# On isosupremic vectorial minimisation problems in $L^{\infty}$ with general
nonlinear constraints

Ed Clark  E. C., Department of Mathematics and Statistics, University of
Reading, Whiteknights Campus, Pepper Lane, Reading RG6 6AX, United Kingdom
<EMAIL_ADDRESS>and  Nikos Katzourakis  N. K., Department of
Mathematics and Statistics, University of Reading, Whiteknights Campus, Pepper
Lane, Reading RG6 6AX, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We study minimisation problems in $L^{\infty}$ for general quasiconvex first
order functionals, where the class of admissible mappings is constrained by
the sublevel sets of another supremal functional and by the zero set of a
nonlinear operator. Examples of admissible operators include those expressing
pointwise, unilateral, integral isoperimetric, elliptic quasilinear
differential, jacobian and null Lagrangian constraints. Via the method of
$L^{p}$ approximations as $p\to\infty$, we illustrate the existence of a
special $L^{\infty}$ minimiser which solves a divergence PDE system involving
certain auxiliary measures as coefficients. This system can be seen as a
divergence form counterpart of the Aronsson PDE system which is associated
with the constrained $L^{\infty}$ variational problem.

###### Key words and phrases:

Calculus of Variations in $L^{\infty}$; PDE-Constrained Optimisation; Euler-
Lagrange equations; Absolute minimisers; Aronsson-Euler systems; Kuhn-Tucker
theory; Isosupremic problems, Isoperimetric problems.

###### 2020 Mathematics Subject Classification:

35J94; 49J20; 49K20; 49K35; 49K99; 35J92.

E.C. has been financially supported through the UK EPSRC scholarship GS19-055

###### Contents

  1. 1 Introduction and main results
  2. 2 Minimisers of $L^{p}$ problems and convergence as $p\to\infty$
  3. 3 The equations for constrained minimisers in $L^{p}$ and in $L^{\infty}$
  4. 4 Explicit classes of nonlinear operators
    1. 4.1 Pointwise constraints, unilateral constraints and inclusions
    2. 4.2 Integral and isoperimetric constraints
    3. 4.3 Quasilinear second order differential constraints
    4. 4.4 Null Lagrangians and determinant constraints

##  1\. Introduction and main results

Let $n,N\in\mathbb{N}$ and let also $\Omega\Subset\mathbb{R}^{n}$ be a bounded
open set with Lipschitz continuous boundary. Consider two functions

(1.1) $f,\,g\ :\ \ \Omega\times\mathbb{R}^{N}\times\mathbb{R}^{N\times
n}\longrightarrow\mathbb{R},$

which will be assumed to satisfy certain natural structural assumptions.
Additionally, let $\bar{p}>n$ be fixed and consider a given nonlinear operator

(1.2) $\begin{split}\mathrm{Q}\ :\ \
W^{1,\bar{p}}_{0}\big{(}\Omega;\mathbb{R}^{N}\big{)}\longrightarrow\textbf{E},\end{split}$

where $(\textbf{E},\|\cdot\|)$ is an arbitrary Banach space. In this paper we
are interested in the following variational problem: given $G\geq 0$ and the
supremal functionals

$\mathrm{F}_{\infty},\mathrm{G}_{\infty}\ :\ \
W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow\mathbb{R},$

defined by

(1.3) $\left\\{\ \
\begin{split}\mathrm{F}_{\infty}(u):&=\,\underset{\Omega}{\mathrm{ess}\sup}\,f(\cdot,u,\mathrm{D}u),\phantom{\Big{|}}\\\
\mathrm{G}_{\infty}(u):&=\,\underset{\Omega}{\mathrm{ess}\sup}\,g(\cdot,u,\mathrm{D}u),\end{split}\right.\
\ $

find $u_{\infty}\in W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})$ such that

(1.4)
$\mathrm{F}_{\infty}(u_{\infty})\,=\,\inf\Big{\\{}\mathrm{F}_{\infty}(u)\ :\
u\in W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N}),\ \mathrm{G}_{\infty}(u)\leq G\
\,\&\,\ \mathrm{Q}(u)=0\Big{\\}}.\\\ $

We are also interested in deriving appropriate differential equations as
necessary conditions that such constrained minimisers $u_{\infty}$ might
satisfy. Hence, the problem under consideration is that of minimising the
supremal functional $\textrm{F}_{\infty}$ within the $G$-sublevel set of the
supremal functional $\textrm{G}_{\infty}$, constrained also by the zero level
set of the mapping $\mathrm{Q}$. Minimisation problems in $L^{\infty}$ with an
$L^{\infty}$ sublevel-set constraints are called _isosupremic_ , a terminology
introduced by Aronsson-Barron in [2], by analogy to the terminology of
_isoperimetric_ constraints utilised for integral functionals.

The assumptions required for our mapping $\mathrm{Q}$ expressing the nonlinear
level-set constraint are very general, allowing for the solvability of a very
large class of problems. In Section 4 we provide a fairly comprehensive list
of explicit examples of operators $\mathrm{Q}$ to which our results apply.
Indeed, $\mathrm{Q}$ may manifest itself as a pointwise, unilateral, or
inclusion constraint (Subsection 4.1), as an integral isoperimetric constraint
(Subsection 4.2), or as a nonlinear PDE constraint, including second order
quasilinear divergence systems (Subsection 4.3), jacobian equations and other
PDEs driven by null Lagrangians (Subsection 4.4).

Minimisation of supremal functionals poses several important challenges.
Firstly, the $L^{\infty}$ norm is neither Gateaux differentiable nor strictly
convex, and the space $L^{\infty}$ is neither reflexive nor separable. A
related additional complication is that the $L^{\infty}$ norm is not additive
but only sub-additive with respect to the domain. As a result, if one solves
the $L^{\infty}$ minimisation existence problem via the direct method, then
the analogue of the Euler-Lagrange equations for the $L^{\infty}$ problem
cannot be derived directly by considering variations, due to the lack of
smoothness of the $L^{\infty}$ norm.

In this paper, we transcend the difficulties illustrated above, by following
the relatively standard strategy of using appropriate approximations by
$L^{p}$ functionals as $p\to\infty$. We solve the desired $L^{\infty}$
variational problem by solving approximating $L^{p}$ variational problems for
all $p$, and obtaining the necessary compactness estimates that allow us to
pass to the limit as $p\to\infty$. The case of finite $p$, which is of
independent interest and a byproduct of our analysis, for us is just the
mechanism to solve the desired $L^{\infty}$ problem. The intuition behind the
use of $L^{p}$ approximations is based on the fact that for a fixed
essentially bounded function on a set of finite measure, the $L^{p}$ norm
tends to the $L^{\infty}$ norm as $p\to\infty$.

Problems involving constraints are relatively new in the Calculus of
Variations in $L^{\infty}$ and previous work has been relatively sparse, even
more so in the vectorial case. To the best of our knowledge, the only work
which directly studies isosupremic problems is [2] by Aronsson-Barron. Among
other questions answered therein, it considers some aspects of the one-
dimensional case for $n=1$, but with no additional constraints of any type
(which amounts to $\mathrm{Q}\equiv 0$ in our setting).

More broadly, very few previous works involve vectorial problems with general
constraints in $L^{\infty}$. Certain vectorial and higher order problems
involving eigenvalues in $L^{\infty}$ have been considered in [26, 28].
Examples of problems with PDE and other constraints are considered in [12, 23,
24, 25]. In the paper [6] of Barron-Jensen, a scalar $L^{\infty}$ constrained
problem was considered, but the constraint was integral. With the exception of
the paper [2], it appears that vectorial variational problems in $L^{\infty}$
involving isosupremic constraints have not been studied before, especially
including additional nonlinear constraints which cover numerous different
cases, as in this work. For assorted interesting works within the wider area
of Calculus of Variations in $L^{\infty}$ we refer to [1, 5, 7, 8, 9, 10, 11,
27, 31, 33, 35, 36].

Let us note that, in this work, we refrain from discussing the question of
defining and studying localised versions of $L^{\infty}$ minimisers on
subdomains. Such minimisers are commonly called _absolute_ in the Calculus of
Variations in $L^{\infty}$, a concept first introduced and studied by Aronsson
in his seminal papers in the 1960s. As already noted in [2], the definition of
constrained absolute minimisers on subdomains is quite problematic in many
respects, even for specific choices of constraints in the scalar case $N=1$,
which has a well developed $L^{\infty}$ theory. In any case, the “non-
intrinsic” method of $L^{p}$ approximations, employed herein, is widely
believed to select always the “best” $L^{\infty}$ minimiser, which in the
scalar unconstrained case can be shown to be indeed an absolute minimiser. In
the vectorial case of $N\geq 2$, the situation is trickier and the “correct”
localised minimality notion is still under discussion, even without any
constraints being involved (see e.g. [3, 4] for work in this direction).

We now state our main results. Our notation is generally standard as e.g. in
[30], or otherwise self-explanatory. As we will only be working with finite
measures, we will be using the normalised $L^{p}$ norms in which we replace
the integral over the domain with the respective average. For our density
functions $f$ and $g$ appearing in (1.1) we will assume the following:

(1.5) $\left\\{\begin{split}&\text{Both $f$ and $g$ are Carath\'{e}odory
functions, namely $f(\cdot,\eta,P)$, $g(\cdot,\eta,P)$ are}\\\
&\text{measurable for all $(\eta,P)$, and $f(x,\cdot,\cdot)$,
$g(x,\cdot,\cdot)$ are continuous for a.e.\
$x$.}\end{split}\right.\\!\\!\\!\\!$ (1.6) $\left\\{\begin{split}&\text{Exists
a continuous
$\mathrm{C}:\overline{\Omega}\times\mathbb{R}^{N}\longrightarrow[0,\infty)$
and an $\alpha>1$:}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt&\hskip
40.0pt\left\\{\ \ \begin{split}0\leq
f(x,\eta,P)\leq\mathrm{C}(x,\eta)\big{(}|P|^{\alpha}+1\big{)},&\\\ 0\leq
g(x,\eta,P)\leq\mathrm{C}(x,\eta)\big{(}|P|^{\alpha}+1\big{)},\end{split}\right.\\\
&\text{for a.e.\ $x$ and all $(\eta,P)$.}\end{split}\right.$ (1.7) For a.e.
$x$ and all $\eta$, $f(x,\eta,\cdot)$ and $g(x,\eta,\cdot)$ are quasiconvex on
$\mathbb{R}^{N\times n}$. (1.8) $\left\\{\begin{split}&\text{{Either} $f$ {or}
$g$ is coercive, namely exist $c,C>0$ such that either}\\\ &\hskip
80.0ptf(x,\eta,P)\,\geq\,c|P|^{\alpha}-C,\\\ &\text{or}\\\ &\hskip
80.0ptg(x,\eta,P)\,\geq\,c|P|^{\alpha}-C,\\\ &\text{for a.e.\ $x$ and all
$(\eta,P)$.}\end{split}\right.$

These assumptions are relatively standard in the Calculus of Variations. We
note only that (1.7) is meant in the sense of Morrey quasiconvexity for
integral functionals (as e.g. in [14]), not in the sense of level-convexity or
of the $L^{\infty}$ notion of “BJW-quasiconvexity” of Barron-Jensen-Wang in
[8], nor in the sense of “$\mathcal{A}$-Young quasiconvexity” of Ansini-
Prinari in [1]. This stronger notion of quasiconvexity is not necessary if one
is interested in merely solving (1.4) by applying the direct method in
$W^{1,\infty}$ _without deriving any PDEs_ , and can indeed be weakened
substantially, but it is needed for Theorems 2 and 3, so we simplify the
exposition by assuming it at the outset. For our operator $\mathrm{Q}$ in
(1.2) we will assume the following:

(1.9) $\mathrm{Q}^{-1}\big{(}\\{0\\}\big{)}\ \text{is weakly closed in}\
W^{1,\overline{p}}_{0}(\Omega;\mathbb{R}^{N}).$

This is a very feeble requirement for $\mathrm{Q}$ and there exist numerous
explicit examples of interest that satisfy (1.9), see Section 4. For $1\leq
p<\infty$, we define the approximating $L^{p}$ functionals
$\mathrm{F}_{p},\mathrm{G}_{p}:W^{1,\alpha
p}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow\mathbb{R}$ by setting

(1.10) $\left\\{\ \ \ \
\begin{split}\mathrm{F}_{p}(u):&=\left(\,{-\displaystyle\int}_{\\!\\!\\!\Omega}f(\cdot,u,\mathrm{D}u)^{p}\,\mathrm{d}\mathcal{L}^{n}\\!\right)^{\\!\\!1/p},\\\
\mathrm{G}_{p}(u):&=\left(\,{-\displaystyle\int}_{\\!\\!\\!\Omega}g(\cdot,u,\mathrm{D}u)^{p}\,\mathrm{d}\mathcal{L}^{n}\\!\right)^{\\!\\!1/p}.\end{split}\right.$

For $p=\infty$, $\mathrm{F}_{\infty}$ and $\mathrm{G}_{\infty}$ are given by
(1.3). For each $p\in[1,\infty]$ and $G\geq 0$ fixed, we define the admissible
minimisation class $\mathfrak{X}^{p}({\Omega})$ on which $\mathrm{F}_{p}$ is
to be minimised, by setting

(1.11) $\mathfrak{X}^{p}(\Omega)\,:=\,\Big{\\{}v\in W^{1,\alpha
p}_{0}(\Omega;\mathbb{R}^{N})\ :\ \mathrm{G}_{p}(v)\leq G\ \text{and}\
\mathrm{Q}(v)=0\Big{\\}}.$

Our first main result concerns the existence of $\mathrm{F}_{p}$-minimisers in
$\mathfrak{X}^{p}(\Omega)$ and the existence of
$\mathrm{F}_{\infty}$-minimisers in $\mathfrak{X}^{\infty}(\Omega)$, obtained
as subsequential limits as $p\to\infty$.

######  Theorem 1 ($\mathrm{F}_{\infty}$-minimisers,
$\mathrm{F}_{p}$-minimisers & convergence as $p\to\infty$).

Suppose that the mappings $f,g$ and $\mathrm{Q}$ satisfy the assumptions (1.5)
through (1.9). If the next compatibility condition is satisfied

(1.12)
$\inf\Big{\\{}\mathrm{G}_{\infty}:\mathrm{Q}^{-1}\big{(}\\{0\\}\big{)}\cap
W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})\Big{\\}}\,<\,G,$

then, for any $p\in[\bar{p},\infty]$, the functional $\mathrm{F}_{p}$ has a
constrained minimiser $u_{p}$ in the admissible class
$\mathfrak{X}^{p}(\Omega)$, namely

(1.13)
$\mathrm{F}_{p}\big{(}u_{p})\,=\,\inf\Big{\\{}\mathrm{F}_{p}\big{(}v)\,:\
v\in\mathfrak{X}^{p}(\Omega)\Big{\\}}.$

Additionally, there exists a subsequence of indices $(p_{j})_{1}^{\infty}$
such that, the sequence of respective $\mathrm{F}_{p_{j}}$-minimisers
satisfies $u_{p}\longrightarrow u_{\infty}$ uniformly on $\overline{\Omega}$,
and $u_{p}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{\infty}$ weakly in
$W^{1,q}_{0}(\Omega;\mathbb{R}^{N})$, for all $q\in(1,\infty)$ fixed, as
$p_{j}\to\infty$. Finally, we have the convergence of minimum values
$\mathrm{F}_{p}(u_{p})\longrightarrow\mathrm{F}_{\infty}(u_{\infty})$ as
$p_{j}\to\infty$.

