b_{i}^{2}}{\left(\sum_{i}w_{i}\,a_{i}\,b_{i}\right)^{2}}\leq\frac{(M+m)^{2}}{4\,M\,m}$

for any choice of $w_{i}\geq 0$ not all identically equal to zero and
$a_{i},b_{i}>0$ such that

$0<m<a_{i}/b_{i}<M.$

Indeed, it suffices to choose $b_{i}=1$, $a_{i}=\lambda_{i}$ and
$w_{i}=v_{i}^{2}$ to obtain that

$I\geq\frac{2\,\sqrt{\lambda_{1}\,\lambda_{N}}}{\lambda_{1}+\lambda_{N}}$

and actually the equality holds true for the vector

$\left(\sqrt{\frac{\lambda_{N}}{\lambda_{1}+\lambda_{N}}},0,\dots,0,\sqrt{\frac{\lambda_{1}}{\lambda_{1}+\lambda_{N}}}\right).$

Since $\lambda_{N}\leq H\,\lambda_{1}$ by assumption and the map $t\mapsto
2\sqrt{t}/(t+1)$ is decreasing, we infer that

$I\geq\frac{2\,\sqrt{\lambda_{1}\,\lambda_{N}}}{\lambda_{1}+\lambda_{N}}=\frac{2\,\sqrt{\lambda_{N}/\lambda_{1}}}{\lambda_{N}/\lambda_{1}+1}\geq\frac{2\,\sqrt{H}}{H+1}$

so that (2.12) is proved for $\delta=2\,\sqrt{H}/(H+1)$, as claimed. The
optimality of this estimate follows by choosing $F(z)=(A\,z,z)$ with $A$ given
in (2.4) (here we use the fact that $A$ is symmetric). ∎

For further reference, we regroup the previous discussion and some other
useful properties of $H$-q. u. c. functions in the following proposition.

######  Proposition 2.6.

Let $F:\mathbb{R}^{N}\to\mathbb{R}$ be a $H$-q. u. c. function. The following
holds:

  1. (1)

$F$ is strictly convex, coercive and $W^{2,N}_{\rm loc}(\mathbb{R}^{N})$.

  2. (2)

Both $DF$ and $DF^{-1}$ are $\eta_{H}$-quasisymmetric and
$2\sqrt{H}/(H+1)$-monotone.

  3. (3)

There exists $C=C(H,N)>0$ such that, if $F(\bar{z})=\min_{\mathbb{R}^{N}}F$,
then

$\displaystyle|DF(z)-DF(\bar{z})|\,|z-\bar{z}|\leq C\,|F(z)-F(\bar{z})|$
(2.13) $\displaystyle\frac{F(w)-F(\bar{z})}{F(z)-F(\bar{z})}\leq
C\,\frac{|w-\bar{z}|}{|z-\bar{z}|}\,\eta_{H}\left(\frac{|w-\bar{z}|}{|z-\bar{z}|}\right)$
(2.14)

for all $z,w\in\mathbb{R}^{N}\setminus\\{\bar{z}\\}$.

  4. (4)

For any $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N};[0,+\infty))$ the function
$F*\varphi(z)+\mu\,|z|^{2}$ is $H$-q. u. c. Moreover, if $\|\varphi\|_{1}=1$,
for all sequences $(\varepsilon_{n})$, $(\mu_{n})$ such that
$\varepsilon_{n}\downarrow 0$, $\mu_{n}\downarrow 0$, the sequence

$F_{n}(z)=F*\varphi_{\varepsilon_{n}}(z)+\frac{\mu_{n}}{2}\,|z|^{2},\qquad\text{with
}\varphi_{\varepsilon_{n}}(z)=\frac{1}{\varepsilon_{n}^{N}}\,\varphi\left(\frac{z}{\varepsilon_{n}}\right),$
(2.15)

is such that $F_{n}\to F$ in $C^{1}_{\rm loc}(\mathbb{R}^{N})$ and
$DF_{n}^{-1}\to DF^{-1}$ in $C^{0}_{\rm loc}(\mathbb{R}^{N})$.

  5. (5)

For any $\delta>0$, the Moreau-Yoshida reguarization $F_{\delta}$ of $F$,
defined as

$F_{\delta}(z)=\inf_{w\in\mathbb{R}^{N}}\Big{\\{}F(w)+\frac{1}{2\,\delta}|w-z|^{2}\Big{\\}},$

is $H$-q. u. c. and it holds $\displaystyle\lambda_{\rm
max}(D^{2}F_{\delta}(z))\leq 1/\delta$.

###### Proof.

The first two properties have already been discussed, so we focus on (3). We
will prove it by exploiting the $\delta$-monotonicity and the quasisymmetry of
$DF$. To this aim we note that, by considering $F(z+\bar{z})-F(\bar{z})$, we
can assume $\bar{z}=0$ and $F(z)\geq F(0)=0$. To prove (2.13), we start by
noticing that the $\delta$-monotonicity of $DF$ gives

$F(z)=\int_{0}^{1}\left(DF(t\,z),z),z\right)\,dt\geq\delta\,|z|\,\int_{1/2}^{1}|DF(t\,z)|\,dt.$
(2.16)

On the other hand, inequality (2.3) applied to $DF$ with $z_{0}=0$ (and since
$DF(0)=0$, too) gives

$\frac{|DF(z)|}{|DF(t\,z)|}\leq C\,\eta_{H}(1/t)\leq
C\,2^{H}\qquad\forall\,t\geq 1/2,$

so that (2.16) can be estimated as

$F(z)\geq\frac{\delta}{C\,2^{H+1}}\,|DF(z)|\,|z|.$

Inequality (2.14) is proven similarly. It holds

$F(w)\leq\int_{0}^{1}|DF(t\,w)|\,|w|\,dt\leq|w|\,\sup\left\\{|DF(x)|:|x|\leq|w|\right\\}$

which combined with (2.13) yields

$\frac{F(w)}{F(z)}\leq
C\,\frac{|w|}{|z|}\,\sup\left\\{\frac{|DF(x)|}{|DF(z)|}:|x|\leq|w|\right\\}.$

Thanks to the quasisymmetry (2.3) of $DF$ we get

$\frac{|DF(x)|}{|DF(z)|}\leq C\,\eta_{H}\left(\frac{|x|}{|z|}\right)\leq
C\,\eta_{H}\left(\frac{|w|}{|z|}\right)$

whenever $|x|\leq|w|$, and thus (2.14) follows.

The first part of assertion (4) has been proved in [23, Proposition 2.3],
while the $C^{1}_{\rm loc}(\mathbb{R}^{N})$ convergence of $F_{n}$ to $F$ is
trivial. Since $DF_{n}$ are homeomorphisms which are locally uniformly
converging to $DF$, the same is true for $DF^{-1}$ thanks to Arens’ theorem
(see [17]). Finally, part (5) has been proved in [23, Proposition 2.3-(iv)],
where it in particular it is shown that if $P_{\delta}=\big{(}{\rm
Id}+\delta\,DF^{-1}\big{)}^{-1}$, for a. e. $z\in\mathbb{R}^{N}$ it holds

$\begin{split}\lambda_{\rm min}(D^{2}F_{\delta}(z))&=\frac{\lambda_{\rm
min}\big{(}D^{2}F(P_{\delta}(z))\big{)}}{1+\delta\,\lambda_{\rm
min}\big{(}D^{2}F(P_{\delta}(z))\big{)}},\\\ \lambda_{\rm
max}(D^{2}F_{\delta}(z))&=\frac{\lambda_{\rm
max}\big{(}D^{2}F(P_{\delta}(z))\big{)}}{1+\delta\,\lambda_{\rm
max}\big{(}D^{2}F(P_{\delta}(z))\big{)}}.\end{split}$

Since

$\sup_{t\geq 0}\frac{t}{1+\delta\,t}=\frac{1}{\delta},$

the claimed bound follows. ∎

Recall that by a smooth strongly elliptic integrand we mean an $F\in
C^{\infty}(\mathbb{R}^{N})$ such that

$\lambda_{\rm min}(D^{2}F(z))\geq\lambda,\qquad\lambda_{\rm
max}(D^{2}F(z))\leq\Lambda$

for some $0<\lambda\leq\Lambda<\infty$. Clearly, strongly elliptic integrands
are $H$-q. u. c. with $H=\Lambda/\lambda$.

######  Corollary 2.7.

Let $H\geq 1$. The cone

${\rm C}_{H}=\\{F:\mathbb{R}^{N}\to\mathbb{R}\text{ s.\,t.\,$F$ is
$H$-q.\,u.\,c.\,or affine}\\}$

is closed with respect to point-wise a. e. convergence and the smooth,
strongly elliptic integrands in ${\rm C}_{H}$ are dense in it. Moreover,
point-wise a. e. convergence of $(F_{n})_{n}$ in ${\rm C}_{H}$ implies
$C^{1}_{\rm loc}(\mathbb{R}^{N})$ convergence of $(F_{n})_{n}$ and $L^{1}_{\rm
loc}(\mathbb{R}^{N})$ convergence of $({\rm det}\,D^{2}F_{n})_{n}$.

###### Proof.

By [30, Lemma 2.5] we know that the point-wise limit of q. u. c. functions is
either q. u. c. or affine and that the point-wise a. e. convergence implies
the $C^{1}_{\rm loc}(\mathbb{R}^{N})$ one. Therefore it suffices to show that

$F\mapsto\operatorname*{ess\leavevmode\nobreak\
sup}_{z\in\mathbb{R}^{N}}\frac{\lambda_{\rm max}(D^{2}F(z))}{\lambda_{\rm
min}(D^{2}F(z))}$

is lower semicontinuous with respect to $C^{1}_{\rm loc}(\mathbb{R}^{N})$
convergence on the cone of q. u. c. functions. Let $F_{n}\in{\rm C}_{H}$ be
such that $F_{n}\to F$ in $C^{1}_{\rm loc}(\mathbb{R}^{N})$ with none of the
$F_{n}$ affine. By Lemma 2.5, $DF_{n}$ is $2\,\sqrt{H}/(H+1)$-monotone hence,
by passing to the limit in the definition od $\delta$-monotonicity, so is
$DF$. By Lemma 2.5 again, this implies that $F$ is $H$-q. u. c., giving the
claimed lower semicontinuity.

To prove the density statement, let $F$ be $H$-q. u. c. for some $H$. For
$\delta_{n},\varepsilon_{n},\mu_{n}\downarrow 0$, the functions

$F_{n}(z)=F_{\delta_{n}}*\varphi_{\varepsilon_{n}}(z)+\frac{\mu_{n}}{2}|z|^{2}$

(here $F_{\delta_{n}}$ denotes the Moreau-Yoshida regularisation of $F$)
constructed through Proposition 2.6, points (4) and (5), are $H$-q. u. c.
integrands that approximate $F$ point-wise. Using (2.6) together with basic
properties of convolution and Proposition 2.6-(5), yields

$\lambda_{\rm
max}(D^{2}F_{n}(z))\leq\frac{1}{\delta_{n}}+\mu_{n},\qquad\lambda_{\rm
min}(D^{2}F_{n}(z))\geq\mu_{n}$

for all $z\in\mathbb{R}^{N}$, hence $F_{n}$ are strongly elliptic integrands.

