#  A nonstandard growth Steklov optimization problem with volume constraint

Ariel Salort, Belem Schvager and Analía Silva  Departamento de Matemática
FCEyN - Universidad de Buenos Aires and IMAS - CONICET. Ciudad Universitaria,
Pabellón I (C1428EGA) Av. Cantilo 2160. Buenos Aires, Argentina.
<EMAIL_ADDRESS>http://mate.dm.uba.ar/ asalort/ Instituto de Matemática
Aplicada San luis (IMASL), Universidad Nacional de San Luis, CONICET. Ejercito
de los Andes 950, D5700HHW , San Luis, Argentina<EMAIL_ADDRESS>https://analiasilva.weebly.com Instituto de Matemática Aplicada San luis
(IMASL), Universidad Nacional de San Luis, CONICET. Ejercito de los Andes 950,
D5700HHW , San Luis, Argentina<EMAIL_ADDRESS>
###### Abstract.

In this article we study an optimal design problem for a nonstandard growth
Steklov eigenvalues ruled by the $g-$Laplacian operator. More precisely, given
$\Omega\subset\mathbb{R}^{n}$ and $\alpha,c>0$ we analyze existence and
symmetry properties of solution of the optimization problem
$\inf\\{\lambda(\alpha,E)\colon E\subset\Omega,|E|=c\\}$, where, for a
suitable function $u(\alpha,E)$, $\lambda(\alpha,E)$ solves

$\begin{cases}-\operatorname{\text{div}}(g(|\nabla u|)\frac{\nabla u}{|\nabla
u|})+(1+\alpha\chi_{E})g(|\nabla u|)\frac{\nabla u}{|\nabla u|}=0&\text{ in
}\Omega,\\\ g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\cdot\eta=\lambda
g(|u|)\frac{u}{|u|}&\text{ on }\partial\Omega\end{cases}$

being $g$ the derivative of a Young function, and $\eta$ the unit outward
normal derivative.

We analyze the behavior of the optimization problem as $\alpha$ approaches
infinity and its connection of the trace embedding for Orlicz-Sobolev
functions.

###### Key words and phrases:

Orlicz-Sobolev spaces, optimization problems

###### 2010 Mathematics Subject Classification:

35J60, 35J66, 35Q93, 46E30

##  1\. Introduction

The literature on optimization problems is very wide, from the classical cases
of isoperimetrical problems to the most recent applications including
elasticity and spectral optimization. Only to mention some references and
motivations, we refer the reader to the books of Allaire [1], Bucur and
Buttazzo [2], Henrot [16], Pironneau [25] and Sokolowski and Zolésio [27],
where a huge amount of shape optimization problems is introduced. Optimization
problems in the more general form can be stated as follows: given a cost
functional $\mathcal{F}$ and a class of admissible domains $\mathcal{D}$,
solve the minimization problem

(1.1) $\min\left\\{\mathcal{F}(D)\colon D\in\mathcal{D}\right\\}.$

In recent years there has been an increasing amount of interest in
optimization problems for power-like functionals, see [1, 2, 3, 4, 8, 9, 11,
12, 13, 16, 21, 25, 27] for instance. Moreover, optimization problems
describing non-local phenomena have been approached recently [5, 11, 14, 24].
However, optimization problems of the form (1.1), where the _state equation_
to be solved on $D$ involves behaviors more general than powers are less
common in the literature. Additional drawbacks can arise in these class of
problems due to the possible lack of homogeneity of the functional. We cite
for instance the articles [6, 22, 23, 30].

Inspired in [3, 11], in this manuscript we study the existence of an optimal
configuration for a minimization problem ruled by the nonlinear degenerate and
possibly not homogeneous operator _$g$ -Laplacian_ defined as
$\Delta_{g}u:=\operatorname{\text{div}}(g(|\nabla u|)\frac{\nabla u}{|\nabla
u|})$, where $G(t)=\int_{0}^{t}g(s)\,ds$ is a Young function fulfilling the
following growth condition

(L) $1<p^{-}\leq\frac{tg(t)}{G(t)}\leq p^{+}<\infty\qquad\text{for all
}t\in\mathbb{R}^{+},$

for fixed constants $p^{+}$ and $p^{-}$.

These kind of problems appears naturally when studying general optimal design
problems, and they are usually formulated as problems of minimization of the
energy, stored in the design under a prescribed loading. Solutions of these
problems are unstable to perturbations of the loading, and the optimal design
problem is formulated as minimization of the stored energy under the most
unfavorable loading. This most dangerous loading is one that maximizes the
stored energy over the class of admissible functions. Finally, the problem is
reduced to minimization involving the behavior on the boundary. See [3, 26]
for more details.

We describe now our problem and main aims. Given a bounded open set
$\Omega\subset\mathbb{R}^{n}$, a fixed subset $E$ of $\Omega$ and a fixed
number $\alpha\in\mathbb{R}$, we consider the following quantity

(1.2) $\lambda(\alpha,E):=\min_{v\in\mathcal{A}}\left\\{\int_{\Omega}G(|\nabla
v|)+G(|v|)\,dx+\alpha\int_{E}G(|v|)\,dx\right\\},$

where the class $\mathcal{A}$ of admissible functions is given by

$\mathcal{A}:=\left\\{v\in W^{1,G}(\Omega)\colon\text{ such that
}\int_{\partial\Omega}G(|v|)\,d\mathcal{H}^{n-1}=1\right\\}.$

See Section 2 for precise definitions. Existence of such a number is
guaranteed by the compactness of the embedding of the Orlicz-Sobolev space
$W^{1,G}(\Omega)$ into $L^{G}(\partial\Omega)$ under suitable hypothesis on
$G$. In fact, (1.2) turns out to be an eigenvalue of the following problem
with Steklov boundary condition:

(1.3)
$\displaystyle\begin{cases}-\Delta_{g}u+(1+\alpha\chi_{E})g(|u|)\frac{u}{|u|}=0&\text{
in }\Omega\\\ g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\cdot\eta=\mu
g(|u|)\frac{u}{|u|}&\text{ on }\partial\Omega,\end{cases}$

where $\eta$ denotes the outer normal to $\partial\Omega$ and $\mu$ is a real
parameter.

Our main goal is to optimize the eigenvalue $\lambda(\alpha,E)$ with respect
to the class of sets $E\subset\Omega$ of fixed volume, i.e., given fixed
values of $\alpha$ and $c\in[0,|\Omega|]$, to analyze the existence of
solution of the optimization problem

(1.4) $\lambda(\alpha,c):=\inf\left\\{\lambda(\alpha,E)\colon
E\subset\Omega,|E|=c\right\\}.$

Such a solution, when exists, will be referred as a _classical solution_ of
(1.4).