Note that, in view of (1.9), the compatibility condition (1.12) guarantees
that $\mathfrak{X}^{p}(\Omega)\neq\emptyset$. Given the existence of
constrained minimisers provided from Theorem 1 above, the next natural
question concerns the deduction of necessary conditions satisfied by
constrained minimisers in the form of PDEs. Firstly, we examine the case of
$p<\infty$. The nonlinear constraint expressed by the zero level-set of the
mapping $\mathrm{Q}$ in (1.11) gives rise to a Lagrange multiplier in the
Euler-Lagrange equations. This can be inferred by employing well-known results
on Lagrange multipliers, see for instance [37]. For this to be possible,
though, one needs improved regularity of the mappings $f,g$ and $\mathrm{Q}$
involved. We will suppose additionally that

(1.14) $\left\\{\ \ \begin{split}&\text{The partial derivatives
$f_{\eta},f_{P},g_{\eta},g_{P}$ of $f$ and $g$ are continuous on \ \ \ }\\\
&\text{$\smash{\overline{\Omega}\times\mathbb{R}^{N}\times\mathbb{R}^{N\times
n}}$, and for $\mathrm{C},\alpha$ as in \eqref{1.6}, we have the bounds}\\\ &\
\ \ \ \ \left\\{\ \
\begin{split}&|f_{\eta}|(x,\eta,P)+|f_{P}|(x,\eta,P)\leq\mathrm{C}(x,\eta)\big{(}|P|^{\alpha-1}+1\big{)},\\\
&|g_{\eta}|(x,\eta,P)+|g_{P}|(x,\eta,P)\leq\mathrm{C}(x,\eta)\big{(}|P|^{\alpha-1}+1\big{)},\end{split}\right.\\\
&\text{for all }(x,\eta,P).\end{split}\right.$

Further, we will assume that:

(1.15) $\left\\{\begin{split}&\text{$\mathrm{Q}$ is continuously
differentiable, and its Fr\'{e}chet derivative}\\\ &\hskip
50.0pt(\mathrm{d}\mathrm{Q})_{\overline{u}}\ :\ \
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow\textbf{E}\\\
&\text{has closed range in $\textbf{E}$, for any
$\overline{u}\in\mathrm{Q}^{-1}(\\{0\\})\subseteq
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$. }\end{split}\right.$

Recall that no regularity was assumed for $\mathrm{Q}$ to obtain the existence
of minimisers in Theorem 1. Finally, for the sake of brevity, for any $u\in
W^{1,1}_{\text{loc}}(\Omega)$, we will employ the following notation

$f[u]\,\equiv\,f(\cdot,u,\mathrm{D}u),\ \ \
g[u]\,\equiv\,g(\cdot,u,\mathrm{D}u),$

and similar notation will be used for the compositions of
$f_{\eta},f_{P},g_{\eta},g_{P}$ of $f$ and $g$ respectively, namely
$f_{\eta}[u],g_{\eta}[u]$, etc. Further, “$\cdot$” and “$:$” will denote the
standard inner products on $\mathbb{R}^{N}$ and $\mathbb{R}^{N\times n}$
respectively.

######  Theorem 2 (The equations in $L^{p}$).

Suppose we are in the setting of Theorem 1 and assumptions (1.5) through (1.9)
are satisfied. Suppose additionally that (1.14) and (1.15) are satisfied.
Then, for any $p\in(\bar{p},\infty)$, there exists Lagrange multipliers

(1.16) $\lambda_{p}\geq 0,\ \ \mu_{p}\geq 0,\ \ \
\psi_{p}\,\in\,\emph{{E}}^{*},$

where $(\emph{{E}}^{*},\|\cdot\|_{*})$ is the dual space of E, such that not
all vanish simultaneously:

(1.17) $|\lambda_{p}|+|\mu_{p}|+\|\psi_{p}\|_{*}\neq 0.$

Then, the minimiser $u_{p}\in\mathfrak{X}^{p}(\Omega)$ satisfies the equation

(1.18)
$\left\\{\begin{split}&\lambda_{p}\,{-\displaystyle\int}_{\\!\\!\\!\Omega}f[u_{p}]^{p-1}\Big{(}f_{\eta}[u_{p}]\cdot\phi\,+\,f_{P}[u_{p}]:\mathrm{D}\phi\Big{)}\,\mathrm{d}\mathcal{L}^{n}\\\
+\
&\mu_{p}\,{-\displaystyle\int}_{\\!\\!\\!\Omega}g[u_{p}]^{p-1}\Big{(}g_{\eta}[u_{p}]\cdot\phi\,+\,g_{P}[u_{p}]:\mathrm{D}\phi\Big{)}\,\mathrm{d}{\mathcal{L}}^{n}\,=\,\big{\langle}\psi_{p},(\mathrm{d}\mathrm{Q})_{u_{p}}(\phi)\big{\rangle},\end{split}\right.$

for all test mappings $\phi\in W^{1,\alpha
p}_{0}\big{(}\Omega;\mathbb{R}^{N}\big{)}$, coupled by the additional
condition

(1.19) $\mu_{p}\big{(}\mathrm{G}_{p}(u_{p})-G\big{)}=0.$

Note that condition (1.19) implies that if $\mathrm{G}_{p}(u_{p})<G$, namely
if the $L^{\infty}$-energy constraint is not realised (i.e. the minimiser
$u_{p}$ lies in the interior of the sublevel set $\\{\mathrm{G}_{p}\leq
G\\}$), then $\mu_{p}=0$ and hence the associated multiplier vanishes.

Now we consider the case of $p=\infty$. In this case, there is not a simple
analogue of the divergence structure Euler Lagrange equations. The equations
are derived by an appropriate passage to limits as $p\to\infty$ in Theorem 2.
A standard approach in Calculus of Variations in $L^{\infty}$ has been to
derive Aronsson-type PDE systems, which are non-divergence counterparts to the
Euler-Lagrange equations, as e.g. done in [2] for the case of $n=1$ and
$\mathrm{Q}\equiv 0$. However, Aronsson-type systems are always non-divergence
and far less tractable than their divergence counterparts. In fact, in the
vectorial case they have discontinuous coefficients and are fully nonlinear in
the higher order case (see for instance [13] and [21, 22, 29] for this
evolving line of development regarding the direct study of generalised
solutions to Aronsson systems).

Nevertheless, there exists an alternative approach which allows to derive
_divergence structure_ PDE systems as necessary conditions. The starting point
of this idea is based on the use of a different scaling in the Euler-Lagrange
equations in $L^{p}$ and has already born substantial fruit in [12, 23, 24,
25, 26, 27, 28]. There is however a toll to be paid for this “forcing” of
divergence structure: certain non-uniquely determined measures arise as
auxiliary parameters in the coefficients of the PDE system, which depend
nonlinearly on the minimisers. For more details on the historical origins of
this alternative approach to deriving $L^{\infty}$ equations for variational
problems we refer to [26].

Our final main result therefore concerns the satisfaction of necessary
conditions for the constrained minimiser in $L^{\infty}$ constructed in
Theorem 1. For this result we will need to impose some natural additional
hypotheses. These hypotheses, although they restrict considerably the classes
of $f,g,\mathrm{Q}$ that were utilised in order to prove existence of
minimisers, they do nonetheless include the interesting case of $F_{\infty}$
being the $L^{\infty}$ norm of the gradient. Firstly, let us introduce some
convenient notation and rewrite (1.18) in a way which will be more appropriate
for the statement and the subsequent proof. By introducing for each
$p\in(\bar{p},\infty)$ the non-negative Radon measures
$\sigma_{p},\tau_{p}\in\mathcal{M}(\overline{\Omega})$ given by

(1.20)
$\begin{split}\sigma_{p}\,&:=\,\left\\{\begin{array}[]{ll}\dfrac{1}{\mathcal{L}^{n}(\Omega)}\bigg{(}\dfrac{f[u_{p}]}{\mathrm{F}_{p}(u_{p})}\bigg{)}^{\\!p-1}\mathcal{L}^{n}\text{\LARGE$\llcorner$}_{\Omega},&\text{
if }\mathrm{F}_{p}(u_{p})>0,\\\ 0,&\text{ if
}\mathrm{F}_{p}(u_{p})=0,\end{array}\right.\\\
\tau_{p}\,&:=\,\left\\{\begin{array}[]{ll}\dfrac{1}{\mathcal{L}^{n}(\Omega)}\bigg{(}\dfrac{g[u_{p}]}{\mathrm{G}_{p}(u_{p})}\bigg{)}^{\\!p-1}\mathcal{L}^{n}\text{\LARGE$\llcorner$}_{\Omega},&\text{
if }\mathrm{G}_{p}(u_{p})>0,\\\ 0,&\text{ if
}\mathrm{G}_{p}(u_{p})=0,\end{array}\right.\end{split}$

and the scaled multipliers

(1.21)
$\begin{split}\hat{\lambda}_{p}\,&:=\,\left\\{\begin{array}[]{ll}\lambda_{p}\mathrm{F}_{p}(u_{p})^{p-1},&\text{
if }\mathrm{F}_{p}(u_{p})>0,\\\ \lambda_{p},&\text{ if
}\mathrm{F}_{p}(u_{p})=0,\end{array}\right.\\\
\hat{\mu}_{p}\,&:=\,\left\\{\begin{array}[]{ll}\mu_{p}\mathrm{G}_{p}(u_{p})^{p-1},&\text{
if }\mathrm{G}_{p}(u_{p})>0,\\\ \mu_{p},&\text{ if
}\mathrm{G}_{p}(u_{p})=0,\end{array}\right.\end{split}$

we can rewrite (1.18) as

(1.22)
$\left\\{\begin{split}&\hat{\lambda}_{p}\int_{\Omega}\Big{(}f_{\eta}[u_{p}]\cdot\phi\,+\,f_{P}[u_{p}]:\mathrm{D}\phi\Big{)}\,\mathrm{d}\sigma_{p}\\\
+\
&\hat{\mu}_{p}\int_{\Omega}\Big{(}g_{\eta}[u_{p}]\cdot\phi\,+\,g_{P}[u_{p}]:\mathrm{D}\phi\Big{)}\,\mathrm{d}\tau_{p}\,=\,\big{\langle}\psi_{p},(\mathrm{d}\mathrm{Q})_{u_{p}}(\phi)\big{\rangle},\end{split}\right.$

Further, let us set

(1.23) $R_{p}\,:=\,\hat{\lambda}_{p}+\hat{\mu}_{p}+\|\psi_{p}\|_{*}$

and note that, by virtue of Theorem 2 and Definition (1.21), we have that
$R_{p}>0$. We may then define the new rescaled multipliers

(1.24) $\Lambda_{p}\,:=\,\frac{\hat{\lambda}_{p}}{R_{p}}\in[0,1],\ \ \
\mathrm{M}_{p}\,:=\,\frac{\hat{\mu}_{p}}{R_{p}}\in[0,1],\ \ \
\Psi_{p}\,:=\,\frac{\psi_{p}}{R_{p}}\in\bar{\mathbb{B}}^{\textbf{E}^{*}}_{1}\\!(0),$

where $\bar{\mathbb{B}}^{\textbf{E}^{*}}_{1}\\!(0)$ is the closed unit ball in
$\textbf{E}^{*}$. Let us finally set

$C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}\,:=\,W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})\bigcap
C^{1}\big{(}\mathbb{R}^{n};\mathbb{R}^{N}\big{)}.$

Now we state the additional assumptions which we need to impose:

(1.25) $\textbf{E}\text{ is a separable Banach space}.$ (1.26) $\left\\{\ \
\begin{split}&\text{The restriction of the differential
$(u,v)\mapsto(\mathrm{d}\mathrm{Q})_{u}(v)$, considered as}\\\ &\hskip
60.0pt\mathrm{d}\mathrm{Q}\ :\ \ \mathrm{Q}^{-1}(\\{0\\})\times
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow\textbf{E},\\\
&\text{satisfies the following conditions:}\\\ &\text{If
}u_{m}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u\text{ in
}\mathrm{Q}^{-1}(\\{0\\})\text{ as }m\to\infty,\text{ and }\phi\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N}),\text{ then }\\\ &\hskip
60.0pt\left\\{\ \
\begin{split}&(\mathrm{d}\mathrm{Q})_{u_{m}}(u_{m})\longrightarrow(\mathrm{d}\mathrm{Q})_{u}(u),\\\
&(\mathrm{d}\mathrm{Q})_{u_{m}}(\phi)\longrightarrow(\mathrm{d}\mathrm{Q})_{u}(\phi),\end{split}\right.\\\
&\text{ as }m\to\infty.\end{split}\right.$

The above assumption requires that $\mathrm{d}\mathrm{Q}$ be weakly-strongly
continuous on the diagonal of
$\mathrm{Q}^{-1}(\\{0\\})\times\mathrm{Q}^{-1}(\\{0\\})$ and on subsets of the
form $\mathrm{Q}^{-1}(\\{0\\})\times\\{\phi\\}$, when
$W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\times
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$ is endowed with its weak topology
and E with its norm topology. We assume further that:

(1.27) $\left\\{\ \ \begin{split}&\text{(i) $g$ does not depend on $P$, namely
$g(x,\eta,P)=g(x,\eta)$,}\\\ &\text{(ii) $f$ is quadratic in $P$ and
independent of $\eta$, namely}\\\ &\text{\hskip 80.0pt
$f(x,\eta,P)=\textbf{A}(x):P\otimes P$,}\\\ &\text{for some continuous
positive symmetric fourth order tensor}\\\
&\text{$\smash{\textbf{A}:\overline{\Omega}\longrightarrow\mathbb{R}^{N\times
n}\otimes\mathbb{R}^{N\times n}}$, which satisfies}\\\ &\text{\hskip
30.0pt$\textbf{A}(x):P\otimes P>0$\ , \ \ $\textbf{A}(x):P\otimes
Q=\textbf{A}(x):Q\otimes P$,}\\\ &\text{for all $x\in\overline{\Omega}$ and
all $P,Q\in\mathbb{R}^{N\times n}\setminus\\{0\\}$.}\end{split}\right.$

The above requirements are compatible with the previous assumptions on $f$. In
fact, by [26, Lemma 4, p. 8] and our earlier assumptions, the positivity and
symmetry requirements for A are superfluous and can be deduced by merely
assuming that $f$ is quadratic in $P$ (up to a replacement of A by its
symmetrisation), but we have added them to (1.27) for simplicity. We may
finally state our last principal result.