If instead $F$ is an affine function of the form $F(z)=(w,z)+c$ for fixed
$w\in\mathbb{R}^{N}$ and $c\in\mathbb{R}$, we can consider $G(z)=|z|^{2}/2$
and set

$F_{n}(z)=n\big{(}G(w+z/n)-G(w)\big{)}+c,\qquad DF_{n}(z)=DG(w+z/n),\qquad
D^{2}F_{n}=\frac{{\rm Id}}{n}.$

Clearly $F_{n}\to\ell$ in $C^{1}_{\rm loc}(\mathbb{R}^{N})$ and the $F_{n}$
are $1$-q. u. c. and strongly elliptic. The last stated property on the
$L^{1}$-convergence of the determinants is contained in [30, Lemma 2.5]. ∎

Actually (see again [30, Lemma 2.5]), the pointwise convergence in the
previous corollary can be weakened to hold on a dense subset of
$\mathbb{R}^{N}$.

######  Remark 2.8.

The previous corollary, as outlined in its proof, implies the lower
semicontinuity of the maximal linear dilatation of monotone, quasiconformal
gradient maps. Lower semicontinuity of the maximal linear dilatation in the
larger class of quasiconformal maps holds true for $N=2$, but fails for $N\geq
3$, due to a famous counterexample of Iwaniec [26].

We conclude this section by introducing a generalisation of the function
$\eta_{H}$ given in (2.2), which will appear in many subsequent computations.
Given $a,b>0$, the auxiliary functions

$\eta_{a,b}(t)=\max\bigl{\\{}t^{a},t^{b}\bigr{\\}},\qquad t\geq 0,$

are increasing homeomorphisms of $\mathbb{R}_{+}$ to itself, and it holds
$\eta_{H}=\eta_{H,1/H}$. Since

$\max\bigl{\\{}t^{a},t^{b}\bigr{\\}}=s\quad\Longleftrightarrow\quad
t=\min\bigl{\\{}s^{1/a},s^{1/b}\bigr{\\}},\qquad\forall\,a,b,t,s>0,$

we see that

$\eta_{a,b}^{-1}(t)=\min\bigl{\\{}t^{1/a},t^{1/b}\bigr{\\}}.$ (2.17)

We collect in the following proposition some elementary properties of the
function $\eta_{a,b}$ and of its inverse.

######  Proposition 2.9.

Let $a,b>0$. Then,

  1. (1)

For all $s,t\geq 0$

$\eta_{a,b}(s\,t)\leq\eta_{a,b}(s)\,\eta_{a,b}(t)\quad\text{as well
as}\quad\eta_{a,b}^{-1}(s\,t)\geq\eta_{a,b}^{-1}(s)\,\eta_{a,b}^{-1}(t)$
(2.18)

  2. (2)

For all $t>0$

$\eta_{a,b}^{-1}(t)\,\eta_{1/a,1/b}(1/t)=1.$ (2.19)

  3. (3)

For all $t>0$ and $\sigma>0$ it holds

$\frac{1}{C}\,\eta_{a,b}(t)\leq\eta_{a,b}(\sigma\,t)\leq C\,\eta_{a,b}(t),$
(2.20)

for positive constants $C=C(a,b,\sigma)$. The same estimate holds true for
$\eta_{a,b}^{-1}$ as well.

  4. (4)

For $a>b>0$ and $c>d>0$

$\eta_{a,b}\circ\eta_{c,d}=\eta_{ac,bd}$ (2.21)

and the same formula holds true for the inverses.

The proof of these facts is elementary and is therefore omitted.

##  3\. Local minimisers and Sobolev regularity of their stress field

######  Definition 3.1.

Let $\Omega\subset\mathbb{R}^{N}$ be open. For all $\mathcal{O}\Subset\Omega$
let us consider the functionals $J\colon W^{1,1}({\mathcal{O}})\to\mathbb{R}$
given by

$J(w,{\mathcal{O}})=\int_{\mathcal{O}}F(Dw)\;dx.$ (3.1)

We say that a function $u\in W^{1,1}_{\rm loc}(\Omega)$ is a local minimiser
of $J$ in $\Omega$ if $F(Du)\in L^{1}_{\rm loc}(\Omega)$ and for any open
${\mathcal{O}}\Subset\Omega$ it holds

$J(u,{\mathcal{O}})=\inf\left\\{J(w,{\mathcal{O}}):\,w\in
u+W^{1,1}_{0}({\mathcal{O}})\right\\}.$

If $F$ is $C^{1}$, strictly convex and coercive, let $\\{\bar{z}\\}={\rm
Argmin}\,(F)$. It is readily checked that any local minimiser $u$ of $J$
defines a local minimiser $\tilde{u}(x)=u(x)-(\bar{z},x)$ for the functional
$\tilde{J}$ given as in (3.1), but with $\tilde{F}(z)$ defined as in (2.9). In
particular, we can assume that (2.7) holds true, i. e. that $\bar{z}=0$. If
$F$ is q. u. c., which will be assumed henceforth, using (2.14) for
$\bar{z}=0$ gives

$F(z)\geq\frac{1}{C}\,F(w)\,|z|\,\eta^{-1}_{H}(|z|)\qquad\forall\,|w|=1,$

so that for all $|z|>1$ it holds

$F(z)\geq\frac{1}{C}\sup_{|w|=1}F(w)\,|z|^{1+\frac{1}{H}}.$ (3.2)

In particular, any local minimiser for $J$ in $\Omega$ belongs to
$W^{1,1+\frac{1}{H}}_{\rm loc}(\Omega)$.

More precisely, despite the fact that q. u. c. integrands are per se
anisotropic, they enjoy, together with their derivative, an isotropic growth
control allowing to treat minimisation problems for $J$ in standard Orlicz-
Sobolev spaces. Indeed, after normalising $F$ so that (2.7) holds true, we can
let for $t\geq 0$

$a(t)=\sup_{|z|\leq t}|DF(z)|,\qquad A(t)=\int_{0}^{t}a(\tau)\,d\tau.$

Notice that $a$ is continuous on $[0,\infty[$ and positive for $t>0$. Given
$0\leq t_{1}<t_{2}$, since $DF$ is a homeomorphism of $\mathbb{R}^{N}$,
$DF(B_{t_{2}})$ is an open neighbourhood of the closed set
$DF(\overline{B_{t_{1}}})$, thus $a(t_{2})>a(t_{1})$, i. e. $a$ is strictly
increasing. By the quasisymmetry of $DF$, choosing $z_{0}=0$ and $|w|=t$ in
(2.3), we obtain that for any $z\in\mathbb{R}^{N}$ it holds

$|DF(z)|\leq C\,a(t)\,\eta_{H}(|z|/t),$

so that for any $k\geq 0$, $t\geq 0$, the monotonicity of $\eta_{H}$ grants

$a(k\,t)\leq C\,\eta_{H}(k)\,a(t).$ (3.3)

Therefore $A$ is a Young function, i. e. it is convex, increasing and fulfils

$\lim_{t\to
0^{+}}\frac{A(t)}{t}=0,\qquad\lim_{t\to\infty}\frac{A(t)}{t}=\infty,$

where the second equality follows from (3.3) for $k=1/t$ and using
$\eta_{H}(1/t)^{-1}=t^{1/H}$ for $t\geq 1$. The inverse of $a$ is readily
checked to be

$a^{-1}(s)=\inf_{|z|\geq s}|DF^{-1}(z)|$

which obeys a similar estimate as (3.3). It follows that $A$ satisfies the
$\Delta_{2}$ condition

$A(2\,t)\leq C\,A(t)\qquad\forall\,t\geq 0$

for $C=C(H,N)$, and the same holds true for its Young conjugate defined as

$A^{*}(s)=\sup_{t\geq
0}\left(s\,t-A(t)\right)=\int_{0}^{s}a^{-1}(\sigma)\,d\sigma$

(see the proof of [4, eq. (16.94)-(16.95)]).

Clearly

$F(z)=\int_{0}^{|z|}(DF(s\,z/|z|),z/|z|)\,ds\leq\int_{0}^{|z|}a(s)\,ds=A(|z|)$

while the $\delta$-monotonicity and quasisymmetry of $DF$ give

$F(z)\geq\delta\,\int_{0}^{|z|}|DF(s\,z/|z|)|\,ds\geq\frac{\delta}{C}\,\int_{0}^{|z|}a(s)\,ds,$

so that

$\frac{1}{C}\,A(|z|)\leq F(z)\leq C\,A(|z|).$ (3.4)

The map $DF$ enjoys similar isotropic bounds, namely,

$\frac{1}{C}\,a(|z|)\leq|DF(z)|\leq a(|z|),$

where the first inequality follows from (2.3) as before, and it is furthermore
possible to show that

$\frac{1}{C}\,A(|z|)\leq A^{*}(DF(z))\leq C\,A(|z|)$ (3.5)

(see the proof of [4, eq. (16.106)]).

Given an open set ${\mathcal{O}}\Subset\Omega$, it follows from (3.4) that

$J(u,{\mathcal{O}})<\infty\quad\text{if and only if}\quad A(|Du|)\in
L^{1}({\mathcal{O}}).$

Using the fact that ${\mathcal{O}}$ has finite measure and $A$ satisfies the
$\Delta_{2}$ condition, the summability of $A(|Du|)$ can be equivalently
stated as

$u\in W^{1,A}({\mathcal{O}}):=\left\\{v\in
W^{1,1}({\mathcal{O}}):A(|v|),A(|Dv|)\in L^{1}({\mathcal{O}})\right\\},$

equipped with the Luxembourg norm

$\|v\|_{L^{A}({\mathcal{O}})}+\||Dv|\|_{L^{A}({\mathcal{O}})},\qquad\|g\|_{L^{A}({\mathcal{O}})}=\inf\left\\{s>0:\int_{\mathcal{O}}A(|g|/s)\,dx\leq
1\right\\}.$

Thanks to the $\Delta_{2}$ condition on $A$, the so-called Orlicz-Sobolev
space $W^{1,A}({\mathcal{O}})$ so defined turns out to be a Banach space.
Existence of local minimiser given, say, a boundary datum $\varphi\in
W^{1,A}(\Omega)$ for $\Omega$ bounded is thus granted by minimising $J$ on
$u+W^{1,A}_{0}(\Omega)$, the latter space being the closed vector space of
those $v\in W^{1,A}(\Omega)$ whose extension at zero outside $\Omega$ belongs
to $W^{1,1}(\mathbb{R}^{N})$. Indeed, thanks to the validity of the
$\Delta_{2}$ condition on both $A$ and $A^{*}$, the Banach space
$W^{1,A}_{0}(\Omega)$ is reflexive by [19, p. 54], while $J$ is convex and
coercive by (3.4). Conversely, a local minimiser $u$ on $\Omega$ minimises
$J(\cdot\,,{\mathcal{O}})$ on $u+W^{1,A}_{0}({\mathcal{O}})$ for any
${\mathcal{O}}\Subset\Omega$ and thus standard methods (see [16, Theorem 2.1])
ensure the validity of the Euler-Lagrange equation

$\int_{\mathcal{O}}(DF(Du),Dw)\,dx=0\qquad\forall\,w\in
W^{1,A}_{0}({\mathcal{O}}).$ (3.6)

Notice that for any $u\in W^{1,A}({\mathcal{O}})$ the inequalities in (3.5)
imply that $|DF(Du)|\in L^{A^{*}}({\mathcal{O}})$ (thus a fortiori
$|DF(Du)|\in L^{1}({\mathcal{O}})$) and therefore Hölder’s inequality in
Orlicz spaces ensures that

$W^{1,A}_{0}({\mathcal{O}})\ni w\mapsto\int_{\mathcal{O}}(DF(Du),Dw)\,dx$

is a well defined continuous linear functional. Thus, if for
${\mathcal{O}}\Subset\Omega$ a function $u\in W^{1,A}({\mathcal{O}})$ fulfils

${\rm div}\,(DF(Du))=0$

in the distributional sense, it does so also in the weak sense (3.6), thanks
to the density of $C^{\infty}_{0}({\mathcal{O}})$ in
$W^{1,A}_{0}({\mathcal{O}})$ granted by [19, Theorem 2.1] (here the
$\Delta_{2}$ condition on $A$ alone is sufficient).

