section subsection subsubsection

0.5 cm A phase-field version
of the Faber–Krahn theorem
Paul Hüttl111Fakulät für Mathematik, Universität Regensburg, 93053 Regensburg,
Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>Patrik Knopf11footnotemark: 1
and Tim Laux222Hausdorff Center for Mathematics, University of Bonn, 53115
Bonn, Germany<EMAIL_ADDRESS>

This is a preprint version of the paper. Please cite as:
P. Hüttl, P. Knopf, and T. Laux [Journal] (2021)
https://doi.org/...

######  Abstract.

We investigate a phase-field version of the Faber–Krahn theorem based on a
phase-field optimization problem introduced in Garcke et al. [21] formulated
for the principal eigenvalue of the Dirichlet–Laplacian. The shape, that is to
be optimized, is represented by a phase-field function mapping into in the
interval $[0,1]$. We show that any minimizer of our problem is a radially
symmetric-decreasing phase-field attaining values close to $0$ and $1$ except
for a thin transition layer whose thickness is of order $\varepsilon>0$. Our
proof relies on radially symmetric-decreasing rearrangements and corresponding
functional inequalities. Moreover, we provide a $\Gamma$-convergence result
which allows us to recover a generalization of the Faber–Krahn theorem for
sets of finite perimeter in the sharp-interface limit.

##  1 Introduction

The Faber–Krahn theorem states that the principal eigenvalue $\lambda(E)>0$ of
the eigenvalue problem

$\displaystyle-\Delta w$ $\displaystyle=\lambda w$ $\displaystyle\quad\text{in
$E$},$ (1.1a) $\displaystyle w$ $\displaystyle=0$ on $\partial E$ (1.1b)

among all sets $E\subset\mathbb{R}^{n}$ with $|E|=1$, becomes minimal if $E$
is a ball. In dimension $n=2$, this result was first conjectured by Lord
Rayleigh [31] and proved independently (under suitable regularity assumptions
on the boundary of $E$) by Faber [18] and Krahn [24]. The result holds true in
every dimension $n\geq 2$ for general open sets $E$ (see, e.g., [23, Section
3.2]). In other words, if $B$ is an open ball in $\mathbb{R}^{n}$, then the
estimate

$\displaystyle|B|^{\frac{2}{n}}\lambda(B)\leq|E|^{\frac{2}{n}}\lambda(E)$
(1.2)

holds for every open set $E\subset\mathbb{R}^{n}$, cf. [20]. The latter result
is referred to as the Faber–Krahn inequality.

In this paper, we prove a diffuse interface version of this celebrated result
where the boundary of such sets $E$ is approximated by a thin interfacial
layer. Phase-fields are natural candidates to relax such minimization problems
for the purpose of numerical implementation in shape and topology
optimization. The diffuse interface relaxation for eigenvalue optimization
problems via phase-fields was introduced by Bogosel and Oudet [7] and Garcke
et al. [21].

In the following, we consider a fixed design domain $\Omega=B_{R}(0)$ that is
an open ball in $\mathbb{R}^{n}$ centered at the origin with a given finite
radius $R>0$. Instead of directly working with open sets $E\subset\Omega$, we
replace any given set $E$ by a function
$\varphi_{\varepsilon}\colon\Omega\to[0,1]$, the so-called phase-field, such
that the region where $\varphi_{\varepsilon}\approx 1$ approximates $E$ and
the region where $\varphi_{\varepsilon}\approx 0$ approximates
$\Omega\backslash E$. These regions are separated by a thin transition layer
whose width is related to the (usually small) interface parameter
$\varepsilon>0$. For more details, we refer to Section 3.