######  Theorem 3 (The equations in $L^{\infty}$).

Suppose we are in the setting of Theorem 2 and that the same assumptions are
satisfied. Additionally we assume that (1.25) through (1.27) hold true. Then,
there exist

(1.28) $\Lambda_{\infty}\in[0,1],\ \ \ \emph{M}_{\infty}\in[0,1],\ \
\Psi_{\infty}\in\bar{\mathbb{B}}^{\emph{{E}}^{*}}_{1}\\!(0),$

which are Lagrange multipliers associated with the constrained minimisation
problem (1.4). There also exist Radon measures

(1.29) $\begin{split}\sigma_{\infty}\in\mathcal{M}({\overline{\Omega}}),\ \ \
\tau_{\infty}\in\mathcal{M}(\overline{\Omega}),\end{split}$

and a Borel measurable mapping
$\mathrm{D}u_{\infty}^{\star}:\overline{\Omega}\longrightarrow\mathbb{R}^{N\times
n}$ which is a version of $\mathrm{D}u_{\infty}\in
L^{\infty}(\Omega;\mathbb{R}^{N\times n})$, such that the minimiser
$u_{\infty}\in\mathfrak{X}^{\infty}(\Omega)$ satisfies the equation

(1.30)
$\begin{split}\Lambda_{\infty}\int_{\overline{\Omega}}f_{P}(\cdot,\mathrm{D}u_{\infty}^{\star}):\mathrm{D}\phi\,\mathrm{d}\sigma_{\infty}\,+\,\emph{M}_{\infty}\int_{\overline{\Omega}}g_{\eta}(\cdot,u_{\infty})\cdot\phi\,\mathrm{d}\tau_{\infty}\,=\,\big{\langle}\Psi_{\infty},(\mathrm{d}\mathrm{Q})_{u_{\infty}}(\phi)\big{\rangle},\end{split}$

for all test maps $\phi\in
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$, coupled by the
condition

(1.31) $\emph{M}_{\infty}\big{(}\mathrm{G}_{\infty}(u_{\infty})-G\big{)}=0.$

Additionally, the map $\mathrm{D}u_{\infty}^{\star}$ can be represented as
follows:

(1.32) $\left\\{\ \ \begin{array}[]{l}\text{For any sequence
$(v_{j})_{1}^{\infty}\subseteq
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$ satisfying}\vskip
6.0pt plus 2.0pt minus 2.0pt\vskip 6.0pt plus 2.0pt minus 2.0pt\\\ \ \ \ \ \ \
\ \ \ \
\left\\{\begin{split}&\lim_{j\to\infty}\|v_{j}-u_{\infty}\|_{(W^{1,1}_{0}\cap
L^{\infty})(\Omega)}=\,0,\\\ &\ \ \
\limsup_{j\to\infty}\mathrm{F}_{\infty}(v_{j})\leq\,\mathrm{F}_{\infty}(u_{\infty}),\end{split}\right.\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \text{exist a subsequence
$(j_{k})_{1}^{\infty}$ such that}\vskip 6.0pt plus 2.0pt minus 2.0pt\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\mathrm{D}u_{\infty}^{\star}(x)\,=\left\\{\begin{array}[]{ll}\underset{k\to\infty}{\lim}\,\mathrm{D}v_{j_{k}}(x),&\text{if
the limit exists},\\\ 0,&\text{otherwise}.\end{array}\right.\vskip 3.0pt plus
1.0pt minus 1.0pt\end{array}\right.$

(Such an explicit sequence $(v_{j})_{1}^{\infty}$ is constructed in the
proof.) Finally, the Lagrange multipliers
$\Lambda_{\infty},\emph{M}_{\infty},\Psi_{\infty}$ and the measures
$\sigma_{\infty},\tau_{\infty}$ can be approximated as follows:

(1.33) $\left\\{\ \
\begin{split}\Psi_{p}\,&\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\,\Psi_{\infty},\
\ \text{ in }\bar{\mathbb{B}}^{\emph{{E}}^{*}}_{1}\\!(0),\\\
\Lambda_{p}\,&\longrightarrow\,\Lambda_{\infty},\ \,\text{ in }[0,1],\\\
\mathrm{M}_{p}\,&\longrightarrow\,\mathrm{M}_{\infty},\ \text{ in
}[0,1],\end{split}\right.$

and

(1.34) $\left\\{\ \
\begin{split}\sigma_{p}\,\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\,\sigma_{\infty},\
\ &\text{ in }\mathcal{M}(\overline{\Omega}),\\\
\tau_{p}\,\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\,\tau_{\infty},\
\ \ &\text{ in }\mathcal{M}(\overline{\Omega}),\end{split}\right.$

along a subsequence $p_{j}\to\infty$.

The weak interpretation of (1.30) is

$-\Lambda_{\infty}\mathrm{div}(f_{P}(\cdot,\mathrm{D}u_{\infty}^{\star})\sigma_{\infty}\big{)}\,+\,\mathrm{M}_{\infty}\,g_{\eta}(\cdot,u_{\infty})\tau_{\infty}\,=\,\big{\langle}\Psi_{\infty},(\mathrm{d}\mathrm{Q})_{u_{\infty}}\big{\rangle},$

in
$\big{(}C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}\big{)}^{*}$,
up to the identifications

$\big{\langle}\Psi_{\infty},(\mathrm{d}\mathrm{Q})_{u_{\infty}}\big{\rangle}\equiv\big{\langle}\Psi_{\infty},(\mathrm{d}\mathrm{Q})_{u_{\infty}}(\cdot)\big{\rangle},\
\ \ g_{\eta}\equiv g_{\eta}\cdot(\cdot),\ \ \ f_{P}\equiv(\cdot)\cdot f_{P}.$

Note that in Theorem 3, the equations obtained depend on certain measures not
a priori known explicitly. Therefore, their significance is understood to be
largely theoretical, rather than computational. For the proof of this result,
we will utilise some machinery developed in the recent paper [26] for some
related work on generalised $\infty$-eigenvalue problems. The main points of
this approach are recalled in the course of the proof, for the convenience of
the reader.

We conclude this lengthy introduction with some comments concerning the
composition of this paper. In Sections 2 and 3 we establish our main results,
Theorems 1, 2 and 3. In Section 4 we provide a rather detailed list of
explicit large classes of nonlinear operators $\mathrm{Q}$ to which our
results apply.

##  2\. Minimisers of $L^{p}$ problems and convergence as $p\to\infty$

In this section we demonstrate Theorem 1. The proof is a consequence of the
next two propositions, utilising the direct method of the Calculus of
Variations.

######  Proposition 4.

In the setting of Theorem 1 and under the same assumptions, for any
$p\in[\bar{p},\infty)$, the functional $\mathrm{F}_{p}$ has a constrained
minimiser $u_{p}\in\mathfrak{X}^{p}(\Omega)$, as claimed in (1.13).

Proof of Proposition 4. Fix $p\geq\bar{p}>n$. We begin by illustrating that
$\mathfrak{X}^{p}(\Omega)\neq\emptyset$. First note that by the compatibility
condition (1.12), the finiteness of the infimum implies that
$\mathrm{Q}^{-1}\big{(}\\{0\\}\big{)}\cap
W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})\neq\emptyset$. Further, there exists
$u_{0}\in W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})$ with $\mathrm{Q}(u_{0})=0$
such that $\mathrm{G}_{\infty}(u_{0})<G$. Hence, by Hölder inequality we have

$\begin{split}\mathrm{G}_{p}(u_{0})\,=\,\left(\,{-\displaystyle\int}_{\\!\\!\\!\Omega}g(\cdot,u_{0},\mathrm{D}u_{0})^{p}\,\mathrm{d}\mathcal{L}^{n}\\!\right)^{\\!\\!1/p}\leq\,\big{\|}g(\cdot,u_{0},\mathrm{D}u_{0})\big{\|}_{L^{\infty}(\Omega)}\,=\,\mathrm{G}_{\infty}(u_{0})\,<\,G.\end{split}$

Consequently, in view of (1.11), both constraints are satisfied by $u_{0}$,
hence $u_{0}\in\mathfrak{X}^{p}(\Omega)\neq\emptyset$. Next, note that $f^{p}$
is a (Morrey) quasiconvex function. To see this, let $h:\mathbb{R}^{N\times
n}\longrightarrow\mathbb{R}$ be an arbitrary quasiconvex function, in our case
we will take $h(P)=f(x,\eta,P)$ for fixed $(x,\eta)$. Then, by assumption
(1.7), for any $\phi\in W^{1,\infty}_{0}(U;\mathbb{R}^{N})$ with
$U\Subset\mathbb{R}^{n}$ open and $P\in\mathbb{R}^{N\times n}$ fixed,

$h(P)\leq-\displaystyle\int_{U}h(P+\mathrm{D}\phi)\
\mathrm{d}\mathcal{L}^{n}.$

Hence, by Jensen’s inequality and the convexity of $t\mapsto t^{p}$, we
conclude that

$h(P)^{p}\leq\bigg{(}\,{-\displaystyle\int}_{\\!\\!\\!U}h(P+\mathrm{D}\phi)\,\mathrm{d}\mathcal{L}^{n}\\!\bigg{)}^{\\!\\!p}\,\leq\,{-\displaystyle\int}_{\\!\\!\\!U}h(P+\mathrm{D}\phi)^{p}\,\mathrm{d}\mathcal{L}^{n}.$

We now proceed to bound $f^{p}$. By (1.6), we estimate

$\begin{split}&0\leq
f(x,\eta,P)^{p}\leq\mathrm{C}(x,\eta)^{p}\big{(}1+|P|^{\alpha}\big{)}^{p}\leq
2^{p-1}\mathrm{C}(x,\eta)^{p}\big{(}1+|P|^{\alpha p}\big{)}.\end{split}$

By standard results (see [14]), $\mathrm{F}_{p}$ is weakly lower
semicontinuous on $W^{1,\alpha p}_{0}(\Omega;\mathbb{R}^{N})$. Let
$(u^{(i)})_{1}^{\infty}\subseteq\mathfrak{X}^{p}(\Omega)$ denote a minimising
sequence. By virtue of (1.6) we have $f\geq 0$, hence clearly
$\inf_{i\in\mathbb{N}}\mathrm{F}_{p}(u^{(i)})\geq 0$. We now show that the
infimum is finite. To this aim, by (1.6) we estimate

$\begin{split}\inf_{i\in\mathbb{N}}\mathrm{F}_{p}(u^{(i)})&\leq\mathrm{F}_{p}(u_{0})\\\
&=\bigg{(}\,{-\displaystyle\int}_{\\!\\!\\!\Omega}f(\cdot,u_{0},\mathrm{D}u_{0})^{p}\
\mathrm{d}{\mathcal{L}}^{n}\bigg{)}^{\\!\\!1/p}\\\
&\leq\bigg{(}\,{-\displaystyle\int}_{\\!\\!\\!\Omega}\Big{(}\mathrm{C}(\cdot,u_{0})(1+|\mathrm{D}u_{0}|^{\alpha})\Big{)}^{p}\,\mathrm{d}{\mathcal{L}}^{n}\bigg{)}^{\\!\\!1/p},\end{split}$

which yields

$\begin{split}\inf_{i\in\mathbb{N}}\mathrm{F}_{p}(u^{(i)})&\leq\Big{\|}\mathrm{C}(\cdot,u_{0})\big{(}1+|\mathrm{D}u_{0}|^{\alpha}\big{)}\Big{\|}_{L^{\infty}(\Omega)}\\\
&\leq\|\mathrm{C}(\cdot,u_{0})\|_{L^{\infty}(\Omega)}\|(1+|\mathrm{D}u_{0}|^{\alpha})\|_{L^{\infty}(\Omega)}\\\
&\leq\|\mathrm{C}(\cdot,u_{0})\|_{L^{\infty}(\Omega)}\big{(}1+\|\mathrm{D}u_{0}\|^{\alpha}_{L^{\infty}(\Omega)}\big{)}\\\
&<\infty.\end{split}$

Hence, the infimum is indeed finite. Now we show that under assumption (1.8),
the minimising sequence $(u^{(i)})_{1}^{\infty}$ is bounded in $W^{1,\alpha
p}(\Omega;\mathbb{R}^{N})$. Let $h$ symbolise either $f$ or $g$, whichever is
coercive. Hence, since $h$ is coercive, we have

${-\displaystyle\int}_{\\!\\!\\!\Omega}\big{|}h(\cdot,u^{(i)},\mathrm{D}u^{(i)})+C\big{|}^{p}\,\mathrm{d}\mathcal{L}^{n}\,\geq\,c\
{-\displaystyle\int}_{\\!\\!\\!\Omega}|\mathrm{D}u^{(i)}|^{\alpha p}\
\mathrm{d}\mathcal{L}^{n}.$

By using the Poincaré and Hölder inequalities, we infer that

$\begin{split}&C\,+\,\bigg{(}\,{-\displaystyle\int}_{\\!\\!\\!\Omega}|h(\cdot,u^{(i)},\mathrm{D}u^{(i)})|^{p}\
\mathrm{d}\mathcal{L}^{n}\bigg{)}^{\\!\\!\frac{1}{p}}\,\geq\,c^{\prime}\,\|u^{(i)}\|^{\alpha}_{W^{1,\alpha
p}(\Omega)}\end{split}$

for some new constant $c^{\prime}>0$ which is independent of $i\in\mathbb{N}$.
If $h=f$, then by the previously derived estimates we have the uniform bound

$\|u^{(i)}\|^{\alpha}_{W^{1,\alpha
p}(\Omega)}\,\leq\,\frac{1}{c^{\prime}}\Big{(}C\,+\,\mathrm{F}_{p}(u^{(i)})\Big{)}\,\leq\,\frac{C}{c^{\prime}}\,+\,\frac{1}{c^{\prime}}\|\mathrm{C}(\cdot,u_{0})\|_{L^{\infty}(\Omega)}\big{(}1+\|\mathrm{D}u_{0}\|^{\alpha}_{L^{\infty}(\Omega)}\big{)},$

and if $h=g$, then by the isosupremic constraint we have the uniform bound

$\|u^{(i)}\|^{\alpha}_{W^{1,\alpha
p}(\Omega)}\,\leq\,\frac{1}{c^{\prime}}\Big{(}C\,+\,\mathrm{G}_{p}(u^{(i)})\Big{)}\,\leq\,\frac{C\,+\,G}{c^{\prime}}.$

In either case, we have that $(u^{(i)})_{1}^{\infty}$ is weakly precompact in
$W^{1,\alpha p}_{0}(\Omega;\mathbb{R}^{N})$. By passing to a subsequence if
necessary, standard strong and weak compactness arguments imply that there
exists a map $u_{p}\in W^{1,\alpha p}_{0}(\Omega;\mathbb{R}^{N})$ and a
subsequence denoted again by $(u^{(i)})_{1}^{\infty}$ such that

$\left\\{\ \begin{split}&u^{(i)}\longrightarrow u_{p},\ \ \ \ \ \ \,\text{in}\
L^{\alpha p}(\Omega;\mathbb{R}^{N}),\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
&\mathrm{D}u^{(i)}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup\mathrm{D}u_{p},\
\ \text{in}\ L^{\alpha p}(\Omega;\mathbb{R}^{N\times n}),\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ \end{split}\right.$

as $i\to\infty$. Further, since $p>n$, by the Morrey estimate we have that
$(u^{(i)})_{1}^{\infty}$ is also bounded in
$C^{0,\gamma}(\overline{\Omega},\mathbb{R}^{N})$ for $\gamma<1-n/(\alpha p)$.
By the compact embedding of Hölder spaces, we conclude that

$u^{(i)}\longrightarrow u_{p}\ \ \ \text{in}\
C(\overline{\Omega};\mathbb{R}^{N}),$

as $i\to\infty$. It remains to show that $\mathfrak{X}^{p}(\Omega)$ is weakly
closed in $W^{1,\alpha p}_{0}(\Omega;\mathbb{R}^{N})$. To this end, we need to
show that the constraints determine a weakly closed subset of $W^{1,\alpha
p}_{0}(\Omega;\mathbb{R}^{N})$. Firstly note that by assumptions (1.5)-(1.7),
$\mathrm{G}_{p}$ is a weakly lower semi-continuous functional on $W^{1,\alpha
p}_{0}(\Omega;\mathbb{R}^{N})$. This can be seen by an analogous argument to
that used to show that $\mathrm{F}_{p}$ is weakly lower-continuous. Since
$(u^{(i)})_{1}^{\infty}\subseteq\mathfrak{X}^{p}(\Omega)$, we have
$\mathrm{G}_{p}(u^{(i)})\leq G$ for all $i\in\mathbb{N}$. Therefore, since
$u^{(i)}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{p}$ in $W^{1,\alpha
p}_{0}(\Omega;\mathbb{R}^{N})$ as $i\to\infty$ and $\mathrm{G}_{p}$ is weakly
lower-continuous, we infer that

$\begin{split}\mathrm{G}_{p}(u_{p})\,\leq\,\liminf_{i\to\infty}\mathrm{G}_{p}(u^{(i)})\,\leq\,G.\end{split}$

Hence, $\mathrm{G}_{p}(u_{p})\leq G$. Recall now that, in view of assumption
(1.9), $\mathrm{Q}^{-1}(\\{0\\})$ is a weakly closed subset of
$W^{1,\bar{p}}(\Omega;\mathbb{R}^{N})$, where $\bar{p}>n$. We now show that
$\mathrm{Q}^{-1}(\\{0\\})$ is also a weakly closed subset of $W^{1,\alpha
p}_{0}(\Omega;\mathbb{R}^{N})$, which will complete the claim that
$\mathfrak{X}^{p}(\Omega)$ is weakly closed in $W^{1,\alpha
p}_{0}(\Omega;\mathbb{R}^{N})$. Note first that since
$(u^{(i)})_{1}^{\infty}\subseteq\mathfrak{X}^{p}(\Omega)$, we have that
$\mathrm{Q}(u^{(i)})=0$ for all $i\in\mathbb{N}$. Recalling that
$u^{(i)}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{p}$ in $W^{1,\alpha
p}_{0}(\Omega;\mathbb{R}^{N})$ as $i\to\infty$ and that $\alpha p>\bar{p}$
because by assumption $\alpha>1$, we deduce that
$u^{(i)}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{p}$ in
$W^{1,\bar{p}}(\Omega;\mathbb{R}^{N})$ as well, as $i\to\infty$. Since
$\mathrm{Q}^{-1}(\\{0\\})$ is weakly closed in
$W^{1,\bar{p}}(\Omega;\mathbb{R}^{N})$, we infer that $\mathrm{Q}(u_{p})=0$,
as desired. Thus, for each $p\in[\bar{p},\infty)$,
$u_{p}\in\mathfrak{X}^{p}(\Omega)$, and

$\mathrm{F}_{p}(u_{p})\,\leq\,\liminf_{i\to\infty}\mathrm{F}_{p}(u^{(i)})\,=\,\inf\big{\\{}\mathrm{F}_{p}\,:\
\mathfrak{X}^{p}(\Omega)\big{\\}}.$

The proposition ensues. ∎

Our next result below establishes the existence of constrained minimisers for
$\mathrm{F}_{\infty}$ and the approximation by minimisers of the
$\mathrm{F}_{p}$ functionals as $p\to\infty$, therefore completing the proof
of Theorem 1.