We summarise the previous discussion in the following proposition.

######  Proposition 3.2.

Let $N\geq 2$ and $F$ be a q. u. c. integrand. A function $u\in W^{1,1}_{\rm
loc}(\Omega)$ is a local minimiser for $J$ in (3.1) if and only if $F(Du)\in
L^{1}_{\rm loc}(\Omega)$ and

${\rm div}\,(DF(Du))=0$

in the distributional sense.

The next proposition is essentially contained in [23, Theorem 3.3], but we
will need the more explicit estimates given in its present form. It revolves
around the Sobolev regularity of the stress field

$V(x)=DF(Du(x)).$ (3.7)

######  Proposition 3.3.

Let $F$ be a $H$-q. u. c. function obeying (2.7) and let $i_{F}$ be as in
(2.8). Let $u$ be a local minimiser for $J$, and let $V$ be given by (3.7).
Then, $V\in W^{1,2}_{\rm loc}(\Omega)$ with estimates

$\displaystyle\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{R}}|DV|_{2}^{2}\,dx\right)^{\frac{1}{2}}$
$\displaystyle\leq\frac{C}{R}\,i_{F}\,\eta_{\frac{H}{H+1},\frac{1}{H+1}}\left(\frac{1}{i_{F}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx\right),$ (3.8a)
$\displaystyle\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{R}}|V|^{2}\,dx\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\,i_{F}\,\eta_{\frac{H}{H+1},\frac{1}{H+1}}\left(\frac{1}{i_{F}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx\right),$ (3.8b)

for a positive constant $C=C(N,H)$ and all balls $B_{R}$ such that
$B_{2R}\subseteq\Omega$.

###### Proof.

From [23, Theorem 3.3, (3.10)] we know that

$\|V\|_{W^{1,2}(B_{R})}\leq C\|V\|_{L^{1}(B_{2R})}$

for a constant $C=C(N,H,R)$. More precisely, setting

$m_{r}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}}|V|\,dx,$ (3.9)

it holds (see [23, proof of Theorem 3.2] or Corollary 5.3 below)

$\|DV\|_{L^{2}(B_{R})}\leq C(N,H)\,R^{\frac{N}{2}-1}\,m_{2R}$ (3.10)

for a constant $C$ which henceforth depends only on $H$ and $N$. On the other
hand, by Poincaré inequality and this last estimate we have

$\begin{split}\|V\|_{L^{2}(B_{R})}&\leq\|V-m_{R}\|_{L^{2}(B_{R})}+(\omega_{N}\,R^{N})^{\frac{1}{2}}m_{R}\\\
&\leq C\,R\,\|DV\|_{L^{2}(B_{R})}+C\,R^{\frac{N}{2}}\,m_{2R}\\\ &\leq
C\,R^{\frac{N}{2}}\,m_{2R}.\end{split}$ (3.11)

We claim that

$m_{2R}\leq
C\,i_{F}\,\eta_{\frac{H}{H+1},\frac{1}{H+1}}\left(\frac{1}{i_{F}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx\right)$ (3.12)

for a constant $C=C(N,H)$. To this aim, for any $t>0$ we proceed as follows

$\begin{split}\int_{B_{2R}}|DF(Du)|\,dx&=\int_{\\{|Du|\leq t\\}\cap
B_{2R}}|DF(Du)|\,dx+\int_{\\{|Du|>t\\}\cap B_{2R}}|DF(Du)|\,dx\\\ &\leq
C\,\sup_{|z|\leq
t}|DF(z)|\,R^{N}+\frac{1}{t}\int_{B_{2R}}|DF(Du)|\,|Du|\,dx.\end{split}$
(3.13)

The first term on the above right-hand side is estimated by using the
quasisymmetry of $DF$ (2.1) as

$|DF(z)|\leq C\,\inf_{|w|=1}|DF(w)|\,\eta_{H}(|z|),$

while on the second term we use (2.13). Therefore (3.13) reduces to

$\int_{B_{2R}}|DF(Du)|\,dx\leq
C\,R^{N}\,i_{F}\,\left(\eta_{H}(t)+\frac{1}{t}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}\frac{F(Du)}{i_{F}}\,dx\right).$

Choose $t$ such that

$t\,\eta_{H}(t)=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}\frac{F(Du)}{i_{F}}\,dx\quad\Longleftrightarrow\quad
t=\eta_{1+\frac{1}{H},1+H}^{-1}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}\frac{F(Du)}{i_{F}}\,dx\right)$

to get through (2.19)

$\int_{B_{2R}}|DF(Du)|\,dx\leq
C\,R^{N}\,i_{F}\,\eta_{\frac{1}{H+1},\frac{H}{H+1}}\left(\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}\frac{F(Du)}{i_{F}}\,dx\right)^{-1}\right)\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}\frac{F(Du)}{i_{F}}\,dx.\\\ $

Finally, notice that

$t\,\eta_{\frac{1}{H+1},\frac{H}{H+1}}(1/t)=\eta_{\frac{1}{H+1},\frac{H}{H+1}}(t)$

for all $t\geq 0$, so that taking (3.9) and (3.7) into account implies the
validity of (3.12). Recalling (3.10) and (3.11) finally gives (3.8a) and
(3.8b), respectively. ∎

##  4\. A regularising procedure

As customary, it will be convenient to work under the assumption that both the
integrand and the minimiser under scrutiny are smooth. In order to do so, we
introduce a family of regularised problems having the required smoothness and
whose integrands and corresponding minimisers converge to the original
problem.

######  Lemma 4.1 (Regularised problems).

Let $F$ be a $H$-q. u. c. function fulfilling (2.7) and let $u$ be a local
minimiser for $J$ in $W^{1,1}(\Omega)$. Then, for any ball
$B_{2R}\Subset\Omega$ there exists a sequence $F_{n}\in
C^{\infty}(\mathbb{R}^{N})$ of $H$-q. u. c. integrands obeying (2.7) and

$\inf_{z\in\mathbb{R}^{N}}\lambda_{\rm min}(D^{2}F_{n}(z))>0$

and a sequence of corresponding minimisers $u_{n}\in C^{\infty}(B_{2R})$ of

$J_{n}(w,B_{2R})=\int_{B_{2R}}F_{n}(Dw)\,dx$

such that

$\displaystyle\int_{B_{2R}}F_{n}(Du_{n})\,dx$
$\displaystyle\to\int_{B_{2R}}F(Du)\,dx,$ (4.1a) $\displaystyle u_{n}$
$\displaystyle\rightharpoonup u\quad\text{in $W^{1,1+1/H}(B_{R})$},$ (4.1b)
$\displaystyle DF_{n}(Du_{n})$ $\displaystyle\rightharpoonup
DF(Du)\quad\text{in $W^{1,2}(B_{R})$}.$ (4.1c)

###### Proof.

We set for brevity $B=B_{2R}$ and $p=1+1/H$. Fix an even $\varphi\in
C^{\infty}_{c}(B_{1};[0,+\infty[)$ such that $\|\varphi\|_{1}=1$ and use the
notation of Proposition 2.6-(4). Notice that for any such $\varphi$ it holds

$z=\int(z-y)\,\varphi(y)\,dy$

so that Jensen inequality gives

$F(z)\leq(\varphi\ast F)(z).$ (4.2)

Set

$F_{n}(z)=(\varphi_{\varepsilon_{n}}*F)(z)+\frac{\mu_{n}}{2}\,|z|^{2},$

with $\varepsilon_{n}\downarrow 0$ and $\mu_{n}\downarrow 0$ to be chosen, so
that $F_{n}$ fulfils for each $n$ all the conditions stated in the lemma.
Notice that thanks to (3.2) and the convergence $F_{n}\to F$ in
$C^{1}_{\text{loc}}(\mathbb{R}^{N})$ we have

$F_{n}(w)\geq\frac{|w|^{p}}{C}\quad\text{as well
as}\quad\frac{1}{C}\leq\inf_{|z|=1}|DF_{n}(z)|\leq C,$ (4.3)

respectively, for constants $C$ independent of $n$.

Let, for sufficiently large $n$,

$\psi_{n}=u*\varphi_{1/n}\in C^{\infty}(B)$

and consider the minimisation problem

$\inf\\{J_{n}(w):w\in{\rm Lip}(\overline{B}),w=\psi_{n}\ \text{on $\partial
B$}\\}.$ (4.4)

According to [36, Theorem 9.2], there is a solution $u_{n}$ of (4.4), which
(see [10, p. 5923] and (3.2)) also solves

$J_{n}(u_{n})=\inf\\{J_{n}(w):w\in\psi_{n}+W^{1,p}_{0}(B)\\}.$

By [8, Theorem 4.1] there are constants $A_{n}\geq 1$ (depending only on $B$
and on the regularity of $\psi_{n}$, but not on $\varepsilon_{n}$, $\mu_{n}$)
such that

${\rm Lip}(u_{n})\leq\frac{A_{n}}{\mu_{n}}.$

The integrand $F$ is therefore strongly elliptic on the range of $Du_{n}$
since $F_{n}\in C^{\infty}(\mathbb{R}^{N})$ and

$\lambda_{\rm min}(D^{2}F_{n}(Du_{n}))\geq\mu_{n},\qquad\lambda_{\rm
max}(D^{2}F_{n}(Du_{n}))\leq C<\infty$

on $\overline{B}$, so that standard regularity theory gives $u_{n}\in
C^{\infty}(\overline{B})$. We first choose $\mu_{n}\downarrow 0$ so that

$\lim_{n}\mu_{n}\int_{B}|D\psi_{n}|^{2}\,dx=0.$ (4.5)

Set

$M_{n}=1+\sup_{B}|D\psi_{n}|+\frac{A_{n}}{\mu_{n}}$

and observe that, since $\varphi_{\varepsilon_{n}}*F\to F$ in $C^{1}_{\rm
loc}(\mathbb{R}^{N})$ as $\varepsilon_{n}\downarrow 0$ and $M_{n}$ is
independent of $\varepsilon_{n}$, we can pick
$(\varepsilon_{n})\subseteq(0,1)$, $\varepsilon_{n}\downarrow 0$, so that

$\|\varphi_{\varepsilon_{n}}*F-F\|_{C^{1}(B_{M_{n}})}\leq\frac{1}{n}.$ (4.6)

Clearly, it still holds $F_{n}\to F$ in $C^{1}_{{\rm loc}}(\mathbb{R}^{N})$.
We claim that (4.1) holds true with such choices.