For any phase-field function $\varphi:\Omega\to[0,1]$, we consider the
eigenvalue problem

$\displaystyle-\Delta w+b^{\varepsilon}(\varphi)w$ $\displaystyle=\lambda w$
$\displaystyle\quad\text{in $\Omega$},$ (1.3a) $\displaystyle w$
$\displaystyle=0$ $\displaystyle\quad\text{on $\partial\Omega$},$ (1.3b)

where $b^{\varepsilon}$ is a suitable coefficient function driving $w$ to zero
in the set $\\{\varphi=0\\}$ (see Section 2.2 and [21]). To optimize the
principal eigenvalue $\lambda_{1}^{\varepsilon,\varphi}$ of this eigenvalue
problem, we want to minimize the cost functional

$\displaystyle
J^{\varepsilon}_{\gamma}(\varphi)=\lambda_{1}^{\varepsilon,\varphi}+\gamma
E^{\varepsilon}(\varphi),$ (1.4)

(inspired by [7] and [21]) over a class of admissible functions $\varphi$. The
expression $\gamma E^{\varepsilon}(\varphi)$ is added as a regularization term
in order to make the optimization problem well-posed. Here, the surface
tension $\gamma$ is a positive constant and $E^{\varepsilon}(\varphi)$ stands
for the Ginzburg–Landau energy associated with $\varphi$ (see (3.6)). We point
out that the Ginzburg–Landau energy can be regarded as a diffuse interface
approximation of the perimeter functional, cf. [28, 27]. The idea of perimeter
regularization in shape optimization was first introduced in [3].

Our main results show that all minimizers $\varphi_{\varepsilon}$ of this
optimization problem are radially symmetric-decreasing functions which indeed
exhibit a phase-field structure as described above (see Theorem 3.7 and
Theorem 3.8). This radial symmetry of the phase-fields is the natural analogue
to the radial symmetry of the balls in the Faber–Krahn inequality. Combining
our symmetry result with the $\Gamma$-convergence of Garcke et al. [21], which
we extend to the case of homogeneous Dirichlet boundary data for the phase-
field variable by exploiting arguments of Bourdin and Chambolle [8], we
recover a generalized version of the Faber–Krahn theorem in the framework of
$BV$ functions (see Theorem 3.15 and Corollary 3.16) in the sharp-interface
limit. However, our result states the stronger fact that the phase-field
approximation exactly captures the symmetry properties of the sharp-interface
limit at any fixed scale $\varepsilon>0$, see also Theorem 3.5. To obtain the
$\Gamma$-convergence in Theorem 3.17 we shortly revisit the proof in [33] and
adapt it to the case of general potentials, allowing for a simultaneous
treatment of smooth double well and non-smooth double obstacle potentials. As
we consider homogeneous Dirichlet boundary data, we can construct a recovery
sequence in the spirit of Modica [27] and Sternberg [33] by using the profile
resulting from the aforementioned potential and cut it off as in [8].

The proof of the main result presented in Theorem 3.7 relies on a
symmetrization technique where the phase-fields and the corresponding
eigenfunctions of (1.3) are compared with their radially symmetric-decreasing
rearrangements (which are also referred to as Schwarz rearrangements in the
literature). These rearrangements are at the heart of the proofs [18, 24] and
have led to breakthrough results such as the quantitative isoperimetric
inequality [19]. In our diffuse interface setting, we show that the principal
eigenvalue $\lambda_{1}^{\varepsilon,\varphi}$ and the Ginzburg–Landau energy
$E^{\varepsilon}(\varphi)$, which constitute the cost functional, are non-
increasing under radially symmetric-decreasing rearrangements. This can be
used to establish our phase-field version of the Faber–Krahn theorem. As a
byproduct, we also obtain a phase-field version of the Euclidean isoperimetric
problem, which states that among all measurable sets of fixed volume, the ball
has minimal perimeter (cf. [26]).