######  Proposition 5.

In the setting of Theorem 1 and under the same assumptions, the functional
$\mathrm{F}_{\infty}$ has a constrained minimiser
$u_{\infty}\in\mathfrak{X}^{\infty}(\Omega)$, as claimed in (1.13).
Additionally, the claimed modes of convergence hold true for a subsequence of
minimisers $(u_{p_{j}})_{j=1}^{\infty}$ as $j\to\infty$.

Proof of Proposition 5. We continue from the proof of Proposition 4. The
element $u_{0}\in\mathfrak{X}^{p}(\Omega)$ provides an energy bound uniform in
$p$, and also $u_{0}\in\mathfrak{X}^{\infty}(\Omega)$. Fix $p,q>1$ with $p\geq
q\geq\bar{p}$. By the Hölder inequality, minimality and the definition of the
constrained class, we have the estimates

$\left\\{\ \
\begin{split}\mathrm{F}_{q}(u_{p})\,&\leq\,\mathrm{F}_{p}(u_{p})\,\leq\,\mathrm{F}_{p}(u_{0})\,\leq\,\|\mathrm{C}(\cdot,u_{0})\|_{L^{\infty}(\Omega)}\big{(}1+\|\mathrm{D}u_{0}\|^{\alpha}_{L^{\infty}(\Omega)}\big{)},\\\
\mathrm{G}_{q}(u_{p})\,&\leq\,\mathrm{G}_{p}(u_{p})\leq G,\end{split}\right.$

with right hand side bounds which are uniform in $p,q$. We now argue in a
similar fashion to that used in the proof of Proposition 4. We first show that
under assumption (1.8), the family of minimisers $(u_{p})_{p\geq\bar{p}}$ is
bounded in $W^{1,q}(\Omega;\mathbb{R}^{N})$, for any $q\in(1,\infty)$ fixed.
Let $h$ symbolise either $f$ or $g$, whichever is coercive. We then we have
that

${-\displaystyle\int}_{\\!\\!\\!\Omega}\big{|}h(\cdot,u_{p},\mathrm{D}u_{p})+C\big{|}^{q}\,\mathrm{d}\mathcal{L}^{n}\,\geq\,c\
{-\displaystyle\int}_{\\!\\!\\!\Omega}|\mathrm{D}u_{p}|^{\alpha q}\
\mathrm{d}\mathcal{L}^{n}.$

Since $\alpha>1$, by the Hölder inequality we infer

$C\,+\,\bigg{(}\,{-\displaystyle\int}_{\\!\\!\\!\Omega}|h(\cdot,u_{p},\mathrm{D}u_{p})|^{q}\
\mathrm{d}\mathcal{L}^{n}\bigg{)}^{\\!\\!\frac{1}{q}}\,\geq\,c^{\frac{1}{q}}\,\|\mathrm{D}u_{p}\|^{\alpha}_{L^{q}(\Omega)},$

for the constants $c,C>0$ of (1.8) (which are independent of $p$ and $q$). If
$h=f$, by applying our earlier estimates we deduce the uniform bound

$\|\mathrm{D}u_{p}\|^{\alpha}_{L^{q}(\Omega)}\,\leq\,\frac{1}{c}\Big{(}C\,+\,\mathrm{F}_{q}(u_{p})\Big{)}\,\leq\,\frac{C}{c}\,+\,\frac{1}{c}\|\mathrm{C}(\cdot,u_{0})\|_{L^{\infty}(\Omega)}\big{(}1+\|\mathrm{D}u_{0}\|^{\alpha}_{L^{\infty}(\Omega)}\big{)}.$

If $h=g$, then again as in our earlier estimates we have the uniform bound

$\|\mathrm{D}u_{p}\|^{\alpha}_{L^{q}(\Omega)}\,\leq\,\frac{1}{c}\Big{(}C\,+\,\mathrm{G}_{q}(u_{p})\Big{)}\,\leq\,\frac{C\,+\,G}{c}.$

In either case, we see that under (1.8), our estimates above imply that

$\|\mathrm{D}u_{p}\|_{L^{q}(\Omega)}\,\leq\,K,$

for some constant $K>0$ independent of $p,q$. Further, by the Poincaré
inequality, we deduce that

$\|u_{p}\|_{W^{1,q}(\Omega)}\,\leq\,K\big{(}1+C(q)\big{)},$

where $C(q)$ is the constant of the Poincaré inequality in
$W^{1,q}_{0}(\Omega;\mathbb{R}^{N})$. Hence, the sequence of minimisers
$(u_{p})_{p\geq\bar{p}}$ is bounded in $W^{1,q}_{0}(\Omega;\mathbb{R}^{N})$
for any fixed $q\in(1,\infty)$, and therefore it is weakly precompact in this
scale of spaces. By a standard diagonal argument, there exists a sequence
$(p_{j})_{1}^{\infty}$ and a mapping

$u_{\infty}\,\in\,\bigcap_{\bar{p}<q<\infty}W^{1,q}_{0}(\Omega;\mathbb{R}^{n}),$

such that $u_{p}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{\infty}$ in
$W^{1,q}_{0}(\Omega;\mathbb{R}^{n})$ as $p_{j}\to\infty$, for any fixed
$q\in(\bar{p},\infty)$. By standard compactness arguments in Sobolev and
Hölder spaces, we infer that

$\left\\{\ \begin{split}&u_{p}\longrightarrow u_{\infty},\ \ \ \ \ \
\,\text{in}\ C\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)},\\\
&\mathrm{D}u_{p}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup\mathrm{D}u_{\infty},\
\ \text{in}\ L^{q}(\Omega;\mathbb{R}^{N\times n}),\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\ \end{split}\right.$

as $p_{j}\to\infty$, for any $q\in(\bar{p},\infty)$. We will now show that
$u_{\infty}\in\mathfrak{X}^{\infty}(\Omega)$. In view of (1.11), we need to
show that $u_{\infty}\in W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})$ and that
$\mathrm{G}_{\infty}(u_{\infty})\leq G$ and also $\mathrm{Q}(u_{\infty})=0$.
By the weak lower semi-continuity of the $L^{q}$ norm for $q\geq\bar{p}$
fixed, we have

$\|\mathrm{D}u_{\infty}\|_{L^{q}(\Omega)}\,\leq\,\liminf_{p_{j}\to\infty}\|\mathrm{D}u_{p}\|_{L^{q}(\Omega)}\,\leq\,K.$

By letting $q\to\infty$, this yields that $\mathrm{D}u_{\infty}\in
L^{\infty}(\Omega;\mathbb{R}^{N})$. By the Poincaré inequality in
$W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})$, we infer that $u_{\infty}\in
W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})$. Next, since
$\mathrm{G}_{p}(u_{p})\leq G$ for all $p\in(\bar{p},\infty)$, via the Hölder
inequality and weak lower semi-continuity, we have

$\begin{split}\mathrm{G}_{\infty}(u_{\infty})=\lim_{q\to\infty}\mathrm{G}_{q}(u_{\infty})&\leq\liminf_{q\to\infty}\Big{(}\liminf_{p_{j}\to\infty}\mathrm{G}_{q}(u_{p})\Big{)}\,\leq\,\liminf_{p_{j}\to\infty}\mathrm{G}_{p}(u_{p})\,\leq\,G,\end{split}$

yielding that indeed $\mathrm{G}_{\infty}(u_{\infty})\leq G$. We now show that
$\mathrm{Q}(u_{\infty})=0$. We have already shown in Proposition 4 that
$\mathrm{Q}^{-1}(\\{0\\})$ is a weakly closed subset of
$W^{1,q}_{0}(\Omega;\mathbb{R}^{N})$ for any $q\in(\bar{p},\infty)$. Since
$\mathrm{Q}(u_{p})=0$ for all $p\geq\bar{p}$ and
$u_{p}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u_{\infty}$ in
$W^{1,q}_{0}(\Omega;\mathbb{R}^{N})$ as $p_{j}\to\infty$, we deduce that
$\mathrm{Q}(u_{\infty})=0$, as desired.

It remains to show that $u_{\infty}$ is indeed a minimiser of
$\mathrm{F}_{\infty}$ in $\mathfrak{X}^{\infty}(\Omega)$, and additionally
that the energies converge. Fix an arbitrary
$u\in\mathfrak{X}^{\infty}(\Omega)$. By minimality and by noting that
$\mathfrak{X}^{\infty}(\Omega)\subseteq\mathfrak{X}^{p}(\Omega)$ for any
$p\in[\bar{p},\infty]$, we have the estimate

$\begin{split}\mathrm{F}_{\infty}(u_{\infty})\,&=\,\smash{\lim_{q\to\infty}\mathrm{F}_{q}(u_{\infty})}\\\
&\leq\,\liminf_{q\to\infty}\Big{(}\liminf_{p_{j}\to\infty}\mathrm{F}_{q}(u_{p})\Big{)}\\\
&\leq\,\liminf_{p_{j}\to\infty}\mathrm{F}_{p}(u_{p})\\\
&\leq\,\limsup_{p_{j}\to\infty}\mathrm{F}_{p}(u_{p})\\\
&\leq\,\limsup_{p_{j}\to\infty}\mathrm{F}_{p}(u)\\\
&=\,\mathrm{F}_{\infty}(u),\end{split}$

for any $u\in\mathfrak{X}^{\infty}(\Omega)$. Hence $u_{\infty}$ is a minimiser
of $\mathrm{F}_{\infty}$ over $\mathfrak{X}^{\infty}(\Omega)$ and (1.4)
follows. The choice $u=u_{\infty}$ in the above implies
$\mathrm{F}_{p}(u_{p})\longrightarrow\mathrm{F}_{\infty}(u_{\infty})$, as
$p_{j}\to\infty$. This completes the proof of the proposition and therefore of
Theorem 1. ∎

##  3\. The equations for constrained minimisers in $L^{p}$ and in
$L^{\infty}$

In this section we establish the proofs of Theorem 2 and Theorem 3. The proof
of the former is a relatively simple consequence of deep results in the
generalised Kuhn-Tucker theory from [37], whilst the proof of the latter is
based on applying an appropriate liming process to the former result through
compactness estimates.

Proof of Theorem 2. Fix $p\in[\bar{p},\infty)$. We begin with the simplifying
observation that, the minimisation problem (1.13) can be rewritten as

$\frac{1}{p}\mathrm{F}_{p}(u_{p})^{p}\,=\,\bigg{\\{}\frac{1}{p}\mathrm{F}_{p}(u)^{p}\
:\ u\in W^{1,\alpha p}_{0}(\Omega;\mathbb{R}^{N}),\
\frac{1}{p}\mathrm{G}_{p}(u)^{p}-\frac{G^{p}}{p}\leq 0\ \&\
\mathrm{Q}(u)=0\bigg{\\}}.$

This reformulation is a labour-saving device, drastically shortening the proof
of this result. In view of assumption (1.14), first we will show that the
following functionals are Fréchet differentiable

$\begin{split}\frac{1}{p}(\mathrm{F}_{p})^{p}\ &:\ \ W^{1,\alpha
p}_{0}\big{(}\Omega;\mathbb{R}^{N}\big{)}\longrightarrow\mathbb{R},\\\
\frac{1}{p}(\mathrm{G}_{p})^{p}-\frac{G^{p}}{p}\ &:\ \ W^{1,\alpha
p}_{0}\big{(}\Omega;\mathbb{R}^{N}\big{)}\longrightarrow\mathbb{R}.\end{split}$

A direct computation gives the next formal expressions for their Gateaux
derivatives

$\begin{split}\bigg{(}\mathrm{d}\Big{[}\frac{1}{p}(\mathrm{F}_{p})^{p}\Big{]}\bigg{)}_{\\!\\!u}(v)\,&=\,{-\displaystyle\int}_{\\!\\!\\!\Omega}f[u]^{p-1}\big{(}f_{\eta}[u]\cdot
v+f_{P}[u]:\mathrm{D}v\big{)}\,\mathrm{d}{\mathcal{L}}^{n},\\\
\bigg{(}\mathrm{d}\Big{[}\frac{1}{p}(\mathrm{G}_{p})^{p}-\frac{G^{p}}{p}\Big{]}\bigg{)}_{\\!\\!u}(v)\,&=\,{-\displaystyle\int}_{\\!\\!\\!\Omega}g[u]^{p-1}\big{(}g_{\eta}[u]\cdot
v+g_{P}[u]:\mathrm{D}v\big{)}\,\mathrm{d}{\mathcal{L}}^{n},\end{split}$

for all $u,v\in W^{1,\alpha p}_{0}(\Omega;\mathbb{R}^{N})$. We will now show
the above formal expressions indeed define Fréchet derivatives, by employing
relatively standard estimates through the Hölder inequality. We argue only for
$\frac{1}{p}(\mathrm{F}_{p})^{p}$, as the estimates for
$\frac{1}{p}\big{(}(\mathrm{G}_{p})^{p}-{G^{p}}\big{)}$ are identical. Since
$\alpha>1$ and $p\geq\bar{p}>n$, by Morrey’s estimate we have

$\begin{split}\bigg{|}\,{-\displaystyle\int}_{\\!\\!\\!\Omega}f[u]&{}^{p-1}\big{(}f_{\eta}[u]\cdot
v+f_{P}[u]:\mathrm{D}v\big{)}\,\mathrm{d}{\mathcal{L}}^{n}\bigg{|}\\\
&\leq\,{-\displaystyle\int}_{\\!\\!\\!\Omega}|f[u]|^{p-1}\Big{(}|f_{\eta}[u]||v|+|f_{P}[u]||\mathrm{D}v|\Big{)}\,\mathrm{d}{\mathcal{L}}^{n}\\\
&\leq\,{-\displaystyle\int}_{\\!\\!\\!\Omega}C(\cdot,u)^{p}\big{(}1+|\mathrm{D}u|^{\alpha}\big{)}^{p-1}\big{(}1+|\mathrm{D}u|^{\alpha-1}\big{)}(|v|+|\mathrm{D}v|)\,\mathrm{d}{\mathcal{L}}^{n}\\\
&\leq\,2^{p}\big{\|}|v|C(\cdot,u)^{p}\big{\|}_{L^{\infty}(\Omega)}\,{-\displaystyle\int}_{\\!\\!\\!\Omega}\Big{(}1+|\mathrm{D}u|^{\alpha-1}+|\mathrm{D}u|^{\alpha
p-\alpha}+|\mathrm{D}u|^{\alpha p-1}\Big{)}\,\mathrm{d}{\mathcal{L}}^{n}\\\ &\
\
+\,2^{p}\big{\|}C(\cdot,u)^{p}\big{\|}_{L^{\infty}(\Omega)}\,{-\displaystyle\int}_{\\!\\!\\!\Omega}\Big{(}1+|\mathrm{D}u|^{\alpha-1}+|\mathrm{D}u|^{\alpha
p-\alpha}+|\mathrm{D}u|^{\alpha
p-1}\Big{)}|\mathrm{D}v|\,\mathrm{d}{\mathcal{L}}^{n}\\\
&\leq\,2^{p}\big{\|}|v|C(\cdot,u)^{p}\big{\|}_{L^{\infty}(\Omega)}\Big{(}1+\|\mathrm{D}u\|_{L^{\alpha
p}(\Omega)}^{\alpha-1}+\|\mathrm{D}u\|_{L^{\alpha p}(\Omega)}^{\alpha
p-\alpha}+\|\mathrm{D}u\|_{L^{\alpha p}(\Omega)}^{\alpha p-1}\Big{)}\\\ &\ \
+\,2^{p}\big{\|}C(\cdot,u)^{p}\big{\|}_{L^{\infty}(\Omega)}\bigg{(}\|\mathrm{D}v\|_{L^{\alpha
p}(\Omega)}\\\ &\ \ +\sum_{t\in\\{\alpha-1,\alpha p-\alpha,\alpha
p-1\\}}\|\mathrm{D}u\|_{L^{\frac{t\alpha p}{\alpha
p-1}}(\Omega)}^{t}\|\mathrm{D}v\|_{L^{\alpha p}(\Omega)}\bigg{)}.\end{split}$