By the minimality of $u_{n}$, it holds $J_{n}(u_{n})\leq J_{n}(\psi_{n})$,
while from (4.5) we get

$\varlimsup_{n}J_{n}(u_{n})\leq\varlimsup_{n}J_{n}(\psi_{n})=\varlimsup_{n}\int_{B}(\varphi_{\varepsilon_{n}}*F)(D\psi_{n})\,dx.$
(4.7)

In order to estimate the rightmost term of (4.7) we use (4.6) to have

$\int_{B}(\varphi_{\varepsilon_{n}}*F)(D\psi_{n})\,dx\leq\frac{|B|}{n}+\int_{B}F(D\psi_{n})\,dx,$
(4.8)

while the vector-valued Jensen inequality implies

$\begin{split}\int_{B}F(D\psi_{n})\,dx&=\int_{B}F\left(\int\varphi_{1/n}(x-y)\,Du(y)\,dy\right)\,dx\\\
&\leq\int_{B}\left(\int\varphi_{1/n}(x-y)F(Du(y))\,dy\right)dx\\\
&\leq\int_{B}\varphi_{1/n}*F(Du)\,dx.\end{split}$

Therefore from (4.8) we have

$\varlimsup_{n}\int_{B}(\varphi_{\varepsilon_{n}}*F)(D\psi_{n})\,dx\leq\varlimsup_{n}\int_{B}\varphi_{1/n}*F(Du)\,dx=\int_{B}F(Du)\,dx$

which, inserted into (4.7), gives

$\varlimsup_{n}J_{n}(u_{n})\leq J(u).$ (4.9)

In particular, $J_{n}(u_{n})$ is bounded and (4.3) then implies a uniform
bound on $Du_{n}$ in $L^{p}(B)$. Moreover, since $u_{n}-\psi_{n}\in
W^{1,p}_{0}(B)$, then Poincaré’s inequality gives

$\begin{split}\|u_{n}\|_{L^{p}(B)}&\leq\|u_{n}-\psi_{n}\|_{L^{p}(B)}+\|\psi_{n}\|_{L^{p}(B)}\\\
&\leq
C\,\big{(}\|D(u_{n}-\psi_{n})\|_{L^{p}(B)}+\|\psi_{n}\|_{L^{p}(B)}\big{)}\\\
&\leq
C\,\big{(}\|Du_{n}\|_{L^{p}(B)}+\|\psi_{n}\|_{W^{1,p}(B)}\big{)},\end{split}$

so that $(u_{n})$ is bounded in $L^{p}(B)$ as well. Therefore $(u_{n})$
possesses a (not relabeled) subsequence weakly converging in $W^{1,p}(B)$ to
some $v$ and it is readily checked that $v\in u+W^{1,p}_{0}(B)$. This in turn
implies that (4.1b) is satisfied.

Now, since the map

$w\mapsto\int_{B}F(Dw)\,dx$

is weakly lower semicontinuous in $W^{1,p}(B)$ and $F_{n}\geq F$ thanks to
(4.2), we get

$J(v)\leq\varliminf_{n}\int_{B}F(Du_{n})\,dx\leq\varliminf_{n}\int_{B}F_{n}(Du_{n})\,dx=\varliminf_{n}J_{n}(u_{n}).$
(4.10)

Coupling the latter with (4.9) gives $J(v)\leq J(u)$, implying $v=u$ by the
strict convexity of $F$. In particular, up to subsequences,

$Du_{n}\rightharpoonup Du\qquad\text{in $L^{p}(B)$}.$ (4.11)

Therefore, from (4.10) and (4.9) we infer

$\int_{B}F(Du_{n})\,dx\to\int_{B}F(Du)\,dx,$ (4.12)

that is (4.1a). We now apply Proposition 3.3 to $DF_{n}(Du_{n})$, thus
obtaining

$\|DF_{n}(Du_{n})\|_{W^{1,2}(B_{R})}\leq C,$

where the uniform bound holds thanks to equations (4.3) and (4.12), being
$B=B_{2R}$. Setting

$V_{n}=DF_{n}(Du_{n})$ (4.13)

we can pick a subsequence such that $V_{n}\to V$ weakly in $W^{1,2}(B_{R})$,
strongly in $L^{2}(B_{R})$, and pointwise a. e. in $B_{R}$, for a suitable
$V\in W^{1,2}(B_{R})$. Since $DF_{n}^{-1}\to DF^{-1}$ locally uniformly thanks
to Proposition 2.6-(4), we infer that $DF_{n}^{-1}(V_{n})\to DF^{-1}(V)$ a.
e.. Then (4.13) implies that $Du_{n}\to DF^{-1}(V)$ a. e., and in turn (4.11)
allows the identification $V=DF(Du)$. This shows (4.1c) and the proof is thus
complete. ∎

##  5\. Caccioppoli inequality

In this section we prove a Caccioppoli-type inequality. To this aim, let us
first show the following result, which is a simplification of [23, Lemma 3.1].

######  Lemma 5.1.

Let $P$ and $S$ be $N\times N$ symmetric matrices, with $P$ positive definite.
Then,

$\left(P\,S,S\,P\right)_{2}\geq\frac{\lambda_{\rm min}(P)}{\lambda_{\rm
max}(P)}|P\,S|_{2}^{2}.$ (5.1)

###### Proof.

Let us observe that both sides of (5.1) are invariant by orthogonal change of
basis, so we can assume that $P$ is diagonal, with positive eigenvalues
$\lambda_{1},\dots,\lambda_{N}$ obeying

$\lambda_{j}\geq\frac{1}{\beta}\,\lambda_{i},\quad\text{where
}\beta=\frac{\lambda_{\rm max}(P)}{\lambda_{\rm min}(P)},$

for any $i,j=1,\dots,N$. Then, if $S=(a_{ij})$, we have

$\left(P\,S,S\,P\right)_{2}=\sum_{i,j=1}^{N}\lambda_{i}\,a_{ij}\,\lambda_{j}\,a_{ji}=\sum_{i,j=1}^{N}\lambda_{i}\,\lambda_{j}\,a_{ij}^{2}\geq\frac{1}{\beta}\sum_{i,j=1}^{N}\lambda_{i}^{2}\,a_{ij}^{2}=\frac{1}{\beta}\,|P\,S|_{2}^{2},$

as claimed. ∎

######  Lemma 5.2 (Caccioppoli Inequality).

Let $F$ be a smooth $H$-q. u. c. function and let $u\in C^{\infty}(B_{R})$.
For $V$ given by (3.7), assume that

${\rm div}\,V=0\qquad\text{in $B_{R}$}.$ (5.2)

Moreover, for a given affine function $\ell:\mathbb{R}^{N}\to\mathbb{R}$, and
all $\rho<R$, set

$A(\ell,\rho)=\left\\{x\in B_{\rho}:F(Du(x))\geq\ell(Du(x))\right\\}.$ (5.3)

Then, for all $\rho<R$, it holds

$\int_{A(\ell,\rho)}|DV|_{2}^{2}\,dx\leq\left(\frac{\pi\,H}{R-\rho}\right)^{2}\,\int_{A(\ell,R)}|V-D\ell|^{2}\,dx.$
(5.4)

###### Proof.

Let $\varepsilon>0$ and set
$V_{\varepsilon}=V+\varepsilon\,Du=DF(Du)+\varepsilon\,Du$. Differentiating
the latter gives

$DV_{\varepsilon}=\big{(}D^{2}F(Du)+\varepsilon\,{\rm Id}\big{)}\,D^{2}u,$

where the matrices $D^{2}F(Du)+\varepsilon\,{\rm Id}$ and $D^{2}u$ are
symmetric positive definite and symmetric, respectively. By (2.5), we have

$\begin{split}\lambda_{\rm max}(D^{2}F(Du)+\varepsilon\,{\rm
Id})&\leq\lambda_{\rm max}(D^{2}F(Du))+\varepsilon\\\ &\leq H\,\lambda_{\rm
min}(D^{2}F(Du))+\varepsilon\\\ &\leq H\,\lambda_{\rm
min}(D^{2}F(Du)+\varepsilon\,{\rm Id})\end{split}$

hence (5.1) gives $|DV_{\varepsilon}|_{2}^{2}\leq
H\,(DV_{\varepsilon},DV_{\varepsilon}^{t})_{2}$. Letting
$\varepsilon\downarrow 0$ provides

$|DV|_{2}^{2}\leq H\,(DV,DV^{t})_{2}.$

We want to integrate the above inequality over $A(\ell,r)$, for $r<R$,
assuming for the moment that $\\{F(Du)=\ell(Du)\\}$ is a $C^{1}$ hypersurface
in $B_{R}$. Using the divergence theorem gives

$\begin{split}\int_{A(\ell,r)}|DV|_{2}^{2}\,dx&\leq
H\,\int_{A(\ell,r)}\left(DV,DV^{t}\right)_{2}\,dx\\\
&=H\,\int_{A(\ell,r)}\sum_{i,j=1}^{N}V^{i}_{j}\,(V^{j}-\ell_{j})_{i}\,dx\\\
&=H\,\int_{A(\ell,r)}\sum_{i,j=1}^{N}\Big{(}V^{i}_{j}\,(V^{j}-\ell_{j})\Big{)}_{i}-\sum_{j=1}^{N}(V^{j}-\ell_{j})\,\operatorname{div}V_{j}\,dx\\\
&=H\,\int_{\partial
A(\ell,r)}\sum_{i,j=1}^{N}V^{i}_{j}\,(V^{j}-\ell_{j})\,n^{i}\,d{\mathcal{H}}^{N-1},\end{split}$
(5.5)

where $n$ denotes the exterior normal to $\partial A(\ell,r)$ and where we
used the fact that (5.2) implies $\operatorname{div}V_{j}=0$ for all
$j=1,\dots,N$. To compute the last integral in (5.5), notice that

$\partial A(\ell,r)=\bigl{(}\partial
B_{r}\cap\\{F(Du)\geq\ell(Du)\\}\bigr{)}\bigcup\,\bigl{(}B_{r}\cap\\{F(Du)=\ell(Du)\\}\bigr{)},$

so that

$n(x)=\begin{cases}\dfrac{x}{|x|}&\text{if $x\in\partial
B_{r}\cap\\{F(Du)\geq\ell(Du)\\}$}\\\\[15.0pt]
-\dfrac{D\big{(}(F-\ell)(Du(x))\big{)}}{|D\big{(}(F-\ell)(Du(x))\big{)}|}&\text{if
$x\in B_{r}\cap\\{F(Du)=\ell(Du)\\}$}.\end{cases}$

Let $x\in B_{r}\cap\\{F(Du)=\ell(Du)\\}$ and compute

$U^{i}(x):=\big{(}(F-\ell)(Du(x))\big{)}_{i}=\sum_{k=1}^{N}(V^{k}-\ell_{k})\,u_{ki}\quad\text{as
well as}\quad V^{i}_{j}=\sum_{h=1}^{N}F_{ih}(Du)\,u_{hj},$

so that

$\begin{split}\sum_{i,j=1}^{N}V^{i}_{j}\,(V^{j}-\ell_{j})\,n^{i}&=-\frac{1}{|U|}\sum_{i,j,k=1}^{N}V^{i}_{j}\,(V^{j}-\ell_{j})\,(V^{k}-\ell_{k})\,u_{ki}\\\
&=-\frac{1}{|U|}\,\sum_{i,j,k,h=1}^{N}F_{ih}(Du)\,u_{hj}\,(V^{j}-\ell_{j})\,u_{ik}\,(V^{k}-\ell_{k})\\\
&=-\frac{1}{|U|}\,\big{(}D^{2}F(Du)\,U,U\big{)}<0,\end{split}$

being $D^{2}F$non-negative definite. Then the previous inequality implies that
the rightmost integral in (5.5) evaluated on $B_{r}\cap\\{F(Du)=\ell(Du)\\}$
has a negative sign and hence (5.5) reduces to

$\begin{split}\int_{A(\ell,r)}|DV|_{2}^{2}\,dx&\leq H\,\int_{\partial
B_{r}\cap\\{F(Du)\geq\ell(Du)\\}}\sum_{i,j=1}^{N}V^{i}_{j}\,(V^{j}-\ell_{j})\,n^{i}\,d{\mathcal{H}}^{N-1}\\\
&\leq H\,\int_{\partial
B_{r}\cap\\{F(Du)\geq\ell(Du)\\}}|DV|_{2}\,|V-D\ell|\,d{\mathcal{H}}^{N-1},\end{split}$
(5.6)

where in the last step we used the Schwarz inequality.