Variants of the Faber–Krahn theorem also for other types of boundary
conditions have been studied from a large variety of different viewpoints. An
interesting approach from the perspective of free boundary value problems is
given in the alternative proof of the classical Faber–Krahn inequality by
Bucur and Freitas [12]. Therein, methods developed by Alt and Cafarelli [2]
(see also [14] for a comprehensive overview) are exploited to analyze the free
boundary $\partial\left\\{w>0\right\\}$, where $w$ is an eigenfunction to the
principal eigenvalue of the Dirichlet Laplacian. Furthermore, the authors do
not rely on symmetric rearrangements but rather on reflection arguments to
prove the radial symmetry of optimal shapes. For Robin-type boundary
conditions, Daners [16] established the Faber–Krahn inequality via a level-set
characterization of the cost functional. This result was further generalized
by Bucur and Daners [11] for the $p$-Laplacian subject to a Robin boundary
condition. In order to avoid Lipschitz regularity in the class of admissible
sets Bucur and Giacomini [13] interpret the Faber–Krahn inequality for the
Robin Laplacian as a free discontinuity problem in the space $SBV$.

An interesting open question is a quantitative version of our result, i.e.,
the stability of the phase-field version of the Faber–Krahn inequality. In the
sharp-interface case, this question has been settled by Brasco, De Philippis,
and Velichkov [9] who show that the deficit in (1.2) can be bounded from below
by the squared Fraenkel asymmetry. This optimal result is achieved by
generalizing the selection principle of Cicalese and Leonardi [15] from the
context of isoperimetric problems to the case of eigenvalues. It should be
expected that the symmetrization procedure applied in our work would result in
a (suboptimal) quantitative version of our inequality, just as in the case of
the sharp interface by Fusco, Maggi, and Pratelli [20], who rely on
symmetrization and their quantitative isoperimetric inequality [19].

Finally, let us also mention another interesting variant of the Faber–Krahn
inequality, namely the Pólya–Szegő conjecture (see [30, p. 159]), which states
that among all planar polygons of fixed enclosed area, the regular polygon
minimizes the first Dirichlet–Laplace eigenvalue. Only recently, some
significant progress has been achieved by Bogosel and Bucur [6], which
indicates that also here, the most symmetric configuration yields the smallest
eigenvalue. This approach might lead to a computer-assisted proof of the
conjecture, at least for $n$-gons with moderately small $n$.

Our paper is structured as follows. First we fix our notation and gather some
preliminary results in Section 2. In Section 3, we precisely formulate the
problem and state our main results. Finally, in Section 4, we provide the
proofs of our results.

##  2 Preliminaries and important tools

###  2.1 Notation

We write $\mathbb{R}_{0}^{+}=[0,\infty)$ to denote the interval of non-
negative real numbers. The interval $[0,+\infty]$ is to be understood as a
subset of the extended real numbers
$\overline{\mathbb{R}}=\mathbb{R}\cup\\{\pm\infty\\}$, on which we use the
standard convention $\pm\infty\,\cdot\,0=0$. For any $n\in\mathbb{N}$,
$\mathcal{L}^{n}$ stands for the $n$-dimensional Lebesgue measure and
$\mathcal{H}^{n}$ denotes the $n$-dimensional Hausdorff measure.

###  2.2 Assumptions

The following assumptions on the potential
$\psi:\mathbb{R}\to\mathbb{R}\cup\left\\{+\infty\right\\}$ and the
coefficients $b^{\varepsilon}$ are supposed to hold throughout this paper:

  1. (A1)

$\psi\in C^{2}([0,1])$, $\psi(0)=\psi(1)=0$ and $\psi>0$ in $(0,1)$.

  2. (A2)

The minima of $\psi$ at $0$ and $1$ are non-degenerate in the sense that for
$x\in\left\\{0,1\right\\}$, we either have $\psi^{\prime}(x)\neq 0$ or
$\psi^{\prime}(x)=0<\psi^{\prime\prime}(x)$.

  3. (A3)

For any $\varepsilon>0$, the coefficient $b^{\varepsilon}$ is a function

$\displaystyle b^{\varepsilon}:[0,1]\to[0,\beta^{\varepsilon}]$ (2.1)

for some real number $\beta^{\varepsilon}>0$. We demand that $b^{\varepsilon}$
is continuous, strictly decreasing and surjective onto
$[0,\beta^{\varepsilon}]$.