We now show that the equations that the constrained minimiser satisfies take
the form as given in (1.18) and (1.19). Given the Fréchet derivatives and our
assumption (1.15) on the range of $\mathrm{d}\mathrm{Q}$, we can invoke the
generalised Kuhn-Tucker theory. By applying [37, Theorem 48.B, p.p. 416-417]
with (in the book’s notation)

$\begin{split}F_{0}&:=\frac{1}{p}(\mathrm{F}_{p})^{p},\ \ \
F_{1}:=\frac{1}{p}(\mathrm{G}_{p})^{p}-\frac{G^{p}}{p},\ \ \
F_{3}:=\mathrm{Q},\\\ X&=U=N_{2}:=W^{1,\alpha
p}_{0}\big{(}\Omega;\mathbb{R}^{N}\big{)},\ \ \ Y:=\textbf{E},\ \ \
n=2,\end{split}$

and by noting that $N_{2}$ herein is the entire (vector) space $W^{1,\alpha
p}_{0}\big{(}\Omega;\mathbb{R}^{N}\big{)}$, we readily infer the claims made
in (1.16)-(1.19). The result ensues. ∎

Now we establish our last main result.

Proof of Theorem 3. The proof is divided into several steps.

Step 1. We first confirm that the measures defined by (1.20) are indeed
finite, and show that their total variations are bounded uniformly in
$p\in(\bar{p},\infty)$. This will imply the convergence modes of (1.34) for
some limiting $\mu_{\infty},\nu_{\infty}\in\mathcal{M}(\overline{\Omega})$
along a subsequence $(p_{j})_{1}^{\infty}$ as $j\to\infty$, as a consequence
of the sequential weak* precompactness of bounded sets in the space of Radon
measures. Indeed, if $\mathrm{F}_{p}(u_{p})>0$, then since $f\geq 0$ we have

$\begin{split}\|\sigma_{p}\|(\overline{\Omega})\,&=\,\sigma_{p}(\overline{\Omega})\\\
&=\,\frac{1}{\mathcal{L}^{n}(\Omega)}\int_{\Omega}\frac{f(\cdot,\mathrm{D}u_{p})^{p-1}}{\mathrm{F}_{p}(u_{p})^{p-1}}\,\mathrm{d}\mathcal{L}^{n}\\\
&=\,\frac{1}{\mathrm{F}_{p}(u_{p})^{p-1}}\
{-\displaystyle\int_{\Omega}}f(\cdot,\mathrm{D}u_{p})^{p-1}\,\mathrm{d}\mathcal{L}^{n}\\\
&\leq\,\frac{1}{\mathrm{F}_{p}(u_{p})^{p-1}}\bigg{(}\
{-\displaystyle\int_{\Omega}}f(\cdot,\mathrm{D}u_{p})^{p}\,\mathrm{d}\mathcal{L}^{n}\bigg{)}^{\\!\\!\frac{p-1}{p}}\\\
&=\,1,\phantom{\Big{|}}\end{split}$

whilst if $\mathrm{F}_{p}(u_{p})=0$, then trivially
$\|\sigma_{p}\|(\overline{\Omega})=0$. In both cases,
$\|\sigma_{p}\|(\overline{\Omega})\leq 1$ for all $p\in(\bar{p},\infty)$. The
estimate for $\|\tau_{p}\|(\overline{\Omega})$ is completely analogous,
yielding $\|\tau_{p}\|(\overline{\Omega})\leq 1$ for all
$p\in(\bar{p},\infty)$.

Step 2. By using assumption (1.27) and definition (1.20), we have the
following differential identity: for any fixed $v\in
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$ and any
$p\in(\bar{p},\infty)$ we have

$\begin{split}\int_{\Omega}f(\cdot,\mathrm{D}v-\mathrm{D}u_{p})\,\mathrm{d}\sigma_{p}\,&=\,\int_{\Omega}f(\cdot,\mathrm{D}v)\,\mathrm{d}\sigma_{p}-\int_{\Omega}f(\cdot,\mathrm{D}u_{p})\,\mathrm{d}\sigma_{p}\\\
&\ \
+\,\int_{\Omega}f_{P}(\cdot,\mathrm{D}u_{p}):(\mathrm{D}u_{p}-\mathrm{D}v)\,\mathrm{d}\sigma_{p}.\end{split}$

Indeed, by using that
$f_{P}(x,P)\,=\,\textbf{A}(x)\\!:\\!\big{(}(\cdot)\otimes
P+P\otimes(\cdot)\big{)}$, we may compute

$\begin{split}\int_{\Omega}f(\cdot,\mathrm{D}v&-\mathrm{D}u_{p})\,\mathrm{d}\sigma_{p}\,=\,\int_{\Omega}\textbf{A}:(\mathrm{D}v-\mathrm{D}u_{p})\otimes(\mathrm{D}v-\mathrm{D}u_{p})\,\mathrm{d}\sigma_{p}\\\
&=\,\int_{\Omega}\textbf{A}:\mathrm{D}v\otimes\mathrm{D}v\,\mathrm{d}\sigma_{p}\,-\,\int_{\Omega}\textbf{A}:\mathrm{D}u_{p}\otimes\mathrm{D}u_{p}\,\mathrm{d}\sigma_{p}\\\
&\ \
+\int_{\Omega}\textbf{A}:\Big{(}(\mathrm{D}u_{p}-\mathrm{D}v)\otimes\mathrm{D}u_{p}+\mathrm{D}u_{p}\otimes(\mathrm{D}u_{p}-\mathrm{D}v)\Big{)}\,\mathrm{d}\sigma_{p}\\\
&=\int_{\Omega}f(\cdot,\mathrm{D}v)\,\mathrm{d}\sigma_{p}-\int_{\Omega}f(\cdot,\mathrm{D}u_{p})\,\mathrm{d}\sigma_{p}\,+\,\int_{\Omega}f_{P}(\cdot,\mathrm{D}u_{p}):(\mathrm{D}u_{p}-\mathrm{D}v)\,\mathrm{d}\sigma_{p}.\end{split}$

We also note that the above established identity holds true over
$\overline{\Omega}$ as well, because
$\sigma_{p}(\partial\Omega)=\tau_{p}(\partial\Omega)=0$.

Step 3. For any fixed $p\in(\bar{p},\infty)$, by using (1.20)-(1.24) and
(1.27), we may rewrite (1.18) (obtained in Theorem 2) as

$\begin{split}\Lambda_{p}\int_{\overline{\Omega}}f_{P}(\cdot,\mathrm{D}u_{p}):\mathrm{D}\phi\,\mathrm{d}\sigma_{p}\,+\,\mathrm{M}_{p}\int_{\overline{\Omega}}g_{\eta}(\cdot,u_{p})\cdot\phi\,\mathrm{d}\tau_{p}\,=\,\big{\langle}\Psi_{p},(\mathrm{d}\mathrm{Q})_{u_{p}}(\phi)\big{\rangle},\end{split}$

for all test maps $\phi\in W^{1,\alpha p}_{0}(\Omega;\mathbb{R}^{N})$, whilst
we also have that

$\Lambda_{p}\in[0,1]$, $\mathrm{M}_{p}\in[0,1]$ and
$\Psi_{p}\in\bar{\mathbb{B}}^{E^{*}}_{1}\\!(0)$.

Further, by assumption (1.25), the weak* topology of the dual space
$\textbf{E}^{*}$ is sequentially (pre-) compact on bounded sets. Thus, the
previous memberships imply that, upon passing to a further subsequence as
$j\to\infty$, symbolised again by $(p_{j})_{1}^{\infty}$, there exist

$\Lambda_{\infty}\in[0,1]$, $\mathrm{M}_{\infty}\in[0,1]$ and
$\Psi_{\infty}\in\bar{\mathbb{B}}^{E^{*}}_{1}\\!(0)$,

such that the modes of convergence (1.33) hold true as $p_{j}\to\infty$.

Step 4. By Steps 2 and 3, for $\phi:=u_{p}-v$, where $v\in
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$ is an arbitrary fixed
map, for any fixed $p\in(\bar{p},\infty)$ we have the identity

$\begin{split}\Lambda_{p}\int_{\overline{\Omega}}&f(\cdot,\mathrm{D}u_{p}-\mathrm{D}v)\,\mathrm{d}\sigma_{p}=\,\big{\langle}\Psi_{p},(\mathrm{d}\mathrm{Q})_{u_{p}}(u_{p}-v)\big{\rangle}\,\\\
&-\,\mathrm{M}_{p}\int_{\Omega}g_{\eta}(\cdot,u_{p})\cdot(u_{p}-v)\,\mathrm{d}\tau_{p}\,+\,\Lambda_{p}\bigg{(}\int_{\Omega}f(\cdot,\mathrm{D}v)\,\mathrm{d}\sigma_{p}-\int_{\Omega}f(\cdot,\mathrm{D}u_{p})\,\mathrm{d}\sigma_{p}\bigg{)}.\end{split}$

Step 5. For any fixed $p\in(\bar{p},\infty)$ and $v\in
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$, we have the
relations

$\left\\{\ \
\begin{split}\int_{\overline{\Omega}}f(\cdot,\mathrm{D}u_{p}-\mathrm{D}v)\,\mathrm{d}\sigma_{p}\,&\geq\,\alpha_{0}\int_{\overline{\Omega}}\big{|}\mathrm{D}u_{p}-\mathrm{D}v\big{|}^{2}\,\mathrm{d}\sigma_{p},\\\
\int_{\overline{\Omega}}f(\cdot,\mathrm{D}u_{p})\,\mathrm{d}\sigma_{p}\,&=\,\mathrm{F}_{p}(u_{p}),\end{split}\right.$

where we have symbolised

$\alpha_{0}\,:=\,\min_{x\in\overline{\Omega}}\bigg{\\{}\min_{|Q|=1}\textbf{A}(x):Q\otimes
Q\bigg{\\}}\,>\,0.$

Let us first establish the claimed equality, beginning with the case that
$\mathrm{F}_{p}(u_{p})>0$. By definition (1.20) and assumption (1.27), we may
compute

$\begin{split}\int_{\Omega}f(\cdot,\mathrm{D}u_{p})\,\mathrm{d}\sigma_{p}\,&=\,\frac{1}{\mathcal{L}^{n}(\Omega)}\int_{\Omega}f(\cdot,\mathrm{D}u_{p})\frac{f(\cdot,\mathrm{D}u_{p})^{p-1}}{\mathrm{F}_{p}(u_{p})^{p-1}}\,\mathrm{d}\mathcal{L}^{n}\\\
&=\,\frac{1}{\mathrm{F}_{p}(u_{p})^{p-1}}\,{-\displaystyle\int_{\Omega}}f(\cdot,\mathrm{D}u_{p})^{p}\,\mathrm{d}\mathcal{L}^{n}\\\
&=\,\mathrm{F}_{p}(u_{p}),\phantom{\Big{|}}\end{split}$

whilst for $\mathrm{F}_{p}(u_{p})=0$ the equality follows trivially. To
establish the claimed inequality, it suffices to note that by assumption
(1.27) and by the variational representation of the minimum eigenvalue of the
symmetric linear operator $\textbf{A}(x):\mathbb{R}^{N\times
n}\longrightarrow\mathbb{R}^{N\times n}$, we have that $\alpha_{0}>0$ and the
inequality

$\alpha_{0}|Q|^{2}\,\leq\,\textbf{A}(x):Q\otimes Q$

for all $x\in\overline{\Omega}$ and all $Q\in\mathbb{R}^{N\times n}$, where
$|\cdot|$ is the Euclidean norm on $\mathbb{R}^{N\times n}$.

Step 6. By Steps 1, 3 and 5, and by using that
$\mathrm{F}_{p}(u_{p})\longrightarrow\mathrm{F}_{\infty}(u_{\infty})$ as
$p_{j}\to\infty$ (as shown in Theorem 1), we may invoke Hutchinson’s theory of
measure-function pairs, in particular [19, Sec. 4, Def. 4.1.1, 4.1.2, 4.2.1
and Th. 4.4.2], to infer that there exists a map

$V_{\infty}\
\in\,L^{2}\big{(}\overline{\Omega},\sigma_{\infty};\mathbb{R}^{N\times
n}\big{)}$

such that, along perhaps a further subsequence $(p_{j})_{1}^{\infty}$ we have

$\mathrm{D}u_{p}\sigma_{p}\,\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\,V_{\infty}\sigma_{\infty}\
\text{ in }\mathcal{M}\big{(}\overline{\Omega};\mathbb{R}^{N\times n}\big{)},$

as $p_{j}\to\infty$, with the property that

$\int_{\overline{\Omega}}\Phi(\cdot,V_{\infty})\,\mathrm{d}\sigma_{\infty}\,\leq\,\liminf_{p_{j}\to\infty}\int_{\overline{\Omega}}\Phi(\cdot,\mathrm{D}u_{p})\,\mathrm{d}\sigma_{p}$

for any $\Phi\in C\big{(}\overline{\Omega}\times\mathbb{R}^{N\times n}\big{)}$
such that $\Phi(x,\cdot)$ is convex and of quadratic growth, for all
$x\in\overline{\Omega}$. Further, in view of assumptions (1.14), (1.15),
(1.26), (1.27), and the modes of convergence established in Theorem 1 together
with the convergence
$\Psi_{p}\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\Psi_{\infty}$
in $\textbf{E}^{*}$ as $p_{j}\to\infty$, we have that

$\begin{split}\big{\langle}\Psi_{p},(\mathrm{d}\mathrm{Q})_{u_{p}}(\phi)\big{)}\rangle&\longrightarrow\big{\langle}\Psi_{\infty},(\mathrm{d}\mathrm{Q})_{u_{\infty}}(\phi)\big{\rangle}\
\ \ \,\text{ in }\mathbb{R},\\\
\big{(}f_{P}(\cdot,\mathrm{D}u_{p}):\mathrm{D}\phi\big{)}\sigma_{p}&\,\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\,\big{(}f_{P}(\cdot,V_{\infty}):\mathrm{D}\phi\big{)}\sigma_{\infty}\
\text{ in }\mathcal{M}\big{(}\overline{\Omega}\big{)},\end{split}$

for any fixed $\phi\in
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}\subseteq
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$. Hence, we may let $p_{j}\to\infty$
in Step 3, to deduce the equation

$\begin{split}\Lambda_{\infty}\int_{\overline{\Omega}}f_{P}(\cdot,V_{\infty}):\mathrm{D}\phi\,\mathrm{d}\sigma_{\infty}\,+\,\mathrm{M}_{\infty}\int_{\overline{\Omega}}g_{\eta}(\cdot,u_{\infty})\cdot\phi\,\mathrm{d}\tau_{\infty}\,=\,\big{\langle}\Psi_{\infty},(\mathrm{d}\mathrm{Q})_{u_{\infty}}(\phi)\big{\rangle},\end{split}$

for any fixed $\phi\in
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$. Further, by letting
$p_{j}\to\infty$ in (1.19) we deduce that

$\mathrm{M}_{\infty}\big{(}\mathrm{G}_{\infty}(u_{\infty})-G\big{)}=0.$

Step 7. The equations established in Step 6 will complete the proof of the
theorem, upon establishing that

$\ \ \ \ V_{\infty}\,=\,\mathrm{D}u^{\star}_{\infty}\ \ \
\sigma_{\infty}\text{-a.e.\ on }\overline{\Omega},$

where
$\mathrm{D}u^{\star}_{\infty}:\overline{\Omega}\longrightarrow\mathbb{R}^{N\times
n}$ is some Borel measurable mapping which is a version of
$\mathrm{D}u_{\infty}\in L^{\infty}(\Omega;\mathbb{R}^{N\times n})$, namely
such that

$\ \ \ \ \mathrm{D}u_{\infty}\,=\,\mathrm{D}u^{\star}_{\infty}\ \ \
\mathcal{L}^{n}\text{-a.e.\ on }\overline{\Omega},$

(recall that $\partial\Omega$ is a nullset for the Lebesgue measure
$\mathcal{L}^{n}$). The remaining steps are devoted to establishing this
claim, together with the approximability properties claimed in (1.32) for some
sequence of mappings $(v_{j})_{1}^{\infty}\subseteq
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$, which will be
constructed explicitly.