Let now $\zeta$ be the function given by

$\zeta(r)=\int_{A(\ell,r)}|DV|_{2}\,|V-D\ell|\,dx\quad\text{for all }0<r<R,$

with first derivative

$\zeta^{\prime}(r)=\int_{\partial
B_{r}\cap\\{F(Du)\geq\ell(Du)\\}}|DV|_{2}\,|V-D\ell|\,d{\mathcal{H}}^{N-1}.$
(5.7)

Let $\varphi\in C^{\infty}([0,R])$ with $\varphi(R)=0$. We multiply (5.6) by
$\varphi^{2}$ and integrate in $[0,R]$. Taking (5.7) into account, integrating
by parts and using the fact that $\varphi(R)=\zeta(0)=0$, we get

$\begin{split}\int_{0}^{R}\int_{A(\ell,r)}\varphi^{2}\,|DV|_{2}^{2}\,dx\,dr&\leq
H\,\int_{0}^{R}\varphi^{2}\,\zeta^{\prime}\,dr\\\
&=H\left(\varphi^{2}(R)\,\zeta(R)-\varphi^{2}(0)\,\zeta(0)\right)-2\,H\,\int_{0}^{R}\varphi\,\varphi^{\prime}\,\zeta\,dr\\\
&\leq
2\,H\,\int_{0}^{R}\int_{A(\ell,r)}|\varphi||\varphi^{\prime}||DV|_{2}|V-D\ell|\;dx\,dr\\\
&\leq
2\,H\left({\int_{0}^{R}\int_{A(\ell,r)}\varphi^{2}\,|DV|_{2}^{2}\,dxdr}\right)^{\frac{1}{2}}\left({\int_{0}^{R}\int_{A(\ell,r)}|\varphi^{\prime}|^{2}\,|V-D\ell|^{2}\,dxdr}\right)^{\frac{1}{2}}\end{split}$

which simplifies to

$\int_{0}^{R}\int_{A(\ell,r)}\varphi^{2}\,|DV|_{2}^{2}\,dx\,dr\leq
4\,H^{2}\,\int_{0}^{R}\int_{A(\ell,r)}|\varphi^{\prime}|^{2}\,|V-D\ell|^{2}\,dx\,dr.$

Fix $0<\rho<R$. Taking advantage of the monotonicity of the functions

$r\mapsto\int_{A(\ell,r)}|DV|_{2}^{2}\,dx,\qquad
r\mapsto\int_{A(\ell,r)}|V-D\ell|_{2}^{2}\,dx$

we then have

$\begin{split}\int_{A(\ell,\rho)}|DV|_{2}^{2}\,dx\,\int_{\rho}^{R}\varphi^{2}\,dr&\leq\int_{\rho}^{R}\int_{A(\ell,r)}\varphi^{2}\,|DV|_{2}^{2}\,dx\,dr\\\
&\leq\int_{0}^{R}\int_{A(\ell,r)}\varphi^{2}\,|DV|_{2}^{2}\,dx\,dr\\\ &\leq
4\,H^{2}\,\int_{0}^{R}\int_{A(\ell,r)}|\varphi^{\prime}|^{2}\,|V-D\ell|^{2}\,dx\,dr\\\
&\leq
4\,H^{2}\int_{A(\ell,R)}|V-D\ell|^{2}\,dx\,\int_{0}^{R}|\varphi^{\prime}|^{2}\,dr,\end{split}$

which implies

$\int_{A(\ell,\rho)}|DV|_{2}^{2}\,dx\leq
4\,H^{2}\,I(R,\rho)\,\int_{A(\ell,R)}|V-D\ell|^{2}\,dx$

for

$I(R,\rho):=\inf\left\\{\dfrac{\displaystyle{\int_{0}^{R}|\varphi^{\prime}|^{2}\,dr}}{\displaystyle{\int_{\rho}^{R}\varphi^{2}\,dr}}:\varphi\in
C^{\infty}([0,R])\setminus\\{0\\},\varphi(R)=0\right\\}.$

A minimiser for this problem is found through elementary considerations as

$\varphi(r)=\begin{cases}1&\text{if $r\leq\rho$}\\\\[5.0pt]
\cos\left(\dfrac{\pi}{2}\,\dfrac{r-\rho}{R-\rho}\right)&\text{if
$r\in[\rho,R]$},\end{cases}$

so that an explicit computation gives

$I(R,\rho)=\frac{\pi^{2}}{4\,(R-\rho)^{2}},$

and (5.4) follows.

To remove the assumption that the set $\\{F(Du)=\ell(Du)\\}$ is $C^{1}$ in
$B_{R}$, we invoke Sard’s theorem, which ensures that this property is
satisfied by $\\{F(Du)=\ell(Du)+k\\}$, for a. e. $k\in\mathbb{R}$. Given
$k_{n}\downarrow 0$, we can repeat the same argument as before, starting from
(5.5), thus arriving to

$\int_{A(\ell+k_{n},r)}|DV|_{2}^{2}\;dx\leq\left(\frac{\pi\,H}{R-\rho}\right)^{2}\int_{A(\ell+k_{n},R)}|V-D\ell|^{2}\;dx.$
(5.8)

Notice that $A(\ell+k_{n},\cdot)\searrow A(\ell,\cdot)$, so we can pass to the
limit in (5.8) by monotone (or dominated) convergence, and in turn (5.4) holds
true also in this case. ∎

We report the following consequence, already present in [23, proof of Theorem
3.2], for the sake of completeness.

######  Corollary 5.3.

Let $F$ be a $H$-q. u. c. function, let $u$ be a local minimiser for $J$ in
$\Omega$ in the sense of Definition 3.1 and let $V$ be given by (3.7). Then,

$\int_{B_{R}}|DV|_{2}^{2}\,dx\leq\frac{C}{R^{N+2}}\left(\int_{B_{2R}}|V|\,dx\right)^{2}$
(5.9)

for $C=C(H,N)$ and all balls $B_{R}$ such that $B_{4R}\subseteq\Omega$.

###### Proof.

By using Lemma 4.1 on $B_{2R}$ we can suppose that both $F$ and $u$ are
smooth, since for the regularised problems the left-hand side of (5.9) is
lower semicontinuous with respect to the weak convergence in $W^{1,2}(B_{R})$,
while the right-hand side is (up to subsequences) convergent. We thus apply
Lemma 5.2 for $\ell(z)\equiv\min F$ to get

$\int_{B_{\rho}}|DV|_{2}^{2}\,dx\leq\frac{\widetilde{C}}{(r-\rho)^{2}}\,\int_{B_{r}}|V|^{2}\,dx$
(5.10)

for all $0<\rho<r<2R$. Let $E$ be a continuous extension operator

$E:W^{1,2}(B_{1};\mathbb{R}^{N})\to W^{1,2}_{0}(B_{2};\mathbb{R}^{N}).$

The Gagliardo-Nirenberg inequality [31, Theorem 12.83] ensures that

$\|E(U)\|_{2}\leq
C\,\|E(U)\|_{1}^{\frac{2}{N+2}}\,\|DE(U)\|_{2}^{1-\frac{2}{N+2}}\qquad\forall\,U\in
W^{1,2}(B_{1};\mathbb{R}^{N})$

for $C=C(N)$, so that by the continuity of $E$

$\begin{split}\int_{B_{1}}|U|^{2}\,dx&\leq\int_{B_{2}}|E(U)|^{2}\,dx\leq
C\,\left(\int_{B_{2}}|E(U)|\,dx\right)^{{\frac{4}{N+2}}}\left(\int_{B_{2}}|DE(U)|^{2}\,dx\right)^{{\frac{N}{N+2}}}\\\
&\leq
C\,\left(\int_{B_{1}}|U|\,dx\right)^{\frac{4}{N+2}}\left(\int_{B_{1}}|DU|^{2}\,dx\right)^{{\frac{N}{N+2}}}\end{split}$

for a bigger constant depending on the operator norms of
$E:L^{1}(B_{1};\mathbb{R}^{N})\to L^{1}(B_{2};\mathbb{R}^{N})$ and
$E:W^{1,2}(B_{1};\mathbb{R}^{N})\to W^{1,2}_{0}(B_{2};\mathbb{R}^{N})$. For
any $\varepsilon>0$ we apply to the previous estimate Young’s inequality with
exponents $(N+2)/2$ and $(N+2)/N$ to get

$\int_{B_{1}}|U|^{2}\,dx\leq\varepsilon^{2}\,\int_{B_{1}}|DU|^{2}\,dx+\frac{C}{\varepsilon^{N}}\left(\int_{B_{1}}|U|\,dx\right)^{2},$

which rescales to

$\int_{B_{r}}|U|^{2}\,dx\leq(\varepsilon\,r)^{2}\,\int_{B_{r}}|DU|^{2}\,dx+\frac{C}{(\varepsilon\,r)^{N}}\left(\int_{B_{r}}|U|\,dx\right)^{2}\qquad\forall\,U\in
W^{1,2}(B_{r};\mathbb{R}^{N})$ (5.11)

for some $C=C(N)>0$. Choose

$\varepsilon^{2}=\frac{1}{2}\frac{(r-\rho)^{2}}{\widetilde{C}\,r^{2}}$

and apply (5.11) with $U=V$ and such $\varepsilon$ to (5.10) to get

$\int_{B_{\rho}}|DV|_{2}^{2}\,dx\leq\frac{1}{2}\,\int_{B_{r}}|DV|_{2}^{2}\,dx+\frac{C}{(r-\rho)^{N+2}}\left(\int_{B_{r}}|V|\,dx\right)^{2}$

for all $0<\rho<r<2R$. An application of [20, Lemma 3.1, Ch. 5] shows that
then

$\int_{B_{\rho}}|DV|_{2}^{2}\,dx\leq\frac{C}{(r-\rho)^{N+2}}\left(\int_{B_{r}}|V|\,dx\right)^{2}$

which, for $r=2R$, $\rho=R$, provides (5.9). ∎

##  6\. A family of $1$-homogeneous functions

Let $F$ be a $H$-q. u. c. function. In this section we gather the basic
properties of the function

$G(z)=F(DF^{-1}(z))$ (6.1)

and of the family of Minkowski functionals related to it, namely,

$g_{k}(z)=\inf\\{t>0:\,G(z/t)<k\\},$ (6.2)

for any $k\geq 0$. Note that, since $DF$ is a homeomorphism of
$\mathbb{R}^{N}$, the function in (6.1) is well defined.