  4. (A4)

The numbers $\beta^{\varepsilon}=b^{\varepsilon}(0)$ satisfy

$\displaystyle\underset{\varepsilon\searrow
0}{\lim}\;\beta^{\varepsilon}=+\infty\quad\text{and}\quad\beta^{\varepsilon}=o\big{(}\varepsilon^{-\kappa}\big{)}\quad\text{with}\quad\begin{cases}\kappa\in(0,1)&\text{if}\;n=2,\\\
\kappa=\frac{2}{n}&\text{if}\;n\geq 3.\end{cases}$ (2.2)

Moreover, there exists a limit function

$\displaystyle b^{0}:[0,1]\to[0,+\infty]$ (2.3)

satisfying

     * •

$b^{0}\big{(}\tfrac{1}{2}\big{)}<+\infty$,

     * •

$b^{\varepsilon}\to b^{0}$ pointwise in $[0,1]$ as $\varepsilon\to 0$,

     * •

$b^{\delta}\geq b^{\varepsilon}$ on $[0,1]$ for all
$0\leq\delta\leq\varepsilon$.

$0$$\varphi$$\beta^{\varepsilon}$$\beta^{\delta}$$b^{0}$$b^{\delta}$$b^{\varepsilon}$$1$

$0$$\varphi$$\psi(\varphi)$$\psi(\varphi)$$1$

Figure 1: Sketch of a possible choice for the coefficient functions
$b^{\varepsilon}$ and $b^{0}$ (left) and the potential $\psi$ (right).

######  Remark 2.1.

The two classical choices we have in mind for the potential $\psi$ are either
the smooth quartic double-well potential $\psi(x)=\frac{1}{4}(1-x)^{2}x^{2}$
(which satisfies $\psi^{\prime}(x)=0<\psi^{\prime\prime}(x)$ for
$x\in\\{0,1\\}$) or the non-smooth double-obstacle potential

$\displaystyle\psi(x)=\begin{cases}\frac{1}{2}(1-x)x&\quad\text{if
}x\in[0,1],\\\ +\infty&\quad\text{else}.\end{cases}$

(which satisfies $\psi^{\prime}(x)\neq 0$ for $x\in\\{0,1\\}$). However, the
assumptions (A1) and (A2) allow for very general potentials. In particular,
asymmetric potentials satisfying $\psi^{\prime}(0)\neq 0$ and
$\psi^{\prime}(1)=0<\psi^{\prime\prime}(1)$ (or vice versa) can also be
included.

Note that in the case of a smooth potential as studied in [27, 33], opposed to
their theory, we do not need a growth condition as in [27, Proposition 3(b)]
or [33, Proposition 3] since we additionally incorporate the box constraint
$\varphi\in[0,1]$ in our set of admissible phase-fields. Therefore, depending
on the choice of $\psi$, one of the results [27, Proposition 3(a)], [33,
Remark (1.35)] and [5, Theorem 3.7] can be applied and yields compactness of
the Ginzburg–Landau energy.

The assumptions on $\psi$ in [33, Theorem 1] differ from (A1) only in the fact
that global continuity is assumed. However, due to the box constraint
$\varphi\in[0,1]$, our phase-fields may not leave the interval $[0,1]$ and
thus, such an assumption is not necessary.