In order to prove the existence of an optimal configuration of (1.4), we adopt
the strategy introduced in [11] and we consider a relaxed version of the
problem. However, now we have an extra difficulty due to the loss of
homogeneity of the equation. Indeed, instead of minimizing eigenvalues of
(1.3), we consider eigenvalues of the relaxed eigenvalue problem

(1.5)
$\displaystyle\begin{cases}-\Delta_{g}u+(1+\alpha\phi)g(|u|)\frac{u}{|u|}=0&\text{
in }\Omega\\\ g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\cdot\eta=\Lambda
g(|u|)\frac{u}{|u|}&\text{ on }\partial\Omega.\end{cases}$

where the function $\phi$ belongs to the class

$\mathcal{B}:=\left\\{\phi\in L^{\infty}(\Omega)\text{ such that
}0\leq\phi\leq 1\text{ and }\int_{\Omega}\phi\,dx=c\right\\}.$

That is, instead of (1.2) we consider the eigenvalue $\Lambda(\alpha,\phi)$
given by

(1.6)
$\Lambda(\alpha,\phi):=\min_{v\in\mathcal{A}}\left\\{\int_{\Omega}G(|\nabla
v|)+G(|v|)\,dx+\alpha\int_{\Omega}\phi G(|v|)\,dx\right\\},$

and the relaxed version of (1.4) becomes to optimize (1.6) over the class
$\mathcal{B}$, i.e.,

(1.7)
$\Lambda(\alpha,c):=\inf\\{\Lambda(\alpha,\phi)\colon\phi\in\mathcal{B}\\}.$

Existence of a solution $u=u(\alpha,\phi)\in\mathcal{A}$ of (1.5) is derived
from the direct method of the calculus of variations, and the Harnack’s
inequality in this setting ensure the strict positive of $u$ in
$\overline{\Omega}$. Any minimizer of (1.7) will be called an _optimal
configuration_ of the data $(\alpha,c)$. If $\phi$ is an optimal configuration
and $u$ satisfies (1.6), then $(u,\phi)$ is called an _optimal pair_ (or
solution).

Our first result established that an optimal pair of the relaxed problem
allows to recover a solution of the original problem (1.4). More precisely, if
$(u,\phi)$ is an optimal pair, then $\phi=\chi_{D}$, for some measurable set
$D\subset\Omega$. Moreover, the set $D$ is shown to be a sublevel set of $u$.

######  Theorem 1.1.

For any $\alpha>0$ and $c\in[0,|\Omega|]$ there exists an optimal pair.
Moreover, any optimal pair $(u,\phi)$ has the following properties:

  * (i)

$u\in C^{1,\gamma}{(\overline{\Omega})}$ for same $\gamma\in(0,1)$.

  * (ii)

There exists an optimal configuration $\phi=\chi_{D}$, where $D$ is a sublevel
set of $u$, that is, there is a number $t\geq 0$ such that $D=\\{u\leq t\\}$.

  * (iii)

Every level set $\\{u=s\\}$, has Lebesgue measure zero.

Theorem 1.1 clearly boils down to a minimization problem ruled by the
$p-$Laplacian operator when $G(t)=t^{p}$, $p>1$, and recovers the main result
of [11].

When in particular $\Omega$ is the unit ball $B_{1}$, we obtain existence of a
spherically symmetric optimal configuration:

######  Theorem 1.2.

Fix $\alpha>0$ and $c\in(0,|B_{1}|)$, there exists an optimal pair
$(u,\chi_{D})$ of (1.6) such that $u$ and $D$ are spherically symmetric.

Once the set $E$ is fixed in (1.2), it is easy to see that when
$\alpha\to\infty$ the quantity $\lambda(\alpha,E)$ converge to to the
minimizer of the problem with $E$ as a hole in $\Omega$ (that is, the
minimizing function vanishes on $E$), i.e.,

(1.8)
$\lambda(\infty,E):=\lim_{\alpha\to\infty}\lambda(\alpha,E)=\inf_{v\in\mathcal{A},v|_{E}\equiv
0}\left\\{\int_{\Omega}G(|\nabla v|)+G(|v|)\,dx\right\\}.$

The natural limit optimization problem in this case is

(1.9) $\lambda(\infty,c):=\inf\\{\lambda(\infty,E)\colon
E\subset\Omega,|E|=c\\}.$

A natural question is whether the optimal configuration of (1.4) converges to
these of (1.9) when $\alpha\to\infty$. The following result answers positively
to that issue.

######  Theorem 1.3.

For any sequence $\alpha_{k}\to\infty$ and optimal pairs $(D_{k},u_{k})$ of
(1.4) there exists a subsequence, still denoted $\alpha_{k}$, and an optimal
pair $(D,u)$ of (1.9) such that

$\displaystyle\lim_{k\to\infty}\chi_{D_{k}}=\chi_{D}$
$\displaystyle\quad\text{weakly* in }L^{\infty}(\Omega),$
$\displaystyle\lim_{k\to\infty}u_{k}=u$ $\displaystyle\quad\text{strongly in
}W^{1,G}(\Omega).$

Moreover, $u>0$ in $\Omega\setminus D$.

The paper is organized as follows. In Section 2 we introduce the notation and
basic facts on Orlicz-Sobolev spaces used along the manuscript. Section 3 is
devoted to study the link between the minimization problems (1.2) and (1.6)
and the eigenvalue problems (1.3) and (1.5). In Sections 4 and 5 we prove our
main results stated in Theorems 1.1, 1.2 and 1.3. Finally in Section 6 we
state some possible generalizations.

##  2\. Preliminaries

In this section we introduce some notation and basic results on Orlicz-Sobolev
spaces that we will use in this paper.

###  2.1. Young functions

An application $G\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is said to be a _Young
function_ if it admits the integral formulation
$G(t)=\int_{0}^{t}g(\tau)\,d\tau$, where the right continuous function $g$
defined on $[0,\infty)$ has the following properties:

($g_{1}$) $\displaystyle g(0)=0,\quad g(t)>0\text{ for }t>0,$ ($g_{2}$)
$\displaystyle g\text{ is non-decreasing on }(0,\infty),$ ($g_{3}$)
$\displaystyle\lim_{t\to\infty}g(t)=\infty.$

From these properties it is easy to see that a Young function $G$ is
continuous, non negative, strictly increasing and convex on $[0,\infty)$.

We will assume the following growth condition on Young functions: there exist
fixed constants $p^{\pm}$ such that

(2.1) $1\leq p^{-}\leq\frac{tg(t)}{G(t)}\leq p^{+}<\infty,\qquad\text{ for all
}t>0.$

The following properties on Young functions are well-known. See for instance
[17] for the proof of these results.

######  Lemma 2.1.

Let $G$ be a Young function satisfying (2.1) and $a,b\geq 0$. Then

($L_{1}$) $\displaystyle\min\\{a^{p^{-}},a^{p^{+}}\\}G(b)\leq
G(ab)\leq\max\\{a^{p^{-}},a^{p^{+}}\\}G(b),$ ($L_{2}$) $\displaystyle
G(a+b)\leq\mathbf{C}(G(a)+G(b))\quad\text{with }\mathbf{C}:=2^{p^{+}},$
($L_{3}$) $\displaystyle G\text{ is Lipschitz continuous}.$

Condition (2.1) is known as the _$\Delta_{2}$ condition_ or _doubling
condition_ and, as it is showed in [17, Theorem 3.4.4], it is equivalent to
the right hand side inequality in (2.1).