Step 8. If $\Lambda_{\infty}=0$, then Step 6 completes the proof of Theorem 3
as the first term involving $V_{\infty}$ vanishes. Hence, we may henceforth
assume that $\Lambda_{\infty}>0$. Therefore, by passing perhaps to a further
subsequence if necessary, we may assume that

$\ \ \ \ \ \Lambda_{p_{j}}\,\geq\,\frac{\Lambda_{\infty}}{2}\,>\,0,\text{ for
all }j\in\mathbb{N}.$

Step 9. By Steps 3, 4, 5 and 8, the absolute continuity
$\tau_{p}<<\mathcal{L}^{n}\text{\LARGE$\llcorner$}_{\Omega}$ and the bounds
$0\leq\mathrm{M}_{p}\leq 1$ and
$0\leq\tau_{p}\big{(}\overline{\Omega}\big{)}\leq 1$, we have the estimate

$\begin{split}\frac{\alpha_{0}\Lambda_{\infty}}{2}\int_{\overline{\Omega}}\big{|}\mathrm{D}u_{p}-\mathrm{D}v\big{|}^{2}\,\mathrm{d}\sigma_{p}\
\leq&\
\,\Big{\langle}\Psi_{p},\,(\mathrm{d}\mathrm{Q})_{u_{p}}(u_{p})-(\mathrm{d}\mathrm{Q})_{u_{p}}(v)\Big{\rangle}\\\
&+\,\|u_{p}-v\|_{L^{\infty}(\Omega)}\Big{\\{}\sup_{j\in\mathbb{N}}\,\big{\|}g_{\eta}(\cdot,u_{p_{j}})\big{\|}_{L^{\infty}(\Omega)}\Big{\\}}\\\
&+\,\Lambda_{p}\bigg{(}\int_{\overline{\Omega}}f(\cdot,\mathrm{D}v)\,\mathrm{d}\sigma_{p}\,-\,\mathrm{F}_{p}(u_{p})\bigg{)},\end{split}$

along the sequence $(p_{j})_{1}^{\infty}$. By letting $j\to\infty$ in the
above estimate, in view of Steps 1, 3 and 6 (for the choice
$\Phi(x,Q):=|\mathrm{D}v(x)-Q|^{2}$) and assumption (1.26), we infer that

$\begin{split}\frac{\alpha_{0}\Lambda_{\infty}}{2}\int_{\overline{\Omega}}\big{|}V_{\infty}-\mathrm{D}v\big{|}^{2}\,\mathrm{d}\sigma_{\infty}\
\leq&\
\,\Big{\langle}\Psi_{\infty},\,(\mathrm{d}\mathrm{Q})_{u_{\infty}}(u_{\infty})-(\mathrm{d}\mathrm{Q})_{u_{\infty}}(v)\Big{\rangle}\\\
&+\,\|u_{\infty}-v\|_{L^{\infty}(\Omega)}\Big{\\{}\sup_{j\in\mathbb{N}}\,\big{\|}g_{\eta}(\cdot,u_{p_{j}})\big{\|}_{L^{\infty}(\Omega)}\Big{\\}}\\\
&+\,\Lambda_{\infty}\bigg{(}\int_{\overline{\Omega}}f(\cdot,\mathrm{D}v)\,\mathrm{d}\sigma_{\infty}\,-\,\mathrm{F}_{\infty}(u_{\infty})\bigg{)},\end{split}$

for any fixed mapping $v\in
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$.

Step 10. Let $(v_{j})_{1}^{\infty}\subseteq
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$ be any sequence of
mappings satisfying the assumptions in (1.32). We claim that there exists a
subsequence of indices $(j_{k})_{1}^{\infty}$ such that

$\left\\{\ \ \begin{array}[]{lll}\mathrm{D}v_{j}\longrightarrow
V_{\infty},&\text{ in
}L^{2}\big{(}\overline{\Omega},\sigma_{\infty};\mathbb{R}^{N\times
n}\big{)},&\text{ and $\sigma_{\infty}$-a.e.\ on }\overline{\Omega},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\mathrm{D}v_{j}\longrightarrow\mathrm{D}u_{\infty},&\text{ in
}L^{q}\big{(}{\Omega};\mathbb{R}^{N\times n}\big{)},\,q\in[1,\infty),&\text{
and $\mathcal{L}^{n}$-a.e.\ on }{\Omega},\end{array}\right.$

as $j_{k}\to\infty$. Therefore, if $\mathrm{D}u^{\star}_{\infty}$ is defined
as in (1.32), then from the above we infer

$\left\\{\begin{split}\mathrm{D}u^{\star}_{\infty}\,&=\,V_{\infty},\ \ \
\,\sigma_{\infty}\text{-a.e.\ on }\overline{\Omega},\\\
\mathrm{D}u^{\star}_{\infty}\,&=\,\mathrm{D}u_{\infty},\ \
\mathcal{L}^{n}\text{-a.e.\ on }{\Omega},\end{split}\right.$

which completes the proof (subject to showing that at least one sequence of
mapping $(v_{j})_{1}^{\infty}$ with the desired properties exists). Let us now
establish the above claims. If $(v_{j})_{1}^{\infty}$ satisfies (1.32), then
by Step 5 and the resulting bound

$\alpha_{0}\|\mathrm{D}v_{j}\|_{L^{\infty}(\Omega)}^{2}\,\leq\,\mathrm{F}_{\infty}(v_{j})\,\leq\,\mathrm{F}_{\infty}(u_{\infty})\,+\,o(1)_{j\to\infty}$

in conjunction with the Vitaly convergence theorem, it follows that

$\ \ \ \ \ \ \left\\{\ \ \begin{array}[]{ll}v_{j}\longrightarrow
u_{\infty},&\text{ in }L^{\infty}({\Omega};\mathbb{R}^{N}),\vskip 6.0pt plus
2.0pt minus 2.0pt\\\
\mathrm{D}v_{j}\longrightarrow\mathrm{D}u_{\infty},&\text{ in
}L^{q}({\Omega};\mathbb{R}^{N\times n}),\ q\in[1,\infty),\end{array}\right.$

along a subsequence of indices $(j_{k})_{1}^{\infty}$ as $k\to\infty$.
Consequently, by the estimate of Step 9, we have

$\begin{split}\frac{\alpha_{0}\Lambda_{\infty}}{2}\int_{\overline{\Omega}}\big{|}V_{\infty}-\mathrm{D}v_{j}\big{|}^{2}\,\mathrm{d}\sigma_{\infty}\
\leq&\
\,\Big{\langle}\Psi_{\infty},\,(\mathrm{d}\mathrm{Q})_{u_{\infty}}(u_{\infty})-(\mathrm{d}\mathrm{Q})_{u_{\infty}}(v_{j})\Big{\rangle}\\\
&+\,\|u_{\infty}-v_{j}\|_{L^{\infty}(\Omega)}\Big{\\{}\sup_{j\in\mathbb{N}}\,\big{\|}g_{\eta}(\cdot,u_{p_{j}})\big{\|}_{L^{\infty}(\Omega)}\Big{\\}}\\\
&+\,\Lambda_{\infty}\bigg{(}\Big{\\{}\sup_{\overline{\Omega}}f(\cdot,\mathrm{D}v_{j})\Big{\\}}\,\sigma_{\infty}\big{(}\overline{\Omega}\big{)}\,-\,\mathrm{F}_{\infty}(u_{\infty})\bigg{)}.\end{split}$

Since by Step 1 we have
$0\leq\sigma_{\infty}\big{(}\overline{\Omega}\big{)}\leq 1$, by noting that
$\sup_{\overline{\Omega}}f(\cdot,\mathrm{D}v_{j})=\mathrm{F}_{\infty}(v_{j})$
(due to the continuity of $\mathrm{D}v_{j}$ on $\overline{\Omega}$) and that
$v_{j}\longrightarrow u_{\infty}$ strongly in
$W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$, from the last estimate and
assumption (1.32) we deduce that

$\begin{split}\limsup_{j_{k}\to\infty}\int_{\overline{\Omega}}\big{|}V_{\infty}-\mathrm{D}v_{j}\big{|}^{2}\,\mathrm{d}\sigma_{\infty}\,\leq\,0.\end{split}$

Step 11. To complete the proof of Theorem 3, it remains to show that at least
one sequence of mapping $(v_{j})_{1}^{\infty}\subseteq
C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$ exists, which
satisfies the modes of convergence required by (1.32). To this end we utilise
(for the first time) the assumption that the bounded domain $\Omega$ has
Lipschitz boundary $\partial\Omega$, and we invoke the regularisation scheme
introduced and utilised in the recent paper [26]. This method is based on
results on the geometry of Lipschitz domains proved in Hofmann-Mitrea-Taylor
[18], and is inspired by the regularisation schemes employed in Ern-Guermond
[15]. If $\mathrm{n}\in
L^{\infty}(\partial\Omega,\mathcal{H}^{n-1};\mathbb{R}^{n})$ be the outer unit
normal vector field on $\partial\Omega$, by [18, Sec. 2, 4], there exists a
smooth vector field $\xi\in C^{\infty}_{c}(\mathbb{R}^{n};\mathbb{R}^{n})$
which is globally transversal to $\mathrm{n}$ on $\partial\Omega$ with respect
to the surface measure, namely exists $c>0$ such that

$\ \ \ \ \ \xi\cdot\mathrm{n}\geq c,\quad\mathcal{H}^{n-1}\text{-a.e.\ on
}\partial\Omega.$

Additionally, $\xi$ can be chosen to satisfy $|\xi|\equiv 1$ in an open
tubular neighbourhood $\\{\mathrm{dist}(\cdot,\partial\Omega)<r\\}$ around
$\partial\Omega$ for some $r>0$, whilst vanishing on
$\\{\mathrm{dist}(\cdot,\partial\Omega)>2r\\}$. (In the special case that
$\partial\Omega$ happens to be a compact $C^{\infty}$ manifold, then we can
simply choose $\xi$ to be a smooth extension of $\mathrm{n}$, and the
transversality condition is satisfied with $c=1$.) Further, it can be shown
that there exists $\varepsilon_{0},\ell>0$ such that, for all
$\varepsilon\in(0,\varepsilon_{0})$ we have

$\ \ \
\inf_{x\in\partial\Omega}\mathrm{dist}\big{(}x+\varepsilon\ell\xi(x),\,\partial\Omega\big{)}\geq\,2\varepsilon.$

We now define our adapted global mollifiers, taken from [26]. Let us select
any function $\varrho\in C^{\infty}_{c}(\mathbb{B}_{1}(0))$ which satisfies
$\varrho\geq 0$ and $\|\varrho\|_{L^{1}(\mathbb{R}^{n})}=1$ (for instance the
“standard” mollifying kernel as in [30]). For any $v\in
L^{\infty}(\Omega;\mathbb{R}^{N})$, extended to
$\mathbb{R}^{n}\setminus\Omega$ by zero, we define for any
$\varepsilon\in(0,\varepsilon_{0})$ the map
$\mathrm{K}^{\varepsilon}v:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{N}$, by
setting

$(\mathrm{K}^{\varepsilon}v)(x)\,:=\,\int_{\mathbb{R}^{n}}v\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\,\varrho(y)\,\mathrm{d}y.$

Then, by [26, Prop. 12, p. 18], for any $u\in
W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})$ and
$\varepsilon\in(0,\varepsilon_{0})$:

$\bullet$ We have that $\mathrm{K}^{\varepsilon}u\in
C^{\infty}_{0}\big{(}\Omega;\mathbb{R}^{N}\big{)}$, and the identity

$\mathrm{D}(\mathrm{K}^{\varepsilon}u)\,=\,\mathrm{K}^{\varepsilon}(\mathrm{D}u)\,+\,\varepsilon\ell(\mathrm{K}^{\varepsilon}(\mathrm{D}u))(\mathrm{D}\xi)^{\top},$

everywhere on $\overline{\Omega}$.

$\bullet$ We have $\mathrm{K}^{\varepsilon}u\longrightarrow u$ in
$W^{1,q}_{0}(\Omega;\mathbb{R}^{N})$ for all $q\in[1,\infty)$, and in
$C^{\gamma}(\overline{\Omega};\mathbb{R}^{N})$ for all $\gamma\in(0,1)$, as
$\varepsilon\to 0$. Further,
$\mathrm{K}^{\varepsilon}u\,\overset{\,*_{\phantom{|}}}{{\smash{\,-\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup}}\,}\,u$
in $W^{1,\infty}_{0}(\Omega;\mathbb{R}^{N})$, as $\varepsilon\to 0$.