######  Proposition 6.1.

Let $F$ be a $H$-q. u. c. function fulfilling (2.7). Then, $G$ is coercive and
for any $k\geq 0$ the set $\\{G\leq k\\}$ is star-shaped with respect to the
origin.

###### Proof.

We apply (2.13) with the vector $DF^{-1}(z)$ and use (2.7) to have

$G(z)\geq|DF^{-1}(z)|\,|z|/C.$

Furthermore, the quasisymmetry (2.3) of $DF^{-1}$ together with $DF^{-1}(0)=0$
give

$|DF^{-1}(z)|\geq\frac{\sup_{|v|=1}|DF^{-1}(v)|}{C\,\eta_{H}(1/|z|)}.$

Gathering the previous inequalities and making use of (2.19) and (2.17) we
have

$G(z)\geq\frac{\sup_{|v|=1}|DF^{-1}(v)|}{C}\,|z|^{1+1/H}\quad\text{for all
}|z|>1,$

thus proving the coercivity of $G$. In particular for any $k>0$ the set
$\\{G\leq k\\}$ is compact with nonempty interior.

Let us now show that $\\{G\leq k\\}$ is star shaped with respect to the
origin. Since $G(0)=0$ by the normalisation (2.7), it suffices to show that
for any $z\in\mathbb{S}^{N-1}$ the function $t\mapsto G(t\,z)$ is
nondecreasing. To this end, suppose first that $F\in C^{2}(\mathbb{R}^{N})$
fulfils

$\lambda_{\rm min}(D^{2}F(z))\geq\mu$ (6.3)

for some $\mu>0$ independent of $z$. Then $DF$ is a diffeomorphism and it
holds

$DG(z)=z\,D(DF^{-1})(z)=(D^{2}F)^{-1}(DF^{-1}(z))\,z,$ (6.4)

therefore

$\frac{d}{dt}G(t\,z)=\left(DG(t\,z),z\right)=t\left((D^{2}F)^{-1}(DF^{-1}(t\,z))\,z,z\right)\geq
0,$

proving the claim. In the general case, thanks to Proposition 2.6-(4), we know
that any $H$-q. u. c. function $F$ can be approximated in $C^{1}$ by a
sequence $(F_{n})$ of $C^{2}$ and $H$-q. u. c. functions obeying (6.3), and
such that $DF_{n}^{-1}\to DF^{-1}$ locally uniformly. Hence, $G_{n}=F_{n}\circ
DF_{n}^{-1}\to G$ locally uniformly, too. It follows that $t\mapsto G(t\,z)$
is nondecreasing, being the pointwise limit of a sequence of nondecreasing
functions. The proof is thus complete. ∎

######  Remark 6.2.

In general, the sets $\\{w\in\mathbb{R}^{N}:G(w)\leq k\\}=\\{DF(z):F(z)\leq
k\\}$ are not convex, so that the functions $g_{k}$ defined in (6.2) may fail
to be norms. Moreover, by employing the approximation in (2.15) in the
previous proof, we see that the $H$-q. u. convexity assumption is not needed
to prove the second statement of the previous proposition and, given any
convex $F\in C^{1}(\mathbb{R}^{N})$ obeying (2.7), the sets $\\{DF(z):F(z)\leq
k\\}$, while not convex in general, are always star-shaped with respect to the
origin.

######  Proposition 6.3.

Let $F$ be a $C^{2}$ and $H$-q. u. c. function obeying (2.7) and

$\inf_{z\in\mathbb{R}^{N}}\lambda_{\rm min}(D^{2}F(z))>0.$

Then, the following holds:

  1. (1)

$g_{k}$ is a Lipschitz $1$-homogeneous function with

${\rm Lip}\,(g_{k})\leq H\,\sup_{\mathbb{S}^{N-1}}g_{k}.$ (6.5)

  2. (2)

For some $C=C(H,N)>0$ it holds

$\sup_{\mathbb{S}^{N-1}}g_{k}\leq C\,\inf_{\mathbb{S}^{N-1}}g_{k}.$ (6.6)

  3. (3)

There exists $C=C(H,N)>0$ such that for any $k\geq h>0$

$\inf_{\\{G\geq
k\\}}(g_{h}-1)\geq\frac{1}{C}\,\min\left\\{\left(\frac{k-h}{h}\right)^{H},\left(\frac{k-h}{h}\right)^{\frac{1}{H(H+1)}}\right\\}.$
(6.7)

###### Proof.

Since $DG$ given in (6.4) doesn’t vanish outside the origin, the implicit
function theorem implies that the boundary $\partial\\{G\leq k\\}=\\{G=k\\}$
is a $C^{1}$-hypersurface and therefore $g_{k}$ is $C^{1}$ outside the origin.
By construction $g_{k}$ is positively $1$-homogeneous, hence it fulfils

$\displaystyle\big{(}Dg_{k}(z),z\big{)}$
$\displaystyle=g_{k}(z),\quad\forall\,z\neq 0,$ (6.8) $\displaystyle
Dg_{k}(\lambda\,z)$ $\displaystyle=Dg_{k}(z),\quad\forall\,\lambda>0,z\neq 0,$
(6.9)

while by construction it holds

$G\left(\frac{z}{g_{k}(z)}\right)=k\qquad\forall\,z\neq 0.$ (6.10)

From the last property, in particular, it holds $g_{k}\equiv 1$ on
$\\{G=k\\}$, and therefore the Lagrange multiplier rule gives
$Dg_{k}(z)=\alpha(z)\,DG(z)$ for some $\alpha(z)\geq 0$ and all
$z\in\\{G=k\\}$.  This, together with (6.8), implies that on $\\{G=k\\}$ we
have

$1=g_{k}(z)=(Dg_{k}(z),z)=\alpha(z)\,(DG(z),z),$

and in turn

$Dg_{k}(z)=\frac{DG(z)}{\big{(}DG(z),z\big{)}}.$

Let $z\in\\{G=k\\}$. Thanks to (6.4) we get

$|Dg_{k}(z)|=\frac{|DG(z)|}{\big{(}DG(z),z\big{)}}=\frac{|z\,(D^{2}F)^{-1}(w)|}{\big{(}z\,(D^{2}F)^{-1}(w),z\big{)}},\qquad\text{with
}w=DF^{-1}(z).$ (6.11)

Let $\Lambda(w)$ and $\lambda(w)$ be the maximum and minimum eigenvalues of
$D^{2}F$ at $w$, respectively. Then it holds

$|z\,(D^{2}F)^{-1}(w)|\leq\frac{1}{\lambda(w)}|z|\quad\text{as well
as}\quad\big{(}z\,(D^{2}F)^{-1}(w),z\big{)}\geq\frac{1}{\Lambda(w)}\,|z|^{2}.$
(6.12)

Therefore, thanks to (6.11), (6.12), and (2.5) we have

$|Dg_{k}(z)|=\frac{|z\,(D^{2}F)^{-1}(w)|}{\big{(}z\,(D^{2}F)^{-1}(w),z\big{)}}\leq\frac{|z|\Lambda(w)}{|z|^{2}\lambda(w)}\leq\frac{H}{|z|}\qquad\text{on
$\\{G=k\\}$}.$ (6.13)

Using (6.9) and (6.10) gives

$|Dg_{k}(z)|=\left|Dg_{k}\left(\frac{z}{g_{k}(z)}\right)\right|.$

Therefore we can use (6.13) with the vector $\displaystyle\frac{z}{g_{k}(z)}$,
thus obtaining

$|Dg_{k}(z)|\leq H\,\frac{g_{k}(z)}{|z|}\leq
H\,\sup_{\mathbb{S}^{N-1}}g_{k}\qquad\forall\,z\neq 0.$

This proves (6.5). To verify (6.6), notice that under the stated assumptions
on $F$ the map $t\mapsto G(t\,z)$ is strictly increasing on $]0,+\infty[$,
hence $\\{g_{k}=1\\}=\\{G=k\\}$. By the $1$-homogeneity of $g_{k}$, it then
holds

$\sup_{\mathbb{S}^{N-1}}g_{k}=\sup_{\\{G=k\\}}|z|\quad\text{as well
as}\quad\inf_{\mathbb{S}^{N-1}}g_{k}=\inf_{\\{G=k\\}}|z|.$

Since $\\{G\leq k\\}$ is star-shaped with respect to the origin, we can choose
$\bar{x}$ and $\bar{y}$ in $\\{G=k\\}$ such that

$|\bar{x}|=\sup_{\\{G=k\\}}|z|\quad\text{as well
as}\quad|\bar{y}|=\inf_{\\{G=k\\}}|z|.$

By (2.3) applied to $DF$ and (2.7) we have

$\frac{\sup_{\mathbb{S}^{N-1}}g_{k}}{\inf_{\mathbb{S}^{N-1}}g_{k}}=\frac{|\bar{x}|}{|\bar{y}|}=\frac{|DF(DF^{-1}(\bar{x}))|}{|DF(DF^{-1}(\bar{y}))|}\leq
C\,\eta_{H}\left(\frac{|DF^{-1}(\bar{x})|}{|DF^{-1}(\bar{y})|}\right).$ (6.14)

By the definition of $G$, it holds
$DF^{-1}(\bar{x}),\,DF^{-1}(\bar{y})\in\\{F=k\\}$, hence (2.14) gives

$1=\frac{F(DF^{-1}(\bar{y}))}{F(DF^{-1}(\bar{x}))}\leq
C\,\frac{|DF^{-1}(\bar{y})|}{|DF^{-1}(\bar{x})|}\,\eta_{H}\left(\frac{|DF^{-1}(\bar{y})|}{|DF^{-1}(\bar{x})|}\right)=C\,\eta_{1+H,1+1/H}\left(\frac{|DF^{-1}(\bar{y})|}{|DF^{-1}(\bar{x})|}\right)$

so that

$\frac{|DF^{-1}(\bar{x})|}{|DF^{-1}(\bar{y})|}\leq\frac{1}{\eta_{1+H,1+1/H}^{-1}(1/C)}.$

Inserting this estimate into (6.14) gives us the conclusion.