The _complementary_ Young function of a Young function $G$ is defined as

$\displaystyle\tilde{G}(t):=\sup_{s\geq 0}\\{st-G(s)\\}.$

It is easy to see that the left hand side inequality in (2.1) is equivalent to
assume that $\tilde{G}$ satisfies the $\Delta_{2}$ condition.

###  2.2. Orlicz-Sobolev spaces

Given a Young function $G$ and a bounded set $\Omega$ we consider the spaces

$\displaystyle L^{G}(\Omega)$
$\displaystyle:=\\{u:\mathbb{R}\to\mathbb{R}\text{ measurable such that
}\Phi_{G,\Omega}(u)<\infty\\},$ $\displaystyle L^{G}(\partial\Omega)$
$\displaystyle:=\\{u:\mathbb{R}\to\mathbb{R}\text{ measurable such that
}\Phi_{G,\partial\Omega}(u)<\infty\\},$ $\displaystyle W^{1,G}(\Omega)$
$\displaystyle:=\\{u\in L^{G}(\Omega)\text{ such that }\Phi_{G,\Omega}(|\nabla
u|)<\infty\\},$

where modulars are defined as

$\Phi_{G,\Omega}(u):=\int_{\Omega}G(|u|)\,dx,$
$\Phi_{G,\partial\Omega}(u):=\int_{\partial\Omega}G(|u|)\,d\mathcal{H}^{n-1}.$

These spaces are endowed with the so called Luxemburg norm defined as follows

$\displaystyle\|u\|_{L^{G}(\Omega)}$
$\displaystyle=\inf\left\\{\lambda>0:\Phi_{G,\Omega}\left(\frac{u}{\lambda}\right)\leq
1\right\\},$ $\displaystyle\|u\|_{L^{G}(\partial\Omega)}$
$\displaystyle=\inf\left\\{\lambda>0:\Phi_{G,\partial\Omega}\left(\frac{u}{\lambda}\right)\leq
1\right\\},$ $\displaystyle\|u\|_{W^{1,G}(\Omega)}$
$\displaystyle=\|u\|_{L^{G}(\Omega)}+\|\nabla u\|_{L^{G}(\Omega)},$

and are reflexive and separable Banach spaces if and only if $G$ y $\tilde{G}$
satisfies the $\Delta_{2}$ condition.

For the sake of simplicity, given $u\in W^{1,G}(\Omega)$ and $\phi\in
L^{\infty}(\Omega)$ we denote

$\displaystyle\Phi_{G,\phi,\Omega}(u):=\int_{\Omega}\phi G(|u|)\,dx,$
$\displaystyle\Phi_{1,G,\Omega}(u):=\int_{\Omega}G(|u|)+G(|\nabla u|)\,dx.$

We conclude this subsection by recalling that under our assumptions,
boundedness of the modular implies boundedness of the norm (see for instance
[7, Lemma 2.1.12]).

######  Lemma 2.2.

Let $G$ be a Young function satisfying (2.1). Then, if
$\Phi_{G,\partial\Omega}(u)$ (resp. $\Phi_{G,\Omega}(u)$) is bounded, then
$\|u\|_{L^{G}(\partial\Omega)}$ (resp. $\|u\|_{L^{G}(\Omega)}$) is bounded.

###  2.3. Some embedding results

In order to guarantee compact embeddings the following conditions will be
assumed

(2.2)
$\int_{0}^{1}\frac{G^{-1}(s)}{s^{1+\frac{1}{n}}}ds<\infty\quad\text{and}\quad\int_{1}^{\infty}\frac{G^{-1}(s)}{s^{1+\frac{1}{n}}}ds=\infty.$

For any Young function satisfying (2.2), the _Sobolev critical function_ is
defined as

$G_{*}^{-1}(t)=\int_{0}^{t}\frac{G^{-1}(s)}{s^{\frac{n+1}{n}}}ds.$

Combining [17, Theorems 7.4.6 and 7.4.6] with [6, Example 6.3] the following
embedding holds.

######  Proposition 2.3.

Let $G$ be a Young function satisfying (2.1) and (2.2). Let
$\Omega\subset\mathbb{R}^{n}$ be a $C^{0,1}$ bounded open subset. Then the
embeddings

$W^{1,G}(\Omega)\hookrightarrow L^{G}(\Omega),\qquad
W^{1,G}(\Omega)\hookrightarrow L^{G}(\partial\Omega)$

are compact.

##  3\. The minimization problems and their related eigenvalue problems

From now on, we will always assume that $G$ is a Young function satisfying
(2.1) and (2.2).

We start this section by studying some basic properties of the relaxed
minimization problem stated in (1.6).

Given $\alpha>0$ and a function $\phi\in L^{\infty}(\Omega)$, in light of the
definition of the set $\mathcal{A}$ observe that the quantity defined in (1.6)
can be reformulated as

(3.1)
$\Lambda(\alpha,\phi):=\inf_{v\in\mathcal{A}}\frac{\mathcal{I}(v)}{\mathcal{J}(v)}$

where the functionals $\mathcal{I},\mathcal{J}\colon
W^{1,G}(\Omega)\to\mathbb{R}$ are defined as

$\displaystyle\mathcal{I}(v)$
$\displaystyle:=\Phi_{1,G,\Omega}(v)+\alpha\Phi_{G,\phi,\Omega}(v),$
$\displaystyle\mathcal{J}(v)$ $\displaystyle:=\Phi_{G,\partial\Omega}(v).$

Since $G$ and its conjugated function $\tilde{G}$ satisfy the $\Delta_{2}$
condition, it is straightforward to see that $\mathcal{I}$ and $\mathcal{J}$
are class $C^{1}(\Omega)$ and their Frechét derivatives $\mathcal{I}^{\prime}$
and $\mathcal{J}^{\prime}$ are defined from $W^{1,G}(\Omega)$ into its dual
space and are given by

$\displaystyle\langle\mathcal{I}^{\prime}(u),v\rangle$
$\displaystyle=\int_{\Omega}g(|\nabla u|)\frac{\nabla u}{|\nabla
u|}\cdot\nabla
v\,dx+\int_{\Omega}(1+\alpha\phi)g(|u|)\frac{u}{|u|}v\,dx,\quad\forall u,v\in
W^{1,G}(\Omega)$ $\displaystyle\langle\mathcal{J}^{\prime}(u),v\rangle$
$\displaystyle=\int_{\partial\Omega}g(|u|)\frac{u}{|u|}v\,d\mathcal{H}^{n-1}.$

We start by proving that the infimum in (3.1) is attained.

######  Proposition 3.1.

Given $\alpha>0$ and $\phi\in L^{\infty}(\Omega)$ there exists
$u\in\mathcal{A}$ solving (3.1).