$\bullet$ For any $\Theta\in
C\big{(}\overline{\Omega}\times\mathbb{R}^{N\times n}\big{)}$, satisfying for
any $x\in\overline{\Omega}$ that $\Theta(x,\cdot)$ is convex on
$\mathbb{R}^{N\times n}$ with $\Theta(x,\cdot)\geq\Theta(x,0)=0$, and also
that the partial derivative $\Theta_{P}$ exists and is continuous on
$\overline{\Omega}\times\mathbb{R}^{N\times n}$, we will show that there
exists a modulus of continuity $\omega\in
C\big{(}[0,\infty);[0,\infty)\big{)}$ with $\omega(0)=0$ which is independent
of $\varepsilon$ and $x$, such that

$\Theta\big{(}x,\mathrm{D}(\mathrm{K}^{\varepsilon}u)(x)\big{)}\,\leq\,\underset{\Omega\cap\mathbb{B}_{\varepsilon}(x+\varepsilon\ell\xi(x))}{\mathrm{ess}\sup}\,\Theta(\cdot,\mathrm{D}u)\,+\,\omega(\varepsilon),$

for any $x\in\Omega$. In particular, the above estimate implies that

$\big{\|}\Theta\big{(}\cdot,\mathrm{D}(\mathrm{K}^{\varepsilon}u)\big{)}\big{\|}_{L^{\infty}(\Omega)}\,\leq\,\|\Theta(\cdot,\mathrm{D}u)\|_{L^{\infty}(\Omega)}\,+\,\omega(\varepsilon).$

(This was already established in [26], but without $x$-dependence for
$\Theta$.) As a result,

$\limsup_{\varepsilon\to
0}\big{\|}\Theta\big{(}\cdot,\mathrm{D}(\mathrm{K}^{\varepsilon}u)\big{)}\big{\|}_{L^{\infty}(\Omega)}\,\leq\,\|\Theta(\cdot,\mathrm{D}u)\|_{L^{\infty}(\Omega)}.$

Let us now establish the claimed estimate. For any fixed $R>0$ such that

$R\,>\,\big{(}\|\mathrm{D}\xi\|_{L^{\infty}(\mathbb{R}^{n})}+1\big{)}\big{(}\|\mathrm{D}u\|_{L^{\infty}(\Omega)}+1\big{)},$

we set

$\omega(t)\,:=\,t\ell
R\|\Theta_{P}\|_{C(\overline{\Omega}\times\bar{\mathbb{B}}_{R}(0))}\,+\underset{|z-x|\leq
t\ell R}{\sup_{|Q|\leq
R}}\Big{|}\Theta\big{(}x,Q\big{)}-\Theta\big{(}z,Q\big{)}\Big{|},\ \ \ t\geq
0.$

Then, we have

$\begin{split}\Theta\big{(}x,\mathrm{D}(\mathrm{K}^{\varepsilon}u)(x)\big{)}&=\Theta\Big{(}x,\mathrm{K}^{\varepsilon}(\mathrm{D}u)(x)\,+\,\varepsilon\ell\mathrm{K}^{\varepsilon}(\mathrm{D}u)(x)(\mathrm{D}\xi(x))^{\top}\Big{)}\\\
&\leq\Theta\big{(}x,\mathrm{K}^{\varepsilon}(\mathrm{D}u)(x)\big{)}+\|\Theta_{P}(x,\cdot)\|_{C(\bar{\mathbb{B}}_{R}(0))}\big{\|}\varepsilon\ell\mathrm{K}^{\varepsilon}(\mathrm{D}u)(\mathrm{D}\xi)^{\top}\big{\|}_{L^{\infty}(\Omega)}\\\
&\leq\Theta\big{(}x,\mathrm{K}^{\varepsilon}(\mathrm{D}u)(x)\big{)}+\varepsilon\ell\|\Theta_{P}\|_{C(\overline{\Omega}\times\bar{\mathbb{B}}_{R}(0))}\|\mathrm{D}u\|_{L^{\infty}(\Omega)}\|\mathrm{D}\xi\|_{L^{\infty}(\mathbb{R}^{n})}\\\
&\leq\Theta\big{(}x,\mathrm{K}^{\varepsilon}(\mathrm{D}u)(x)\big{)}\,+\,\varepsilon\ell
R\|\Theta_{P}\|_{C(\overline{\Omega}\times\bar{\mathbb{B}}_{R}(0))},\end{split}$

for any $x\in\Omega$. Further, since $\varrho\mathcal{L}^{n}$ is a probability
measure on $\mathbb{R}^{n}$, by Jensen’s inequality, we have

$\begin{split}\Theta\big{(}x,\mathrm{K}^{\varepsilon}(\mathrm{D}u)(x)\big{)}\,&=\,\Theta\left(x,\int_{\mathbb{R}^{n}}\mathrm{D}u\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\,\varrho(y)\,\mathrm{d}y\right)\\\
&\leq\,\int_{\mathbb{R}^{n}}\Theta\Big{(}x,\mathrm{D}u\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\Big{)}\varrho(y)\,\mathrm{d}y\\\
&\leq\,\underset{y\in\mathbb{B}_{1}(0)}{\mathrm{ess}\sup}\,\Theta\Big{(}x,\mathrm{D}u\big{(}x+\varepsilon\ell\xi(x)-\varepsilon
y\big{)}\Big{)}\\\
&=\,\underset{z\in\mathbb{B}_{\varepsilon}(x+\varepsilon\ell\xi(x))}{\mathrm{ess}\sup}\,\Theta\big{(}x,\mathrm{D}u(z)\big{)}\\\
&\leq\,\underset{z\in\mathbb{B}_{\varepsilon}(x+\varepsilon\ell\xi(x))}{\mathrm{ess}\sup}\,\Theta\big{(}z,\mathrm{D}u(z)\big{)}\\\
&\ \ \
+\underset{|z-x|\leq\varepsilon\ell\|\mathrm{D}\xi\|_{L^{\infty}(\mathbb{R}^{n})}}{\sup_{|Q|\leq\|\mathrm{D}u\|_{L^{\infty}(\Omega)}}}\Big{|}\Theta\big{(}x,Q\big{)}-\Theta\big{(}z,Q\big{)}\Big{|},\end{split}$

for any $x\in\Omega$. By using that $\mathrm{D}u\equiv 0$ on
$\mathbb{R}^{n}\setminus\Omega$ and our assumptions on $\Theta$, the previous
two estimates yield that

$\begin{split}\Theta\big{(}x,\mathrm{D}(\mathrm{K}^{\varepsilon}u)(x)\big{)}\,&\leq\,\underset{\mathbb{B}_{\varepsilon}(x+\varepsilon\ell\xi(x))}{\mathrm{ess}\sup}\,\Theta(\cdot,\mathrm{D}u)\,+\,\omega(\varepsilon)\\\
&=\,\max\bigg{\\{}\underset{\Omega\cap\mathbb{B}_{\varepsilon}(x+\varepsilon\ell\xi(x))}{\mathrm{ess}\sup}\,\Theta(\cdot,\mathrm{D}u),\
\underset{\mathbb{B}_{\varepsilon}(x+\varepsilon\ell\xi(x))\setminus\Omega}{\mathrm{ess}\sup}\,\Theta(\cdot,\mathrm{D}u)\bigg{\\}}\,+\,\omega(\varepsilon)\\\
&=\,\max\bigg{\\{}\underset{\Omega\cap\mathbb{B}_{\varepsilon}(x+\varepsilon\ell\xi(x))}{\mathrm{ess}\sup}\,\Theta(\cdot,\mathrm{D}u),\
\underset{\mathbb{B}_{\varepsilon}(x+\varepsilon\ell\xi(x))\setminus\Omega}{\mathrm{ess}\sup}\,\Theta(\cdot,0)\bigg{\\}}\,+\,\omega(\varepsilon)\\\
&=\,\underset{\Omega\cap\mathbb{B}_{\varepsilon}(x+\varepsilon\ell\xi(x))}{\mathrm{ess}\sup}\,\Theta(\cdot,\mathrm{D}u)\,+\,\omega(\varepsilon),\end{split}$

for any $x\in\Omega$.

In conclusion, the above establish the existence of an approximating sequence
as in (1.32), by taking

$\ \ \ \
v_{j}\,:=\,\mathrm{K}^{\varepsilon_{j}}u_{\infty}\in\,C^{1}_{0}\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)},\
\ j\in\mathbb{N},$

along any infinitesimal sequence $(\varepsilon_{j})_{1}^{\infty}$ satisfying
$\varepsilon_{j}\to 0$, for the choice $\Theta:=f$, which is admissible
because of our hypotheses (1.5), (1.6) and (1.27).

Step 12. By putting together all the previous, the proof of Theorem 3 has now
been completed. ∎

##  4\. Explicit classes of nonlinear operators

In this section we provide various examples of nonlinear operators
$\mathrm{Q}$ as in (1.2), satisfying our assumptions. Note that each our main
results, Theorems 1, 2 and 3, have been obtained with progressively stronger
assumption on the operator $\mathrm{Q}$ which expresses one of constraints in
the admissible class. For the sake of clarity, in the next table we list in a
concise way which assumptions are required to be satisfied by $\mathrm{Q}$, in
order to obtain the corresponding result (assuming that $f,g$ satisfy
separately their respective required assumptions, the table concerns
$\mathrm{Q}$ solely).

$\underset{\text{Table 1. The list of assumptions on $\mathrm{Q}$ to obtain
the main
results.}}{\left|\\!\begin{array}[]{rrrr}\hline\cr\phantom{\Big{|}}\eqref{1.9}&\phantom{a}&\phantom{a}&\Mapsto\text{Theorem
\ref{theorem1}}\\\ \hline\cr\phantom{\Big{|}}\eqref{1.9}&\&\ \
\eqref{1.16}&\phantom{a}&\Mapsto\text{Theorem \ref{theorem2}}\\\
\hline\cr\phantom{\Big{|}}\eqref{1.9}&\&\ \ \eqref{1.16}&\&\ \ \eqref{1.25}\ \
\&\ \ \eqref{1.26}&\Mapsto\text{Theorem \ref{theorem3}}\\\
\hline\cr\end{array}\\!\right|}$

###  4.1. Pointwise constraints, unilateral constraints and inclusions

The nonlinear operator of (1.2) we are using in the admissible class of (1.4),
can include the following model cases:

Case 1. $\Pi(x,u(x))=0$ for a.e. $x\in\Omega$, where
$\Pi:\Omega\times\mathbb{R}^{N}\longrightarrow\mathbb{R}^{M}$ is given.

Case 2. $\Pi(x,u(x))\leq 0$ for a.e. $x\in\Omega$, where
$\Pi:\Omega\times\mathbb{R}^{N}\longrightarrow\mathbb{R}$ is given.

Case 3. $u(x)\in\mathcal{K}$ for a.e. $x\in\Omega$, where
$\mathcal{K}\subseteq\mathbb{R}^{N}$ is a given closed set.

Constraints as in Case 1 are sometimes called _holonomic_ (see for instance
[2]). We now elaborate on the assumptions required to be fulfilled in each of
these cases.

######  Proposition 6 (Case 1).

Let $\Pi\in C^{1}(\overline{\Omega}\times\mathbb{R}^{N};\mathbb{R}^{M})$,
$M\in\mathbb{N}$. By defining

(4.1) $\mathrm{Q}\ :\ \
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow
L^{1}(\Omega;\mathbb{R}^{M}),\ \ \ \ \mathrm{Q}(u)\,:=\,\Pi(\cdot,u),$

and setting $\emph{{E}}:=L^{1}(\Omega;\mathbb{R}^{M})$, we have the following:

_(i)_ The zero set of $\mathrm{Q}$ equals

$\mathrm{Q}^{-1}(\\{0\\})\,=\,\Big{\\{}v\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\ :\ \Pi(x,v(x))=0,\text{ a.e.
}x\in\Omega\Big{\\}},$

and assumption (1.9) is always satisfied.

_(ii)_ If for any $x\in\Omega$ we have

$\big{\\{}\Pi(x,\cdot)=0\big{\\}}\,\subseteq\,\big{\\{}\Pi_{\eta}(x,\cdot)=0\big{\\}},$

namely when all points in the zero set are critical points, then $\mathrm{Q}$
satisfies (1.15).

_(iii)_ Assumptions (1.25) and (1.26) are always satisfied.

The choice of E is deliberately made “as large as possible”, as then the
Lagrange multipliers of Theorems 2 and 3 are valued in the smaller space
$\textbf{E}^{*}=L^{\infty}(\Omega;\mathbb{R}^{M})$.

Proof of Proposition 6. (i) Follows directly from the definitions, by the
continuity of $\Pi$ and by Morrey’s estimate, because $\bar{p}>n$.

(ii) Indeed, since

$(\mathrm{d}\mathrm{Q})_{u}(\phi)\,=\,\Pi_{\eta}(\cdot,u)\cdot\phi,$

if $u\in\mathrm{Q}^{-1}(\\{0\\})$, then $\Pi(\cdot,u)=0$ a.e. on $\Omega$ and
therefore $\Pi_{\eta}(\cdot,u)=0$ a.e. on $\Omega$, which implies that
$(\mathrm{d}\mathrm{Q})_{u}=0$, hence its image is the closed trivial subspace
$\\{0\\}\subseteq L^{1}(\Omega;\mathbb{R}^{M})$.

(iii) Note first that $L^{1}(\Omega;\mathbb{R}^{M})$ is separable. Also, if we
have $u_{m}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u$ and
$\phi_{m}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup\phi$ in
$W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$ as $m\to\infty$, then by Morrey’s
theorem and the compactness of the imbedding of Hölder spaces we have
$u_{m}\longrightarrow u$ and also $\phi_{m}\longrightarrow\phi$ in
$C\big{(}\overline{\Omega};\mathbb{R}^{N}\big{)}$ as $m\to\infty$. Hence, we
have as $m\to\infty$ that

$(\mathrm{d}\mathrm{Q})_{u_{m}}(\phi_{m})=\Pi_{\eta}(\cdot,u_{m})\cdot\phi_{m}\,\longrightarrow\,\Pi_{\eta}(\cdot,u)\cdot\phi=(\mathrm{d}\mathrm{Q})_{u}(\phi),$

in $C\big{(}\overline{\Omega};\mathbb{R}^{M}\big{)}$, which a fortiori implies
strong convergence in $L^{1}(\Omega;\mathbb{R}^{M})$. ∎

We note that the proof of (iii) above is immediate if one assumes the
additional hypothesis of (ii), since then $(\mathrm{d}\mathrm{Q})_{u_{m}}=0$
for any sequence $(u_{m})_{1}^{\infty}\subseteq\mathrm{Q}^{-1}(\\{0\\})$.

######  Proposition 7 (Case 2).

Let $\Pi\in C^{1}(\overline{\Omega}\times\mathbb{R}^{N})$ and let us define
$\pi:\mathbb{R}\longrightarrow\mathbb{R}$ by

(4.2) $\pi(t)\,:=\,\left\\{\begin{array}[]{ll}0,&t\leq 0,\\\
t^{2},&t>0.\end{array}\right.$

By defining the operator

$\mathrm{Q}\ :\ \ W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow
L^{1}(\Omega),\ \ \ \ \mathrm{Q}(u)\,:=\,\pi(\Pi(\cdot,u)),$

for $\emph{{E}}:=L^{1}(\Omega)$, we have the following:

_(i)_ The zero set of $\mathrm{Q}$ equals

$\mathrm{Q}^{-1}(\\{0\\})\,=\,\Big{\\{}v\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\ :\ \Pi(x,v(x))\leq 0,\text{ a.e.
}x\in\Omega\Big{\\}},$

and assumption (1.9) is always satisfied.

_(ii)_ If for any $x\in\Omega$ it holds that

$\big{\\{}\Pi(x,\cdot)=0\big{\\}}\,\subseteq\,\big{\\{}\Pi_{\eta}(x,\cdot)=0\big{\\}},$

then $\mathrm{Q}$ satisfies assumption (1.15).

_(iii)_ Assumptions (1.25) and (1.26) are always satisfied.

Proof of Proposition 7. (i) Follows as in the proof of Proposition 6(i), upon
noting that $\\{\pi\leq 0\\}=(-\infty,0]$.

(ii) Since

$(\mathrm{d}\mathrm{Q})_{u}(\phi)\,=\,\pi^{\prime}\big{(}\Pi(\cdot,u)\big{)}\Pi_{\eta}(\cdot,u)\cdot\phi,$

if $u\in\mathrm{Q}^{-1}(\\{0\\})$, then $\Pi(\cdot,u)\leq 0$ a.e. on $\Omega$
and therefore $\pi^{\prime}(\Pi(\cdot,u))=0$ a.e. on $\Omega$ because
$\\{\pi^{\prime}=0\\}=(-\infty,0]$, which implies that
$(\mathrm{d}\mathrm{Q})_{u}=0$, hence its image is the closed trivial subspace
$\\{0\\}\subseteq L^{1}(\Omega;\mathbb{R}^{M})$ and (1.15) is satisfied.

(iii) Similar to the proof of Proposition 6(iii), using the different
expression for the differential $\mathrm{d}\mathrm{Q}$ as above. ∎

######  Proposition 8 (Case 3).

Let $\mathcal{K}\subseteq\mathbb{R}^{N}$ be a closed set with
$\mathcal{K}\neq\emptyset$. Then, there exists $\Pi\in
C^{\infty}(\mathbb{R}^{N})$ satisfying
$\mathcal{K}=\\{\Pi=0\\}\subseteq\\{\Pi_{\eta}=0\\}$. Further, if one defines

$\mathrm{Q}\ :\ \ W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow
L^{1}(\Omega),\ \ \ \ \mathrm{Q}(u)\,:=\,\Pi(u),$

and $\emph{{E}}:=L^{1}(\Omega)$, then we have

$\mathrm{Q}^{-1}(\\{0\\})\,=\,\Big{\\{}v\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\ :\ v(x)\in\mathcal{K},\text{ a.e.
}x\in\Omega\Big{\\}},$

and $\mathrm{Q}$ satisfies (1.9), (1.15), (1.25) and (1.26).