Finally, to prove the last assertion, let us fix $k>h$, choose $\tilde{x}$
such that

$G(\tilde{x})=k\quad\text{as well as}\quad g_{h}(\tilde{x})=\inf_{G\geq
k}g_{h},$

and set

$\tilde{y}=\frac{\tilde{x}}{g_{h}(\tilde{x})}.$

Then in holds $g_{h}(\tilde{y})=1$, so that $\tilde{y}\in\\{G=h\\}$. Set
furthermore

$x:=DF^{-1}(\tilde{x})\in\\{F=k\\}\quad\text{as well as}\quad
y:=DF^{-1}(\tilde{y})\in\\{F=h\\}.$ (6.15)

By the definition of $\tilde{x}$ and $\tilde{y}$ and the $1$-homogeneity of
$g_{h}$ it follows that

$\inf_{\\{G\geq
k\\}}(g_{h}-1)=\frac{g_{h}(\tilde{x})-g_{h}(\tilde{y})}{g_{h}(\tilde{y})}{=\frac{|\tilde{x}-\tilde{y}|}{|\tilde{y}|}.}$
(6.16)

Using the distortion estimate (2.3) to the map $DF^{-1}$ gives

$\frac{|x-y|}{|y|}\leq
C\,\eta_{H}\left(\frac{|\tilde{x}-\tilde{y}|}{|\tilde{y}|}\right)$

so that, by (2.20) for the function $\eta_{H}^{-1}$ we obtain

$\frac{|\tilde{x}-\tilde{y}|}{|\tilde{y}|}\geq\eta_{H}^{-1}\left(\frac{1}{C}\frac{|x-y|}{|y|}\right)\geq\frac{1}{C}\,\eta_{H}^{-1}\left(\frac{|x-y|}{|y|}\right),$

which inserted into (6.16) reads as

$\inf_{\\{G\geq
k\\}}(g_{h}-1)\geq\frac{1}{C}\,\eta^{-1}_{H}\left(\frac{|x-y|}{|y|}\right).$
(6.17)

By (6.15) we have

$\begin{split}k-h&=F(x)-F(y)\leq\int_{0}^{1}|DF(y+t\,(x-y))|\,|x-y|\,dt\\\
&\leq|x-y|\,\left[\int_{0}^{1}|DF(y+t\,(x-y))-DF(y)|\,dt+|DF(y)|\right].\end{split}$
(6.18)

To estimate the rightmost integral in the above inequality we use (2.3) for
the map $DF$ (recalling that $DF(0)=0$) and (2.18) to have

$\begin{split}\int_{0}^{1}|DF(y+t\,(x-y))-DF(y)|\,dt&\leq
C\,|DF(y)|\,\int_{0}^{1}\eta_{H}\left(t\,\frac{|x-y|}{|y|}\right)\,dx\\\ &\leq
C\,|DF(y)|\,\eta_{H}\left(\frac{|x-y|}{|y|}\right)\,\int_{0}^{1}\eta_{H}(t)\,dt\\\
&\leq\frac{CH}{H+1}\,|DF(y)|\,\eta_{H}\left(\frac{|x-y|}{|y|}\right).\end{split}$

Therefore (6.18) simplifies to

$k-h\leq
C\,|DF(y)|\,|x-y|\,\left[1+\eta_{H}\left(\frac{|x-y|}{|y|}\right)\right].$

On the other hand, (2.13) implies

$h=F(y)\geq|y|\,|DF(y)|/C.$

Gathering the previous two inequalities gives

$\frac{k-h}{h}\leq
C\,\frac{|x-y|}{|y|}\,\left[1+\eta_{H}\left(\frac{|x-y|}{|y|}\right)\right].$
(6.19)

Since

$t\,(1+\eta_{H}(t))\leq 2\,\eta_{1,H+1}(t),$

we can rewrite (6.19) as

$\frac{|x-y|}{|y|}\geq\eta_{1,H+1}^{-1}\left(\frac{k-h}{C\,h}\right)$

and recalling (6.17) gives

$\inf_{\\{G\geq
k\\}}(g_{h}-1)\geq\frac{1}{C}\,\eta^{-1}_{H}\left(\eta_{1,H+1}^{-1}\left(\frac{k-h}{h}\right)\right)=\frac{1}{C}\,\eta^{-1}_{1/H,H(H+1)}\left(\frac{k-h}{h}\right),$

where we used (2.20), (2.21) to clean up the estimate. Using (2.17) gives
(6.7) and then the conclusion. ∎

##  7\. Proof of the main result

######  Theorem 7.1.

Let $F$ be a $H$-q. u. c. function obeying (2.7) and let $u$ be a local
minimiser for $J$ in $\Omega$ in the sense of Definition 3.1. Then, there
exists a constant $C=C(H,N)>0$ such that, for any $B_{R}$ with
$B_{2R}\subseteq\Omega$, $u$ is locally Lipschitz and satisfies the estimate

$\sup_{B_{R/2}}|F(Du)|\leq C\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx.$ (7.1)

###### Proof.

Let us suppose for the moment that both $F$ and $u$ are smooth in $B_{2R}$.
Moreover, by considering $F/i_{F}$ (see (2.8)), we can also suppose that

$\inf_{|z|=1}|DF(z)|=1.$ (7.2)

Fix $R>r>0$ and set $\bar{R}=\dfrac{R+r}{2}$. Choose $\varphi\in
C^{\infty}_{c}(B_{\bar{R}},[0,1])$ such that

${\rm supp}\,(\varphi)\subseteq B_{\bar{R}},\qquad\varphi\lfloor_{B_{r}}\equiv
1,\qquad|D\varphi|\leq\frac{C}{R-r}.$ (7.3)

For any $k\geq 0$ we consider the Minkowski functional of $\\{G\leq k\\}$
defined as in (6.2), with $G$ given by (6.1), which satisfies by construction

$g_{k}(z)\leq 1\quad\Longleftrightarrow\quad\text{ $z\in\\{G\leq
k\\}$}\quad\Longleftrightarrow\quad F(DF^{-1}(z))\leq k.$ (7.4)

Let $V$ be given by (3.7). It holds $\varphi\,(g_{k}(V)-1)_{+}\in
W^{1,2}_{0}(B_{\bar{R}})$ and thanks to (7.4)

$(g_{k}(V)-1)_{+}\equiv 0\qquad\text{ on }B_{\bar{R}}\setminus
A\left(k,\bar{R}\right),$

where we set

$A(k,\rho)=\\{x\in B_{\rho}:F(Du(x))\geq k\\}=\\{x\in B_{\rho}:G(V(x))\geq
k\\},$

i. e. equation (5.3) for the affine function $\ell(z)\equiv k$. Then the
Sobolev embedding and the chain rule give

$\begin{split}\left(\int_{A(k,r)}|g_{k}(V)-1|^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}&\leq\left(\int_{B_{\bar{R}}}|\varphi\,(g_{k}(V)-1)_{+}|^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}\\\
&\leq C\int_{B_{\bar{R}}}|D(\varphi\,(g_{k}(V)-1)_{+})|^{2}\,dx\\\ &\leq
C\int_{B_{\bar{R}}}|(g_{k}(V)-1)_{+}|^{2}|D\varphi|^{2}\,dx\\\
&\quad+C\int_{B_{\bar{R}}}|D(g_{k}(V)-1)_{+})|^{2}\varphi^{2}\,dx.\end{split}$
(7.5)

In order to estimate the above right-hand side, for the first term we use
(7.4) and (7.3) to achieve

$\int_{B_{\bar{R}}}|(g_{k}(V)-1)_{+}|^{2}|D\varphi|^{2}\,dx\leq\frac{C}{(R-r)^{2}}\int_{A(k,R)}(g_{k}(V)-1)^{2}\;dx,$
(7.6)

while for the second term we apply the chain rule and inequalities (7.4) and
(6.5) to have

$\begin{split}\int_{B_{\bar{R}}}|D(g_{k}(V)-1)_{+})|^{2}\varphi^{2}\,dx\,&{\leq\int_{A(k,\bar{R})}|Dg_{k}(V)|^{2}|DV|_{2}^{2}\,dx}\\\
&\leq
H^{2}\,\sup_{\mathbb{S}^{N-1}}g_{k}^{2}\int_{A(k,\bar{R})}|DV|_{2}^{2}\,dx.\end{split}$
(7.7)

To further estimate the last integral, we use the Caccioppoli inequality (5.4)
with $\rho=\bar{R}<R$, which results in

$H^{2}\,\sup_{\mathbb{S}^{N-1}}g_{k}^{2}\int_{A(k,\bar{R})}|DV|_{2}^{2}\,dx\leq\frac{C}{(R-r)^{2}}\,\sup_{\mathbb{S}^{N-1}}g_{k}^{2}\,\int_{A(k,R)}|V|^{2}\,dx.$
(7.8)

Moreover, we observe that $g_{k}(V)\geq 1$ on $A(k,R)$, hence the
$1$-homogeneity of $g_{k}$ implies that

$|V|=\frac{|V|}{g_{k}(V)}\,g_{k}(V)\leq\sup_{z\neq
0}\frac{|z|}{g_{k}(z)}\,g_{k}(V)\leq\frac{g_{k}(V)}{\inf_{\mathbb{S}^{N-1}}g_{k}}.$
(7.9)

From (7.8), (7.9) and (6.6) we have that (7.7) simplifies to

$\int_{B_{\bar{R}}}|D(g_{k}(V)-1)_{+})|^{2}\varphi^{2}\,dx\leq\frac{C}{(R-r)^{2}}\int_{A(k,R)}g^{2}_{k}(V)\,dx,$

which, inserted into (7.5) together with (7.6), gives

$\begin{split}\left(\int_{A(k,r)}|g_{k}(V)-1|^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}&\leq\frac{C}{(R-r)^{2}}\int_{A(k,R)}g_{k}^{2}(V)+(g_{k}(V)-1)^{2}\,dx\\\
&\leq\frac{C}{(R-r)^{2}}\left(\int_{A(k,R)}(g_{k}(V)-1)^{2}\,dx+{|A(k,R)|}\right),\end{split}$
(7.10)

with a constant $C=C(H)$.

Let now $0<h<k$ and observe that

$g_{h}(V(x))-1\geq\inf_{\\{G\geq k\\}}(g_{h}-1)\qquad\text{for $x\in
A(k,R)$},$

therefore (6.7) gives

$\frac{1}{C}\,\min\left\\{\left(\frac{k}{h}-1\right)^{H},\left(\frac{k}{h}-1\right)^{\frac{1}{H(H+1)}}\right\\}^{2}\,|A(k,R)|\leq\int_{A(k,R)}(g_{h}(V)-1)^{2}\,dx,$

so that

$|A(k,R)|\leq\frac{C}{\left(\frac{k}{h}-1\right)^{2H}}\,\int_{A(k,R)}(g_{h}(V)-1)^{2}\,dx\qquad\text{if}\quad
h<k<2\,h.$ (7.11)

Inserting this estimate into (7.10) gives for $0<h<k<2\,h$

$\begin{split}&\left(\int_{A(k,r)}|g_{k}(V)-1|^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}\\\
&\leq\frac{C}{(R-r)^{2}}\int_{A(k,R)}(g_{k}(V)-1)^{2}+\frac{C}{\left(\frac{k}{h}-1\right)^{2H}}\,(g_{h}(V)-1)^{2}\,dx\\\
&\leq\frac{C}{\left(\frac{k}{h}-1\right)^{2H}(R-r)^{2}}\int_{A(h,R)}(g_{h}(V)-1)^{2}\,dx,\end{split}$
(7.12)

where in the second inequality we used the fact that

$A(k,R)\subseteq A(h,R)\quad\text{as well as}\quad g_{h}\geq
g_{k}\quad\text{for all $h\leq k$}$ (7.13)

and the bound

$\left(\frac{k}{h}-1\right)^{-2H}\geq 1\qquad\text{for $0<h<k<2h$}.$

Estimate (7.12) in conjunction with (7.11) allows for a De Giorgi-type
iteration, whose proof we include here for completeness. For $0<h<k<2h$, we
apply Hölder’s inequality, (7.12), (7.11) and (7.13) to have

$\begin{split}\int_{A(k,r)}(g_{k}(V)-1)^{2}\,dx&\leq|A(k,r)|^{\frac{2}{N}}\left(\int_{A(k,r)}|g_{k}(V)-1|^{2^{*}}\,dx\right)^{\frac{2}{2^{*}}}\\\
&\leq\frac{C\,|A(k,r)|^{\frac{2}{N}}}{\left(\frac{k}{h}-1\right)^{2H}(R-r)^{2}}\int_{A(h,R)}(g_{h}(V)-1)^{2}\,dx\\\
&\leq\frac{C}{{\left(\frac{k}{h}-1\right)^{2H(1+2/N)}}(R-r)^{2}}\left(\int_{A(h,R)}(g_{h}(V)-1)^{2}\,dx\right)^{1+\frac{2}{N}}.\end{split}$
(7.14)

Let $M>0$ to be chosen and set for $n\geq 0$

$k_{n}=M\,(1-2^{-(n+1)}),\qquad r_{n}=\frac{R}{2}+\frac{R}{2^{n+1}},\qquad
X_{n}=\int_{A(k_{n},r_{n})}(g_{k_{n}}(V)-1)^{2}\,dx,$

so that $\lim_{n\to\infty}k_{n}=M$ and $\lim_{n\to\infty}r_{n}=R/2$. An
elementary computation shows that

$k_{n}\leq k_{n+1}\leq
2\,k_{n},\qquad\frac{k_{n+1}}{k_{n}}-1\geq\frac{1}{2^{n+2}},\qquad(r_{n}-r_{n+1})^{2}=\frac{R^{2}}{4^{n+2}},$

therefore (7.14) with the viable choice $k=k_{n+1}$ and $h=k_{n}$ reads as

$X_{n+1}\leq\frac{C}{R^{2}}\,b^{n}\,X_{n}^{1+2/N},$

for some positive $b=b(H,N)$ and $C=C(H,N)$.