###### Proof.

This is a consequence of the direct method in the calculus of variations.
Indeed, let $\\{u_{k}\\}_{k\in\mathbb{N}}$ be a minimizing sequence for
$\Lambda(\alpha,\phi)$, that is

(3.2)
$u_{k}\in\mathcal{A}\quad\text{and}\quad\lim_{k\to\infty}\Phi_{1,G,\Omega}(u_{k})+\alpha\Phi_{G,\phi,\Omega}(u_{k})=\Lambda(\alpha,\phi).$

Observe that $\\{u_{k}\\}_{k\in\mathbb{N}}$ is bounded in
$L^{G}(\partial\Omega)$ and for $k$ big enough, $\Phi_{1,G,\Omega}(u_{k})\leq
1+\Lambda(\alpha,\phi)$. Therefore, since $W^{1,G}(\Omega)$ is a reflexive
space, due Lemma 2.2 together with the compact embedding stated in Proposition
2.3 there exists a function $u\in W^{1,G}(\Omega)$ and a sub-sequence
$\\{u_{k_{j}}\\}_{j\in\mathbb{N}}\subset\\{u_{k}\\}_{k\in\mathbb{N}}$ such
that

$\displaystyle u_{k_{j}}\rightharpoonup u\;\text{ weakly in }W^{1,G}(\Omega),$
$\displaystyle u_{k_{j}}\to u\;\text{ strongly in }L^{G}(\Omega),$
$\displaystyle u_{k_{j}}\to u\;\text{ strongly in }L^{G}(\partial\Omega),$
$\displaystyle u_{k_{j}}\to u\;\text{ a.e. in }L^{G}(\Omega).$

The strong convergence in $L^{G}(\partial\Omega)$ implies that
$\Phi_{G,\partial\Omega}(u)=1$, and therefore $u\in\mathcal{A}$. Hence, by
definition

$\Lambda(\alpha,\phi)\leq\Phi_{1,G,\Omega}(u)+\alpha\Phi_{G,\phi,\Omega}(u).$

On the other hand, since modulars are lower semi-continuous, by Fatou’s Lemma
we get

$\displaystyle\Phi_{1,G,\Omega}(u)+\alpha\Phi_{G,\phi,\Omega}(u)$
$\displaystyle\leq\liminf_{j\to\infty}\Phi_{1,G,\Omega}(u_{k_{j}})+\alpha\Phi_{G,\phi,\Omega}(u_{k_{j}})=\Lambda(\alpha,\phi).$

From the last two inequalities the lemma follows. ∎

We define the Euler-Lagrange equation corresponding to the constrained
minimization problem (3.1): given $\alpha>0$ and $\phi\in L^{\infty}(\Omega)$
we consider the following eigenvalue problem with Steklov boundary condition

(3.3) $\begin{cases}-\Delta_{g}u+g(|u|)\frac{u}{|u|}+\alpha\phi
g(|u|)\frac{u}{|u|}=0&\text{ in }\Omega\\\ g(|\nabla u|)\frac{\nabla
u}{|\nabla u|}\cdot\eta=\mu g(|u|)\frac{u}{|u|}&\text{ on
}\partial\Omega,\end{cases}$

where $\eta$ denotes the outer unit normal derivative on $\partial\Omega$. We
say that $\mu\in\mathbb{R}$ is an eigenvalue of (3.3) with associated
eigenfunction $u\in\mathcal{A}$ if for any $v\in W^{1,G}(\Omega)$ it holds
that

$\langle\mathcal{I}^{\prime}(u),v\rangle=\mu\langle\mathcal{J}^{\prime}(u),v\rangle.$

We prove now that (3.1) is actually an eigenvalue of the previous Steklov
problem.

######  Proposition 3.2.

The number $\Lambda(\alpha,\phi)$ defined in (3.1) is an eigenvalue of (3.3).

###### Proof.

Proposition 3.1 guarantees that there is $u\in\mathcal{A}$ such that
$\Lambda(\alpha,\phi)=\frac{\mathcal{I}(u)}{\mathcal{J}(u)}$, i.e., $u$ solves
the minimization problem 3.1. Hence, for any $v\in W^{1,G}(\Omega)$ we get

$\displaystyle 0=$
$\displaystyle\left\langle\left(\frac{\mathcal{I}(u)}{\mathcal{J}(u)}\right)^{\prime},v\right\rangle=\left\langle\frac{\mathcal{I}^{\prime}(u)\mathcal{J}(u)-\mathcal{I}(u)\mathcal{J}^{\prime}(u)}{\mathcal{J}(u)^{2}},v\right\rangle$
$\displaystyle=\left\langle\frac{\mathcal{I}^{\prime}(u)}{\mathcal{J}(u)},v\right\rangle-\left\langle\frac{\mathcal{I}(u)\mathcal{J}^{\prime}(u)}{\mathcal{J}(u)^{2}},v\right\rangle=\frac{1}{\mathcal{J}(u)}\langle{\mathcal{I}^{\prime}(u),v}\rangle-\frac{1}{\mathcal{J}(u)}\left\langle\frac{\mathcal{I}(u)}{\mathcal{J}(u)}\mathcal{J}^{\prime}(u),v\right\rangle$

which gives that

$\langle\mathcal{I}^{\prime}(u),v\rangle=\frac{\mathcal{I}(u)}{\mathcal{J}(u)}\langle\mathcal{J}^{\prime}(u),v\rangle=\Lambda(\alpha,\phi)\langle\mathcal{J}^{\prime}(u),v\rangle$

for all $v\in W^{1,G}(\Omega)$, concluding the proof. ∎

The next result states some useful properties of minimizers of the relaxed
problem.

######  Proposition 3.3.

Let $u\in\mathcal{A}$ be a solution of (3.1). Then $u\in
C^{1,\gamma}(\overline{\Omega})$ and $u>0$ in $\overline{\Omega}$.

###### Proof.

Let $u\in\mathcal{A}$ be a solution of (3.1). By the regularity theory of [28,
Corollary 3.1] we have that $u\in C^{1,\gamma}(\overline{\Omega})$ for some
$\gamma\in(0,1)$. Moreover, since $|u|$ also solves (3.1), we can assume that
$u\geq 0$ in $\Omega$, and by the strong maximum principle stated in [19] we
have that either $u>0$ in $\Omega$ or $u$ is constant. If there would exist
$x\in\Omega$ such that $u(x)=0$, then we would have $u=0$ in
$\overline{\Omega}$. Therefore $u>0$ in $\overline{\Omega}$. ∎

We conclude this section by remarking that a completely analogous analysis can
be done to link the quantity $\lambda(\alpha,E)$ defined in (1.2) and the
eigenvalue problem (1.3):

######  Proposition 3.4.

The number $\lambda(\alpha,E)$ defined in (1.2) is an eigenvalue of (1.3).

##  4\. An existence result

This section is devoted to prove the existence of an optimal configuration of
(1.4) and to analyze some properties which it fulfills, namely, Theorem 1.1.

###### Proof of Theorem 1.1.

We first prove existence for a fixed $\alpha$ and $c$. For the sake of
simplicity we set $\Lambda=\Lambda(\alpha,c)$ and
$\Lambda(\phi)=\Lambda(\alpha,\phi)$.