Proof of Proposition 8. It is well-known that for every such set
$\mathcal{K}$, there exists a function $\Pi\in C^{\infty}(\mathbb{R}^{N})$
with the claimed properties. A proof of this fact can be found e.g. in [34,
Sec. 1.1.13, p. 25] (the claimed inclusion is not explicitly stated, but
follows from the method of proof by the smooth Urysohn lemma). The rest
follows from Proposition 6. ∎

###  4.2. Integral and isoperimetric constraints

The nonlinear operator of (1.2) can also cover the following important case of
constraint:

$\int_{\Omega}h(\cdot,u,\mathrm{D}u)\,\mathrm{d}\mathcal{L}^{n}\leq H,$

when $h:\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{N\times
n}\longrightarrow\mathbb{R}$ and $H\in\mathbb{R}$ are given.

######  Proposition 9.

Let $h:\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{N\times
n}\longrightarrow\mathbb{R}$ satisfy the assumptions (1.5)-(1.7) and (1.14)
that $f,g$ are assumed to satisfy, with $\alpha\leq\bar{p}$. Let also
$H\in\mathbb{R}$ be given and let $\pi:\mathbb{R}\longrightarrow\mathbb{R}$ be
as in (4.2). Then, by defining the operator

$\mathrm{Q}\ :\ \
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow\mathbb{R},\ \ \ \
\mathrm{Q}(u)\,:=\,\pi\left(\,\int_{\Omega}h(\cdot,u,\mathrm{D}u)\,\mathrm{d}\mathcal{L}^{n}-H\\!\right),$

and setting $\emph{{E}}:=\mathbb{R}$, we have the following:

_(i)_ The zero set of $\mathrm{Q}$ equals

$\mathrm{Q}^{-1}(\\{0\\})\,=\,\left\\{v\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\ :\
\int_{\Omega}h(\cdot,v,\mathrm{D}v)\,\mathrm{d}\mathcal{L}^{n}\leq H\right\\}$

and assumption (1.9) is satisfied.

_(ii)_ $\mathrm{Q}$ satisfies (1.15), (1.25) and (1.26).

Proof of Proposition 9. (i) If $\mathrm{Q}(u_{m})=0$ and
$u_{m}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u$ in
$W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$ as $m\to\infty$, then since
$\\{\pi\leq 0\\}=(-\infty,0]$, we have

$\int_{\Omega}h(\cdot,u_{m},\mathrm{D}u_{m})\,\mathrm{d}\mathcal{L}^{n}-H\,\leq\,0.$

Since $h$ satisfies (1.5)-(1.7) for $\alpha\leq\bar{p}$, by standard results
(see e.g. [14]), the functional
$u\mapsto\|h(\cdot,u,\mathrm{D}u)\|_{L^{1}(\Omega)}$ is weakly lower-
semicontinuous in $W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$. Hence

$\int_{\Omega}h(\cdot,u,\mathrm{D}u)\,\mathrm{d}\mathcal{L}^{n}-H\,\leq\,\liminf_{m\to\infty}\int_{\Omega}h(\cdot,u_{m},\mathrm{D}u_{m})\,\mathrm{d}\mathcal{L}^{n}-H\,\leq\,0.$

Therefore, $\mathrm{Q}(u)=0$, yielding that $\mathrm{Q}^{-1}(\\{0\\})$ is
weakly closed and hence (1.9) is satisfied.

(ii) By a computation, the Gateaux derivative of $\mathrm{Q}$ is given by

$(\mathrm{d}\mathrm{Q})_{u}(\phi)=\pi^{\prime}\\!\left(\int_{\Omega}h(\cdot,u,\mathrm{D}u)\,\mathrm{d}\mathcal{L}^{n}-H\\!\right)\\!\\!\int_{\Omega}\Big{[}h_{\eta}(\cdot,u,\mathrm{D}u)\cdot\phi\,+\,h_{P}(\cdot,u,\mathrm{D}u):\mathrm{D}\phi\Big{]}\,\mathrm{d}\mathcal{L}^{n},$

and assumption (1.14) for $h$ implies that $\mathrm{d}\mathrm{Q}$ is (jointly)
continuous on $W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\times
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$. Further, if
$u\in\mathrm{Q}^{-1}(\\{0\\})$, then by part (i) we have

$\int_{\Omega}h(\cdot,u,\mathrm{D}u)\,\mathrm{d}\mathcal{L}^{n}-H\,\leq\,0,$

and therefore the first factor of $(\mathrm{d}\mathrm{Q})_{u}(\phi)$ vanishes
because $\\{\pi^{\prime}=0\\}=(-\infty,0]$. Thus,
$(\mathrm{d}\mathrm{Q})_{u}=0$ when $u\in\mathrm{Q}^{-1}(\\{0\\})$, and hence
its image is the closed trivial subspace $\\{0\\}\subseteq\mathbb{R}$,
yielding that (1.15) is satisfied.

(iii) For any sequences $u_{m}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup
u$ in $\mathrm{Q}^{-1}(\\{0\\})\subseteq
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$ and
$\phi_{m}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup\phi$ in
$W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$ as $m\to\infty$, by part (ii) we
have

$(\mathrm{d}\mathrm{Q})_{u_{m}}(\phi_{m})\,=\,0\longrightarrow
0\,=\,(\mathrm{d}\mathrm{Q})_{u}(\phi)$

as $m\to\infty$, hence (1.25) and (1.26) are satisfied. ∎

###  4.3. Quasilinear second order differential constraints

The operator $\mathrm{Q}$ of (1.2) can also cover the case of various types of
nontrivial PDE constraints. As an example, we discuss the case of quasilinear
divergence second order systems of PDE of the form

(4.3)
$\mathrm{div}\big{(}A(\cdot,u,\mathrm{D}u)\big{)}\,=\,B(\cdot,u,\mathrm{D}u)\
\ \text{ in }\Omega,$

where the coefficients maps
$A:\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{N\times
n}\longrightarrow\mathbb{R}^{N\times n}$ and
$B:\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{N\times
n}\longrightarrow\mathbb{R}^{N}$ are given. Given the plethora of
possibilities on the assumptions for such systems, the discussion in this
subsection is less formal and is only aimed as a general indication of the
admissible choices for $\mathrm{Q}$.

Suppose that $A,B$ are $C^{1}$ and satisfy appropriate growth bounds, and also
that $P\mapsto A(\cdot,\cdot,P)$ a monotone map, and that the set of weak
solutions to the system (4.3) is strongly precompact in
$W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$. A sufficient conditions for strong
precompactness in $W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$ for the set of
weak solutions is for example a global $C^{1,\gamma}$ or a $W^{2,1+\gamma}$ a
priori uniform bound on the set of solutions, for some $\gamma\in(0,1)$.
Appropriate assumptions on the coefficients $A,B$ that allow the derivation of
such a priori bounds can be found e.g. in [17] for $N=1$ and in [16] for
$N\geq 2$. Then, by defining the operator

$\mathrm{Q}\ :\ W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow
W^{-1,\bar{p}^{\prime}}(\Omega;\mathbb{R}^{N})$

as

$\langle\mathrm{Q}(u),\psi\rangle\,:=\,\int_{\Omega}\Big{[}A(\cdot,u,\mathrm{D}u):\mathrm{D}\psi\,+\,B(\cdot,u,\mathrm{D}u)\cdot\psi\Big{]}\,\mathrm{d}\mathcal{L}^{n},$

and setting also $\textbf{E}:=W^{-1,\bar{p}^{\prime}}(\Omega;\mathbb{R}^{N})$,
assumptions (1.9), (1.15), (1.25) and (1.26) are satisfied, with

$\mathrm{Q}^{-1}(\\{0\\})\,=\,\bigg{\\{}u\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\ :\
\mathrm{div}\big{(}A(\cdot,u,\mathrm{D}u)\big{)}=B(\cdot,u,\mathrm{D}u)\text{
weakly in }\Omega\bigg{\\}}.$

Note first that the expression of $\mathrm{Q}^{-1}(\\{0\\})$ is immediate by
the definition of the differential operator $\mathrm{Q}$. Next, note that by
assumption, for any sequence of weak solutions $(u_{m})_{1}^{\infty}\subseteq
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$ to (4.3), there exists $u\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$ such that $u_{m}\longrightarrow u$
strongly along a subsequence $m_{j}\to\infty$. By applying this to any
sequence $(u_{m})_{1}^{\infty}\subseteq\mathrm{Q}^{-1}(\\{0\\})$ (namely
sequence of solutions) for which
$u_{m}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u$ as $m\to\infty$, by
passing to the limit in the weak formulation for fixed $\psi\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$, which reads

$\int_{\Omega}\Big{[}A(\cdot,u_{m},\mathrm{D}u_{m}):\mathrm{D}\psi\,+\,B(\cdot,u_{m},\mathrm{D}u_{m})\cdot\psi\Big{]}\,\mathrm{d}\mathcal{L}^{n}\,=\,0,$

we get that $u\in\mathrm{Q}^{-1}(\\{0\\})$, as the convergence is in fact
strong. Hence, (1.9) is satisfied. Further, under appropriate bounds, the
operator $\mathrm{Q}$ is Fréchet differentiable and

$\begin{split}\big{\langle}(\mathrm{d}\mathrm{Q})_{u}(\phi),\psi\big{\rangle}\,&=\,\int_{\Omega}\Big{[}A_{\eta}(\cdot,u,\mathrm{D}u)\cdot\phi\,+\,A_{P}(\cdot,u,\mathrm{D}u):\mathrm{D}\phi\Big{]}:\mathrm{D}\psi\,\mathrm{d}\mathcal{L}^{n}\\\
&+\,\int_{\Omega}\Big{[}B_{\eta}(\cdot,u,\mathrm{D}u)\cdot\phi\,+\,B_{P}(\cdot,u,\mathrm{D}u):\mathrm{D}\phi\Big{]}\cdot\psi\,\mathrm{d}\mathcal{L}^{n}.\end{split}$

To see that the image of
$(\mathrm{d}\mathrm{Q})_{u}:W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})\longrightarrow\textbf{E}$
is closed for any fixed $u\in\mathrm{Q}^{-1}(\\{0\\})$, let
$(T_{m})_{1}^{\infty}\subseteq\mathrm{Rg}\big{(}(\mathrm{d}\mathrm{Q})_{u}\big{)}\subseteq\textbf{E}$
be a sequence in the range with $T_{m}\longrightarrow T$ strongly in E as
$m\to\infty$. Since
$T_{m}\in\mathrm{Rg}\big{(}(\mathrm{d}\mathrm{Q})_{u}\big{)}$, exists
$\phi_{m}\in W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$ solving the following
linear second order system

$\begin{split}-\mathrm{div}\Big{(}A_{\eta}(\cdot,u,\mathrm{D}u)\cdot\phi_{m}\,&+\,A_{P}(\cdot,u,\mathrm{D}u):\mathrm{D}\phi_{m}\Big{)}\\\
&+\,B_{P}(\cdot,u,\mathrm{D}u):\mathrm{D}\phi_{m}\,+\,B_{\eta}(\cdot,u,\mathrm{D}u)\cdot\phi_{m}\,=\,T_{m}.\end{split}$

By the monotonicity of the above system (due to our earlier assumption), under
appropriate conditions one has a uniform bound in
$W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$, yielding the weak compactness of
the sequence of solutions $(\phi_{m})_{1}^{\infty}$, which establishes the
closedness of
$\mathrm{Rg}\big{(}(\mathrm{d}\mathrm{Q})_{u}\big{)}\subseteq\textbf{E}$ and
(1.15) ensues.

Finally, for any sequence
$(u_{m})_{1}^{\infty}\subseteq\mathrm{Q}^{-1}(\\{0\\})$ satisfying
$u_{m}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u$ as $m\to\infty$ and
any $\phi\in W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{N})$, there exists
$m_{j}\to\infty$ such that $u_{m}\longrightarrow u$ as $m_{j}\to\infty$. These
facts imply that
$(\mathrm{d}\mathrm{Q})_{u_{m}}(u_{m})\longrightarrow(\mathrm{d}\mathrm{Q})_{u}(u)$
and also
$(\mathrm{d}\mathrm{Q})_{u_{m}}(\phi)\longrightarrow(\mathrm{d}\mathrm{Q})_{u}(\phi)$,
both strongly in E as $m\to\infty$. Hence, (1.25) and (1.26) are satisfied.

###  4.4. Null Lagrangians and determinant constraints

We close this paper with the observation that Theorem 1 holds true even when
$\mathrm{Q}$ expresses a fully nonlinear pointwise jacobian determinant
constraint, or even a more general pointwise PDE constraint driven by a null
Lagrangian. As an explicit example, let $n=N$ and consider the differential
operator

$\mathrm{Q}\ :\ \ W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{n})\longrightarrow
W^{-1,(\bar{p}/n)^{\prime}}(\Omega),$

by setting

$\mathrm{Q}(u)\,:=\,\det(\mathrm{D}u)-h,$

for a fixed $h\in L^{\bar{p}/n}(\Omega)$, satisfying the necessary
compatibility condition

$\int_{\Omega}h\,\mathrm{d}\mathcal{L}^{n}\,=\,0.$

We also take

$\textbf{E}\,:=\,W^{-1,(\bar{p}/n)^{\prime}}(\Omega)\,=\,\big{(}W_{0}^{1,\bar{p}/n}(\Omega)\big{)}^{*}.$

Then, we have

$\mathrm{Q}^{-1}(\\{0\\})\,=\,\Big{\\{}u\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{n})\ :\ \det(\mathrm{D}u)=h\text{ a.e.\
in }\Omega\Big{\\}}.$

It follows that (1.9) is satisfied by the well-known property of weak
continuity for jacobian determinants (see e.g. [14, Th. 8.20, p. 395]).
However, the situation is more complicated regarding the satisfaction of the
remaining assumptions. If additionally $n=2$, then (1.25) and (1.26) are also
satisfied. Indeed, since

$(\mathrm{d}\mathrm{Q})_{u}(\phi)\,=\,\mathrm{cof}(\mathrm{D}u):\mathrm{D}\phi,$

and since for $u=\phi$ we have the identity

$(\mathrm{d}\mathrm{Q})_{u}(u)\,=\,\mathrm{cof}(\mathrm{D}u):\mathrm{D}u\,=\,n\det(\mathrm{D}u),$

for any $(u_{m})_{1}^{\infty}\subseteq\mathrm{Q}^{-1}(\\{0\\})$ with
$u_{m}\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup u$ as $m\to\infty$, we
have

$\begin{split}(\mathrm{d}\mathrm{Q})_{u_{m}}(u_{m})\,=\,n\det(\mathrm{D}u_{m})\,\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup\,n\det(\mathrm{D}u)\,=\,(\mathrm{d}\mathrm{Q})_{u}(u)\end{split}$

in $L^{\bar{p}/2}(\Omega)$ as $m\to\infty$, whilst for any $\phi\in
W^{1,\bar{p}}_{0}(\Omega;\mathbb{R}^{2})$ we have

$(\mathrm{d}\mathrm{Q})_{u_{m}}(\phi)\,-\\!\\!\\!\\!\\!-\\!\\!\\!\\!\rightharpoonup(\mathrm{d}\mathrm{Q})_{u}(\phi)$

in $L^{\bar{p}/2}(\Omega)$ as $m\to\infty$, by the linearity of the cofactor
operator when $n=2$. Then, the compactness of the imbedding

$L^{\bar{p}/2}(\Omega)\,\Subset\,W^{-1,(\bar{p}/2)^{\prime}}(\Omega)$

implies that the above modes of convergence are in fact strong in
$\textbf{E}=W^{-1,(\bar{p}/2)^{\prime}}(\Omega)$. However, it is not clear
when assumption (1.15) is satisfied, or when (1.26) is satisfied in the case
that $n\geq 3$. This means Theorems 2 and 3 as they stand do not apply to the
case of jacobian constraints. This does _not_ mean that it is impossible to
derive the associated equations, it merely means that in this case of such a
highly nonlinear constraint a different specialised method of proof is
required.

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