Thanks to [22, Lemma 7.1], it holds

$X_{0}\leq\frac{R^{N}}{C^{N/2}\,b^{N^{2}/4}}\quad\Longrightarrow\quad\lim_{n\to\infty}X_{n}=0.$
(7.15)

Moreover, from $X_{n}\to 0$ it follows that

$\int_{A(M,R/2)}(g_{M}(V)-1)^{2}\,dx=0$

which in turn by (7.4) forces $F(Du)\leq M$ a. e. in $B_{R/2}$. Let us use
this information to write explicitly the implication in (7.15). Since
$r_{0}=R$ and $k_{0}=M/2$, it reads as

$\int_{A(M/2,R)}(g_{M/2}(V)-1)^{2}\,dx\leq
c\,R^{N}\quad\Longrightarrow\quad\sup_{B_{R/2}}F(Du)\leq M,$ (7.16)

for some $c=c(H,N)>0$. It remains to choose $M>0$ in such a way that the first
inequality in (7.16) holds true. We aim to obtain an upper bound for $g_{M/2}$
in terms of an upper bound for $G$. To get the latter, we recall that by (2.7)
the following convexity inequality

$F(z)\leq\left(DF(z),z\right)\qquad\forall\,z\in\mathbb{R}^{N}$

holds true. We choose $z=DF^{-1}(y)$ and then use the quasisymmetry (2.3) of
$DF^{-1}$ to obtain

$G(y)\leq\left(y,DF^{-1}(y)\right)\leq|DF^{-1}(y)|\,|y|\leq
C\,|DF^{-1}(w)|\,|y|\,\eta_{H}\left(\frac{|y|}{|w|}\right),$

for some $C=C(H,N)>0$ and for all $y\in\mathbb{R}^{N}$,
$w\in\mathbb{R}^{N}\setminus\\{0\\}$. Choose $w=DF(z)$ for arbitrary $z$ such
that $|z|=1$ to get

$G(y)\leq C\,|y|\,\inf_{|z|=1}\eta_{H}\left(\frac{|y|}{|DF(z)|}\right).$

Moreover, the normalisation (7.2) implies that $1/|DF(z)|\leq 1$ for all $z$
such that $|z|=1$ which, together with the monotonicity of $\eta_{H}$, implies
that the previous inequality reads as

$G(y)\leq
C\,|y|\,\eta_{H}(|y|)=C\,\eta_{H+1,1+\frac{1}{H}}(|y|)\qquad\forall\,y\in\mathbb{R}^{N}.$

Choosing $\displaystyle y=\frac{z}{g_{M/2}(z)}$ yields, by the very definition
of $g_{M/2}$,

$\frac{M}{2}=G\left(\frac{z}{g_{M/2}(z)}\right)\leq
C\,\eta_{H+1,1+\frac{1}{H}}\left(\frac{|z|}{g_{M/2}(z)}\right),$

which, solved with respect to $g_{M/2}(z)$, gives through (2.19) and (2.20)

$g_{M/2}(z)\leq\frac{C\,|z|}{\eta^{-1}_{H+1,1+\frac{1}{H}}(M)}\leq
C\,|z|\,\eta_{\frac{1}{H+1},\frac{H}{H+1}}(1/M).$

Therefore

$\int_{A(M/2,R)}(g_{M/2}(V)-1)^{2}\,dx\leq\int_{B_{R}}g_{M/2}^{2}(V)\,dx\leq
C\,\eta_{\frac{1}{H+1},\frac{H}{H+1}}^{2}(1/M)\int_{B_{R}}|V|^{2}\,dx.$

In order to satisfy the assumption of (7.16), we can then choose $M$ so that

$C\,\eta_{\frac{1}{H+1},\frac{H}{H+1}}^{2}(1/M)\int_{B_{R}}|V|^{2}\,dx\leq
c\,R^{N},$

i. e., rearranging and using (2.19) again,

$\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{R}}|V|^{2}\,dx\right)^{\frac{1}{2}}\leq\frac{c}{\eta_{\frac{1}{H+1},\frac{H}{H+1}}(1/M)}=c\,\eta_{H+1,1+\frac{1}{H}}^{-1}(M)$

for some $c=c(H,N)>0$, which is readily solved through (2.20) as

$M\geq
C\,\eta_{H+1,1+\frac{1}{H}}\left(\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{R}}|V|^{2}\,dx\right)^{\frac{1}{2}}\right).$

Now recall that by (3.8) it holds

$\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{R}}|V|^{2}\,dx\right)^{\frac{1}{2}}\leq
C\,\eta_{\frac{H}{H+1},\frac{1}{H+1}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx\right).$

Then, thanks to (2.21) and again (2.20) it is sufficient to require

$M\geq
C\,\eta_{H+1,1+\frac{1}{H}}\circ\eta_{\frac{H}{H+1},\frac{1}{H+1}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx\right)=C\,\eta_{H}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx\right).$ (7.17)

Choosing $M$ such that the equality in (7.17) holds true gives, thanks to
(7.16), that $F(Du)\leq M$ in $B_{R/2}$, with the estimate

$\sup_{B_{R/2}}|F(Du)|\leq
C\,\eta_{H}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx\right).$

We can then remove assumption (7.2) to get, for a general $i_{F}>0$,

$\sup_{B_{R/2}}|F(Du)|\leq
C\,i_{F}\,\eta_{H}\left(\frac{1}{i_{F}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx\right).$ (7.18)

We next take advantage of the invariances of this estimate. For any
$\lambda>0$, the function $u_{\lambda}(x)=\lambda\,u(x)$ is a local minimiser
in $\Omega$ for the functional $J_{\lambda}$ having integrand

$F_{\lambda}(z)=F(z/\lambda).$

In particular, (7.18) holds true also for such $F_{\lambda},u_{\lambda}$, that
is,

$\sup_{B_{R/2}}|F_{\lambda}(Du_{\lambda})|\leq
C\,i_{F_{\lambda}}\,\eta_{H}\left(\frac{1}{i_{F_{\lambda}}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F_{\lambda}(Du_{\lambda})\,dx\right).$
(7.19)

Notice that $F_{\lambda}$ is still $H$-q. u. c. and that

$i_{F_{\lambda}}=\frac{1}{\lambda}\,\inf_{|z|=1}\left|DF\Big{(}\frac{z}{\lambda}\Big{)}\right|.$
(7.20)

For any $z,w$ of unit norm, the distortion estimate (2.3) applied first to the
vectors $z/\lambda,w$ and then to $w,z/\lambda$ gives

$\left|DF\Big{(}\frac{z}{\lambda}\Big{)}\right|\leq
C\,|DF(w)|\,\eta_{H}(1/\lambda)\quad\text{as well as}\quad|DF(w)|\leq
C\,\left|DF\Big{(}\frac{z}{\lambda}\Big{)}\right|\,\eta_{H}(\lambda),$

respectively, so that

$\frac{i_{F}}{C\,\eta_{H}(\lambda)}\leq\left|DF\Big{(}\frac{z}{\lambda}\Big{)}\right|\leq
C\,i_{F}\,\eta_{H}(1/\lambda)$

for all $|z|=1$. Taking the infimum for such $z$’s and using (7.20) give

$\frac{i_{F}}{C\,\eta_{H}(\lambda)\,\lambda}\leq i_{F_{\lambda}}\leq
C\,i_{F}\,\frac{\eta_{H}(1/\lambda)}{\lambda}.$

Since $DF$ is $H^{N-1}$-quasiconformal, it is also locally $1/H$-Hölder
continuous and so is the map $t\mapsto|DF(t\,z)|$. It follows that
$\lambda\mapsto i_{F_{\lambda}}$ is continuous on $]0,+\infty[$, being an
infimum of equicontinuous functions. Moreover, it fulfils

$\begin{split}\lim_{\lambda\downarrow
0}i_{F_{\lambda}}&\geq\frac{i_{F}}{C}\,\lim_{\lambda\downarrow
0}\frac{1}{\eta_{H}(\lambda)\,\lambda}=\infty\quad\text{as well
as}\quad\lim_{\lambda\uparrow\infty}i_{F_{\lambda}}\leq
C\,i_{F}\,\lim_{\lambda\uparrow\infty}\frac{\eta_{H}(1/\lambda)}{\lambda}=0.\end{split}$

Thanks to the intermediate value theorem we can then choose $\lambda$ so that

$\frac{1}{i_{F_{\lambda}}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F_{\lambda}(Du_{\lambda})\,dx=\frac{1}{i_{F_{\lambda}}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx=1.$

For such $\lambda$ and corresponding $u_{\lambda}$, $F_{\lambda}$, (7.19)
becomes (7.1), since

$\sup_{B_{R/2}}F_{\lambda}(Du_{\lambda})=\sup_{B_{R/2}}F(Du)\quad\text{as well
as}\quad i_{F_{\lambda}}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F(Du)\,dx,$

and

$\eta_{H}\left(\frac{1}{i_{F_{\lambda}}}\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F_{\lambda}(Du_{\lambda})\,dx\right)=\eta_{H}(1)=1.$

This gives us the conclusion in the smooth case.

To remove the smoothness assumption on $F$ and $u$, let $F_{n}$ and $u_{n}$ be
the regularised approximations obtained thanks to Lemma 4.1. Then we can apply
the same argument as before to $F_{n},u_{n}$, thus obtaining

$\sup_{B_{R/2}}|F_{n}(Du_{n})|\leq C\,\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2R}}F_{n}(Du_{n})\;dx.$ (7.21)

Now observe that (4.1) allows to assume (up to subsequences) that
$DF_{n}(Du_{n})\to DF(Du)$ pointwise a. e. in $B_{2R}$, while Proposition
2.6-(4) ensures that $DF_{n}^{-1}\to DF^{-1}$ locally uniformly, so that both
$Du_{n}\to Du$ and $F(Du_{n})\to F(Du)$, pointwise a. e. in $B_{R}$. This
observation and the first estimate in (4.1) allow to pass to the limit in
(7.21), thus obtaining again the conclusion. ∎

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