Let $\\{\phi_{k}\\}_{k\in\mathbb{N}}$ be a minimizing sequence, i.e.,

$0\leq\phi_{k}\leq 1,\quad\int_{\Omega}\phi_{k}\,dx=c\quad\text{and
}\quad\lim_{k\to\infty}\Lambda(\phi_{k})=\Lambda.$

In light of Proposition 3.1, for each $k\in\mathbb{N}$ let
$u_{k}\in\mathcal{A}$ be such that

(4.1)
$\displaystyle\begin{split}\Lambda(\phi_{k})&=\inf\left\\{v\in\mathcal{A}\colon\Phi_{1,G,\Omega}(v)+\alpha\Phi_{G,\phi_{k},\Omega}(v)\right\\}\\\
&=\Phi_{1,G,\Omega}(u_{k})+\alpha\Phi_{G,\phi_{k},\Omega}(u_{k}).\\\
\end{split}$

Since $\phi_{k}\leq 1$ for all $k\in\mathbb{N}$ we have that
$\Lambda(\phi_{k})$ is uniformly bounded in $k$, and then
$\\{u_{k}\\}_{k\in\mathbb{N}}$ is uniformly bounded in $W^{1,G}(\Omega)$.
Therefore, from Proposition 2.3 up to a subsequence, there exist $u\in
W^{1,G}(\Omega)$ and $\phi\in L^{\infty}(\Omega)$ such that

$\displaystyle u_{k}\rightharpoonup u\;\text{ weakly in }W^{1,G}(\Omega),$
$\displaystyle u_{k}\to u\;\text{ strongly in }L^{G}(\Omega),$ $\displaystyle
u_{k}\to u\;\text{ strongly in }L^{G}(\partial\Omega),$
$\displaystyle\phi_{k}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\phi\;\text{
weakly* in }L^{\infty}(\Omega).$

The strong convergence of $u_{k}$ in $L^{G}(\partial\Omega)$ gives that
$\Phi_{G,\partial\Omega}(u)=1$ and the weak* convergence in
$L^{\infty}(\Omega)$ of $\phi_{k}$ gives that $0\leq\phi\leq 1$ and
$\int_{\Omega}\phi\,dx=c$.

Hence, taking limit in (4.1) and using the lower semicontinuity of the modular
we get

(4.2)
$\displaystyle\begin{split}\Lambda&=\lim_{k\to\infty}\Lambda(\phi_{k})=\liminf_{k\to\infty}\Phi_{1,G,\Omega}(u_{k})+\alpha\Phi_{G,\phi_{k},\Omega}(u_{k})\\\
&\geq\Phi_{1,G,\Omega}(u)+\alpha\Phi_{G,\phi,\Omega}(u).\end{split}$

On the other hand, since by definition of $\Lambda$, we have

(4.3)
$\Lambda\leq\Lambda(\phi)\leq\Phi_{1,G,\Omega}(u)+\alpha\Phi_{G,\phi,\Omega}(u).$

By (4.2) and (4.3) we obtain that $(u,\phi)$ is an optimal pair.

The proof of item (i) follows as a consequence of the regularity theory for
quasilinear elliptic equations in Orlicz Sobolev spaces. More precisely,
following the arguments of [15] and [19], in [28, Corollary 3.1] is stated the
boundary regularity for weak solutions of the $g$-Laplacian, which gives that
$u\in C^{1,\gamma}(\overline{\Omega})$ for some $\gamma\in(0,1)$.

Now, by the Bathtub Principle [20, Theorem 1.15], the minimization problem

(4.4) $\inf_{\phi\in\mathcal{B}}\Phi_{G,\phi,\Omega}(u)$

is solved by $\phi(x)=\chi_{D}(x)$, where $D$ is given by $D=\\{x\colon
u<t\\}\cup\mathcal{K}\cdot\\{x\colon u=t\\}$ and the number $\mathcal{K}$ is
chosen such that $\mathcal{K}|\\{x\colon u=t\\}|=c-|\\{x\colon u<t\\}|$, which
gives $|D|=c$. This principle also ensures that

$\\{u<t\\}\subset D\subset\\{u\leq t\\}\quad\text{ where
}t=\sup\\{s\colon|\\{u<s)\\}|\leq c\\}.$

Therefore, (4.2) and (4.4) yield

$\Lambda\geq\inf_{\phi\in\mathcal{B}}\\{\Phi_{1,G,\Omega}(u)+\alpha\Phi_{G,\phi,\Omega}(u)\\}=\Phi_{1,G,\Omega}(u)+\alpha\Phi_{G,\chi_{D},\Omega}(u).$

Moreover, since $\phi$ and $u$ are admissible in the characterization of
$\Lambda$, we get

$\displaystyle\Lambda$
$\displaystyle=\inf_{\phi\in\mathcal{B}}\inf_{v\in\mathcal{A}}\\{\Phi_{1,G,\Omega}(v)+\alpha\Phi_{G,\phi,\Omega}(v)\\}\leq\Phi_{1,G,\Omega}(u)+\alpha\Phi_{G,\chi_{D},\Omega}(u),$

and then $(u,\chi_{D})$ is an optimal pair.

Finally, if $(u,\chi_{D})$ is any solution and
$\mathcal{R}_{s}=\left\\{u=s\right\\}$ for any $s>0$. Since, $\nabla u=0$ a.e.
on $\mathcal{R}_{s}$, we get $\Delta_{g}u=0$ on $\mathcal{R}_{s}$. Then

$(1+\chi_{D})g(|u|)\frac{u}{|u|}=0\text{ a.e. on }\mathcal{R}_{s}.$

Since $u=s>0$ on $\mathcal{R}_{s}$, we have

$(1+\chi_{D})g(|u|)\frac{u}{|u|}>0\text{ on }\mathcal{R}_{s},$

from where $\left|\mathcal{R}_{s}\right|=0$. This proves $(iii)$. In
particular, when $s=t$ we get the last assertion in $(ii)$. ∎

In the following proposition we obtain the derivative of
$\Lambda(\alpha,\phi)$ in the direction of $f$ in the class $\mathcal{F}$
given by

$\mathcal{F}:=\left\\{f\colon f\leq 0\text{ in }\\{\phi=1\\},f\geq 0\text{ in
}\\{\phi=0\\},\int_{\Omega}f\,dx=0\right\\}.$

######  Proposition 4.1.

Let $f\in\mathcal{F}$, then the right derivative of $\Lambda(\alpha,\phi)$ in
the direction of $f\in\mathcal{F}$ is given by

(4.5) $\Lambda^{\prime}(\alpha,\phi)(f)=\lim_{t\to
0}\frac{\Lambda(\alpha,\phi+tf)-\Lambda(\alpha,\phi)}{t}=\alpha\Phi_{G,f,\Omega}(u)$

where $u$ is of solution of $\Lambda(\alpha,\phi).$

###### Proof.

Let us consider the function $\phi_{t}=\phi+tf$, $t\geq 0$. Since
$f\in\mathcal{F}$ and $\phi$ is an admissible function, it follows that
$\phi_{t}$ is admissible for every $t\geq 0$ small enough.

Consider an eigenfunction $u_{t}$ of $\Lambda(\alpha,\phi_{t})$ and an
eigenfunction $u$ of $\Lambda(\alpha,\phi)$. First, using $u_{t}$ in the
variational formulation of $\Lambda(\alpha,\phi)$ we obtain

(4.6)
$\frac{\Lambda(\alpha,\phi_{t})-\Lambda(\alpha,\phi)}{t}\geq\alpha\Phi_{G,f,\Omega}(u_{t}).$

Moreover, taking $u$ in the variational formulation of
$\Lambda(\alpha,\phi_{t})$ we obtain

(4.7)
$\frac{\Lambda(\alpha,\phi_{t})-\Lambda(\alpha,\phi)}{t}\leq\alpha\Phi_{G,f,\Omega}(u).$

Now, using $v=G^{-1}(|\partial\Omega|^{-1})$ as a test function in the
characterization of $\Lambda(\alpha,\phi_{t})$ we obtain that the family
$u_{t}$ is bounded in $W^{1,G}(\Omega)$ for $0<t\leq t_{0}$. Then, from
Proposition 2.3, $u_{t}\rightarrow u$ strongly in $L^{G}(\Omega)$ when
$t\rightarrow 0$. Finally, taking limit as $t\to 0$ in (4.6) and (4.7), we
obtain (4.5) as desired. ∎

As a direct consequence of Proposition 4.5 we can provide for an alternative
proof to the fact the the optimal set is a sublevel set of $u$, claim which
was already proved in the second part of (ii) from Theorem 1.1.

######  Corollary 4.2.

Let $u$ be a solution to (1.4). Then the optimal set D satisfies
$D=\left\\{u\leq t\right\\}$ for some $t\geq 0$.

###### Proof.

Since in the first part of (ii) from Theorem 1.1 we have already proved that
$\chi_{D}$ realizes the minimum of $\Lambda(\alpha,\phi)$, we have that for
all $f\in\mathcal{F}$,

(4.8) $\Lambda^{\prime}(\alpha,\phi)(f)=\alpha\Phi_{G,f,\Omega}(u)\geq 0.$

Given a point $x_{0}\in D$ of positive density i.e., for every
$\varepsilon\geq 0,\left|B(x_{0},\varepsilon)\cap D\right|>0$, and
$x_{1}\in(\Omega\setminus D)$ also with positive density, we can take a
function $f\in\mathcal{F}$ of the form
$f:=M\chi_{T_{1}}-M\chi_{T_{0}}\in\mathcal{F}$ with $T_{0}\subset
B(x_{0},\varepsilon)\cap D$ and $T_{1}\subset
B(x_{1},\varepsilon)\cap(\Omega\setminus D)$ and
$M^{-1}:=\left|T_{0}\right|=\left|T_{1}\right|$. Hence,

$\displaystyle 0\leq\alpha\Phi_{G,f,\Omega}(u)$
$\displaystyle=\alpha\int_{\Omega}(M\chi_{T_{1}}-M\chi_{T_{0}})G(|u|)dx$
$\displaystyle=\alpha\frac{1}{|T_{1}|}\int_{T_{1}}G(|u|)\,dx-\alpha\frac{1}{|T_{0}|}\int_{T_{0}}G(|u|)dx,$

and taking limit when $\varepsilon\to 0$ and using the continuity of $u$ (from
Theorem 1.1 $u\in C^{1,\gamma}(\bar{\Omega})$) we have that $G(u(x_{0}))\leq
G(u(x_{1}))$. Therefore by the monotonicity of $G$, we obtain $u(x_{0})\leq
u(x_{1})$. We conclude that $D=\left\\{u\leq t\right\\}$ as we wanted to
prove. ∎

We conclude this section by proving Theorem 1.2. For that end, we recall some
basic results on _spherical symmetrization_ of functions on Orlicz-Sobolev
spaces.

Given a mensurable set $\Omega\subset\mathbb{R}^{n}$, the spherical
symmetrization $\Omega^{*}$ of $\Omega$ with respect to an axis given by a
unit vector $e_{k}$ reads as follows: for each positive number $r$, take the
intersection $\Omega\cap\partial B(0,r)$ and replace it by the spherical cap
of the same $\mathcal{H}^{n-1}-$measure and center $re_{k}$. Hence,
$\Omega^{*}$ is the union of these caps.

Now, the spherical symmetrization $u^{*}$ of a measurable function
$u\colon\Omega\to\mathbb{R}_{+}$ is constructed by symmetrizing the super-
level sets so that, for all $t$

$\\{u^{*}\geq t\\}=\\{u\geq t\\}^{*}.$

We refer to [18] for more details.

Denoting by $B_{1}$ the ball of unit radius centered at the origin, the
following properties on symmetrization can be found in [6, Theorem 4.1] (see
also [18]).

######  Proposition 4.3.

Let $u\in W^{1,G}(B_{1})$ and $u^{*}$ be its spherical symmetrization. Then,
$u^{*}\in W^{1,G}(B_{1})$. Moreover,

  * (i)

$\Phi_{G,B_{1}}(u^{*})=\Phi_{G,B_{1}}(u)$,

  * (ii)

$\Phi_{G,\partial B_{1}}(u^{*})=\Phi_{G,\partial B_{1}}(u)$,

  * (iii)

$\Phi_{G,B_{1}}(|\nabla u^{*}|)\leq\Phi_{G,B_{1}}(|\nabla u|)$,

  * (iv)

$\Phi_{G,(\alpha\chi_{D})_{*},B_{1}}(u^{*})\leq\Phi_{G,\alpha\chi_{D},B_{1}}(u)$,

where $D\subset B_{1}$ and $(\alpha\chi_{D})_{*}=-(-\alpha\chi_{D})^{*}$.

We are now in position to prove our symmetrization result for solutions.

###### Proof of Theorem 1.2.

Given $\alpha>0$ and $c\in(0,|B_{1}|)$, let $(u,\chi_{D})$ be an optimal pair
to (1.6) and let $u^{*}$ be the spherical symmetrization of $u$. Consider the
set $D^{*}$ defined by $\chi_{D^{*}}=(\chi_{D})_{*}$.

Observe that from item (ii) in Proposition 4.3, $u^{*}\in\mathcal{A}$ since
$u\in\mathcal{A}$. Then, using Proposition 4.3 again we get

$\displaystyle\lambda(\alpha,D^{*})$
$\displaystyle=\inf_{v\in\mathcal{A}}\\{\Phi_{1,G,B_{1}}(v)+\Phi_{G,(\alpha\chi_{D})_{*},B_{1}}(v)\\}$
$\displaystyle\leq\Phi_{1,G,B_{1}}(u^{*})+\Phi_{G,(\alpha\chi_{D})_{*},B_{1}}(u^{*})$
$\displaystyle\leq\Phi_{1,G,B_{1}}(u)+\alpha\Phi_{G,\chi_{D},B_{1}}(u)=\lambda(\alpha,D).$

Finally, since $|D^{*}|=c=|D|$, $(u^{*},\chi_{D^{*}})$ is also an optimal pair
to (1.6). ∎

##  5\. Limit as $\alpha\to\infty$

Fixed a subset $E\subset\Omega\subset\mathbb{R}^{n}$ and
$\alpha\in\mathbb{R}$, in our first result we have focused on the study of the
optimization problem

$\lambda(\alpha,c):=\inf\\{\lambda(\alpha,E)\colon E\subset\Omega,|E|=c\\}$

where the quantity $\lambda(\alpha,E)$ defined in (1.2) is an eigenvalue of
(1.3). In this section, fixed the value of $c$ we analyze the the behavior of
$\lambda(\alpha,c)$ as $\alpha\to\infty$. We recall that the corresponding
limit problem is defined as

$\displaystyle\lambda(\infty,c):$
$\displaystyle=\inf\left\\{\lim_{\alpha\to\infty}\lambda(\alpha,E)\colon
E\subset\Omega,|E|=c\right\\}$
$\displaystyle=\inf\left\\{\lambda(\infty,E)\colon
E\subset\Omega,|E|=c\right\\}.$

Our concern now is to analyze the convergence as $\alpha\to\infty$ of the
optimal configurations $\lambda(\alpha,c)$ obtained in Theorem 1.1 to those of
$\lambda(\infty,c)$. A key result for that purpose is to study a monotonicity
property of $\lambda(\infty,c)$ with respect to the parameter $c$.

######  Lemma 5.1.

$\lambda(\infty,c)$ is strictly monotonically increasing in $c$.

###### Proof.

We split the proof in three steps.
_Step 1._ First we show that

(5.1) $\inf_{|E|=c}\lambda(\infty,E)=\inf_{|E|\geq c}\lambda(\infty,E).$

It is clear by definition of infimum that

$\inf_{|E|=c}\lambda(\infty,E)\geq\inf_{|E|\geq c}\lambda(\infty,E).$

On the other hand, if $v$ is a test function for a set of measure larger or
equal to $c$ it is also a test function for a set of measure $c$. Thus, the
two infima coincide.

_Step 2._ We prove that, if $u$ is a extremal for $\lambda(\infty,c)$ then
$\left|\left\\{x:u(x)=0\right\\}\right|=c$. Suppose by contradiction that $u$
vanishes in a set $E$ with $|E|\geq c$. By taking a subset we may assume that
$E$ is closed. Let us take a small ball $B$ so that $|E\setminus B|\geq c$
with $B$ centered at a point in $\partial E\cap\partial\Omega_{1}$, where
$\Omega_{1}$ is the connected component of $\Omega\setminus E$ such that
$\partial\Omega\subset\partial\Omega_{1}$. We can pick the ball $B$ in such a
way that $\left|E\cap B\right|\geq 0$. In particular
$\left|\left\\{x:u(x)=0\right\\}\cap B\right|\geq 0$. Since that $u$ is an
extremal for $\lambda(\infty,c)$ and $\left|E\setminus B\right|>c$, it is an
extremal for $\lambda(\infty,\left|E\setminus B\right|)$. Thus, it holds that

$-\Delta_{g}u+g(|u|)\frac{u}{|u|}=0\quad\text{ in }\Omega\setminus(E\setminus
B)=(\Omega\setminus E)\cup B.$

Now, as $u\geq 0$, by the strong maximum principle (see [19, Lemma 6.4]) there
holds that either $u\equiv 0$ or $u>0$ in each connected component of
$(\Omega\setminus E)\cup B$. Since $u\neq 0$ on $\partial\Omega$ we get, in
particular, that $u>0$ in $B$. This is a contradiction to the choice of the
ball $B$. Therefore

$\left|\left\\{x:u(x)=0\right\\}\right|=c.$

_Step 3._ Finally, we show that $\lambda(\infty,c)$ is strictly monotonically
increasing in $c$.
From step 1, we get that $\lambda(\infty,c)$ is non decreasing respect to $c$.
On the other hand, let $0<c_{1}<c_{2}<\left|\Omega\right|$ such that
$\lambda(\infty,c_{1})=\lambda(\infty,c_{2})$ and let $u$ be an extremal for
$\lambda(\infty,c_{2})$ then, by step 2
$\left|\left\\{u=0\right\\}\right|=c_{2}$. But since $u$ is an admissible
function for $\lambda(\infty,c_{1})$, it is an extremal for
$\lambda(\infty,c_{1})$ with $\left|\left\\{x:u(x)=0\right\\}\right|>c_{1}$.
This contradicts step 2, and therefore $\lambda(\infty,c)$ is strictly
monotonically increasing in $c$. ∎

Now we are in position to prove our second main result.

###### Proof of Theorem 1.3.

Observe that in light of Theorem 1.1, if $(u_{\alpha},\chi_{D_{\alpha}})$ is
an optimal pair to $\Lambda(\alpha,c)$, then $(u_{\alpha},D_{\alpha})$ is an
optimal pair to $\lambda(\alpha,c)$. Then, if $u_{\alpha}\to u_{\infty}$ and
$D_{\alpha}\to D_{\infty}$ in some suitable sense as $\alpha\to\infty$ , we
would get that $(u_{\infty},D_{\infty})$ is an optimal pair to
$\lambda(\infty,c)$. Let us identify such a limit optimal pair.

We split the proof in two steps.

_Step 1_. We characterized the limit of $\Lambda(\alpha,c)$ as
$\alpha\to\infty$.

Given $c>0$ and $\alpha>0$, let $(u_{\alpha},\chi_{D_{\alpha}})$ be a solution
of

(5.2)
$\Lambda(\alpha,c)=\inf_{\phi\in\mathcal{B}}\inf_{u\in\mathcal{A}}\frac{\Phi_{1,G,\Omega}(u)+\alpha\Phi_{G,\phi,\Omega}(u)}{\Phi_{G,\partial\Omega}(u)}.$

Let $u_{0}\in W^{1,G}(\Omega)$ and $D_{0}\subset\Omega$ be such that
$\left|D_{0}\right|=c$ and $G(\left|u_{0}\right|)\chi_{D_{0}}=0$. Then, we
have that

$\Lambda(\alpha,c)\leq\frac{\Phi_{1,G,\Omega}(u_{0})+\alpha\Phi_{G,\chi_{D_{0}},\Omega}(u_{0})}{\Phi_{G,\partial\Omega}(u_{0})}=\frac{\Phi_{1,G,\Omega}(u_{0})}{\Phi_{G,\partial\Omega}(u_{0})}:=\mathcal{K}$

with $\mathcal{K}$ independent of $\alpha$. Thus ${\Lambda(\alpha,c)}$ is a
bounded sequence in $\mathbb{R}$ and it is clearly increasing. Consequently,
$\\{u_{\alpha}\\}_{\alpha>0}$ is bounded in $W^{1,G}(\Omega)$. Moreover
$\\{\chi_{D_{\alpha}}\\}_{\alpha>0}$ is bounded in $L^{\infty}(\Omega)$.
Therefore for Proposition 2.3, we may choose a sequence $\alpha_{k}$,
$u_{\infty}\in W^{1,G}(\Omega)$ and $\phi_{\infty}\in L^{\infty}(\Omega)$ such
that

(5.3) $u_{\alpha_{k}}\rightharpoonup u_{\infty}\;\text{ weakly in
}W^{1,G}(\Omega),\\\ $ (5.4) $u_{\alpha_{k}}\to u_{\infty}\;\text{ strongly in
}L^{G}(\Omega),\\\ $ (5.5) $u_{\alpha_{k}}\to u_{\infty}\;\text{ strongly in
}L^{G}(\partial\Omega),\\\ $ (5.6)
$\chi_{D_{\alpha_{k}}}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\phi_{\infty}\;\text{
weakly* in }L^{\infty}(\Omega).$

By (5.5) we get that $\left\|u_{\infty}\right\|_{L^{G}(\partial\Omega)}=1$ and
by (5.6) we can assure that $0\leq\phi_{\infty}\leq 1$ and
$\int_{\Omega}\phi_{\infty}dx=c$, i.e., $\phi_{\infty}\in\mathcal{B}$ and
$u_{\infty}\in\mathcal{A}$. Finally, by (5.4) and (5.6), also it holds
$\Phi_{G,\chi_{D_{\alpha_{k}}},\Omega}(u_{\alpha_{k}})\to\Phi_{G,\phi_{\infty},\Omega}(u_{\infty})$
as $k\to\infty$.

Observe that since $u_{\alpha_{k}}$ is minimizer,

$\alpha_{k}\Phi_{G,\chi_{D_{\alpha_{k}}},\Omega}(u_{\alpha_{k}})\leq\Lambda(\alpha_{k},c)\leq\mathcal{K}\qquad\text{
for all }k\in\mathbb{N}.$

Then,
$0\leq\Phi_{G,\chi_{D_{\alpha_{k}}},\Omega}(u_{\alpha_{k}})\leq\frac{\mathcal{K}}{\alpha_{k}}$,
from where, when $k\to\infty$ we get
$\int_{\Omega}\phi_{\infty}G(\left|u_{\infty}\right|)dx=0$, which gives that

(5.7) $\phi_{\infty}G(\left|u_{\infty}\right|)=0\text{ a.e. in }\Omega.$

Moreover, since $\Lambda(\alpha,c)$ is bounded and increasing, there exists
$\Lambda_{\infty}$ such that

$\displaystyle\Lambda_{\infty}=\lim_{\alpha\to\infty}\Lambda(\alpha,c)$
$\displaystyle=\lim_{k\to\infty}\Phi_{1,G,\Omega}(u_{\alpha_{k}})+\alpha_{k}\Phi_{G,\chi_{D_{\alpha_{k}}},\Omega}(u_{\alpha_{k}})$
$\displaystyle\geq\liminf_{k\to\infty}\Phi_{1,G,\Omega}(u_{\alpha_{k}})$
$\displaystyle\geq\Phi_{1,G,\Omega}(u_{\infty})$
$\displaystyle\geq\inf_{\phi\in\mathcal{B}}\inf_{\stackrel{{\scriptstyle
u\in\mathcal{A}}}{{u\phi=0}}}\Phi_{1,G,\Omega}(u)$
$\displaystyle\geq\Lambda(\alpha_{k},c),\qquad\text{for all }k\in\mathbb{N},$

where we have used the lower semicontinuity of the modular with respect to the
weak convergence and the definition of $\Lambda$.

From the previous computations we deduce that

$\Lambda_{\infty}=\inf_{\phi\in\mathcal{B}}\inf_{\stackrel{{\scriptstyle
u\in\mathcal{A}}}{{u\phi=0}}}\Phi_{1,G,\Omega}(u)=\Phi_{1,G,\Omega}(u_{\infty}).$

_Step 2_. We identify the limit optimal pair.
If we consider the set $D_{\infty}=\left\\{\phi_{\infty}>0\right\\}$ we get
that $b:=|D_{\infty}|\geq a$. Suppose that $b>a$, then by Lemma 5.1 we get
$\lambda(\infty,a)<\lambda(\infty,b)$, but, on the other hand

$\displaystyle\lambda(\infty,b)$ $\displaystyle=\inf_{\stackrel{{\scriptstyle
E\subset\Omega}}{{\left|E\right|=b}}}\lambda(\infty,E)\leq\lambda(\infty,D_{\infty})=\inf_{\stackrel{{\scriptstyle
v\in W^{1,G}}}{{v|_{D_{\infty}=0}}}}\Phi_{1,G,\Omega}(v)$
$\displaystyle\leq\Phi_{1,G,\Omega}(u_{\infty})=\Lambda_{\infty}\leq\lambda(\infty,a),$

where we have used (5.7) and the characterization of $\Lambda_{\infty}$ from
the previous step. This contradiction implies that $\left|D_{\infty}\right|=c$
and then $\phi_{\infty}=\chi_{D_{\infty}}$.

Finally, let us prove that $D_{\infty}=\\{u_{\infty}=0\\}$. Indeed, if we take
$x\in D_{\infty}$, from (5.7) we get $G(|u_{\infty}|)=0$ and then
$u_{\infty}(x)=0$, so $D_{\infty}\subseteq\\{u_{\infty}=0\\}$. Let us see that
the claim $D_{\infty}\subset\\{u_{\infty}=0\\}$ leads to a contradiction.
Observe that in the second step of Lemma 5.1 we proved that if $u$ is a
extremal for $\lambda(\infty,c)$ then $\left|\\{x:u(x)=0\right\\}|=c$. This
gives that an optimal pair $(u_{\alpha},\chi_{D_{\alpha}})$ to
$\Lambda(\alpha,c)$ satisfies $|\\{x\colon u_{\alpha}(x)=0\\}|=c$ and because
of the convergences in step i, $|\\{x:u_{\infty}(x)=0\\}|=c$. Then, if we
suppose that $D_{\infty}\subset\left\\{u_{\infty}=0\right\\}$, we obtain that
$c=|D_{\infty}|<|\\{x:u_{\infty}(x)=0\\}|=c$, which is absurd.

The proof is now completed. ∎

##  6\. Generalizations and final remarks

The techniques used in our results are flexible enough to prove existence of
an optimal solution to problems involving eigenvalues with different
homogeneity. More precisely, fixed $\alpha>0$ and $c\in[0,|\Omega|]$, given a
Young function $G$ satisfying (2.1) and (2.2), and a Young function $H$
increasing more slowly than $G$ in the sense that

$\lim_{t\to\infty}\frac{H(t)}{G(\lambda t)}=0$

for any $\lambda>0$, the results of Theorems 1.1, 1.2 and 1.3 can be extended
to optimization problems having the form

$\lambda(\alpha,c):=\inf\\{\lambda(\alpha,E)\colon E\subset\Omega,|E|=c\\}$

where

$\lambda(\alpha,E):=\min_{v\in\mathcal{A}}\left\\{\int_{\Omega}G(|\nabla
v|)+G(|v|)\,dx+\alpha\int_{E}G(|v|)\,dx\right\\},$

being $\mathcal{A}$ the class of admissible functions given by

$\mathcal{A}:=\left\\{v\in W^{1,G}(\Omega)\colon\text{ such that
}\int_{\partial\Omega}H(|v|)\,d\mathcal{H}^{n-1}=1\right\\}.$

## Acknowledgments

AS and AS are members of CONICET, Argentina. BS is a fellow of CONICET. AS and
BS are partially supported by ANPCyT PICT 2017-0704 and 2019-03837 and by UNSL
under grants PROIPRO 03-2420 and PROICO 03-0720. AS is partially supported by
ANPCyT PICT 2017-1119.

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