The crucial difference between [33] and [5] is that in [33], the potentials
need to satisfy (A2) with $\psi^{\prime}(x)=0<\psi^{\prime\prime}(x)$ for
$x\in\\{0,1\\}$, whereas [5] only covers the case $\psi^{\prime}(x)\neq 0$ for
$x\in\\{0,1\\}$. However, we will see that also the mixed case
$\psi^{\prime}(0)\neq 0$ and $\psi^{\prime}(1)=0<\psi^{\prime\prime}(1)$ (or
vice versa) can be handled by combining the proofs of [33, Theorem 1] and [5,
Proposition 3.11]. This is possible since their construction of a recovery
sequence remains practicable as the ODE

$\displaystyle\eta^{\prime}(t)=\sqrt{2\psi(\eta(t))},$ (2.4)

which is used to define the profile at the diffuse interface, possesses a
global solution that is strictly increasing as long as $\eta(t)\in(0,1)$. In
[33], any solution of (2.4) satisfies $\eta(t)\in(0,1)$ for all
$t\in\mathbb{R}$, whereas in [5], there exist $t_{0},t_{1}\in\mathbb{R}$ with
$t_{0}<t_{1}$ such that

$\displaystyle\eta(t)\begin{cases}=0,&\text{if $t\in(-\infty,t_{0}]$},\\\
\in(0,1),&\text{if $t\in(t_{0},t_{1})$},\\\ =1,&\text{if
$t\in[t_{1},\infty)$}.\end{cases}$ (2.5)

In our $\Gamma$-convergence proof, we will take care of both cases
simultaneously. Therefore, proceeding as in [33], we interpolate the solution
$\eta$ of (2.4) in such a way that the interpolated solution exhibits the
behavior described in (2.5). The solvability of (4.17) and further properties
of solutions to this ODE will be analyzed in depth in the proof of Theorem
3.17.

###  2.3 Symmetric-decreasing rearrangements

For functions $f:\mathbb{R}^{n}\to\mathbb{R}_{0}^{+}$ vanishing at infinity
(i.e., the level sets ${\\{x\in\mathbb{R}^{n}\,|\,f(x)>t\\}}$ have finite
Lebesgue-measure for all $t>0$), a definition of their radially symmetric-
decreasing rearrangement can be found in [25, Section 3.3]. We can easily
adapt this definition to functions $f:\Omega\to\mathbb{R}_{0}^{+}$ where
$\Omega=B_{R}(0)$ is an open ball in $\mathbb{R}^{n}$ with radius $R>0$
centered at the origin.

######  Definition 2.2.

Let $\Omega=B_{R}(0)$ be an open ball in $\mathbb{R}^{n}$ centered at the
origin with a given radius $R>0$.

  1. (a)

A measurable function ${f:\Omega\to\mathbb{R}}$ is called (radially)
symmetric-decreasing, if any fixed representative of the equivalence class of
$f$ satisfies the properties

$\displaystyle\left\\{\begin{aligned}
f(x)&=f(y)&&\quad\text{if}\quad\left|x\right|=\left|y\right|,\\\ f(x)&\geq
f(y)&&\quad\text{if}\quad\left|x\right|\leq\left|y\right|\end{aligned}\right.$
(2.6)

for almost all $x,y\in\Omega$. If additionally

$\displaystyle f(x)>f(y)\quad\text{if}\quad\left|x\right|<\left|y\right|$

for almost all $x,y\in\Omega$, then $f$ is called strictly (radially)
symmetric-decreasing.

  2. (b)

For any measurable set $A\subseteq\Omega$ with $\mathcal{L}^{n}(A)<\infty$,
its (radially) symmetric rearrangement $A^{*}$ is defined to be the open ball
centered at the origin whose volume is equal to that of $A$. This means that

$\displaystyle
A^{*}=\big{\\{}x\in\Omega\,:\,\left|x\right|<r\big{\\}}\quad\text{where $r\geq
0$ satisfies}\quad\mathcal{L}^{n}(B^{n})\,r^{n}=\mathcal{L}^{n}(A).$

Here, $B^{n}$ denotes the $n$-dimensional unit ball.

  3. (c)

Let $f:\Omega\to\mathbb{R}_{0}^{+}$ be any measurable function. Then its
(radially) symmetric-decreasing rearrangement $f^{*}$ is defined as

$\displaystyle
f^{*}(x)=\int_{0}^{\infty}\mathds{1}_{{\\{f>t\\}}^{*}}(x)\;\mathrm{d}t$

for all $x\in\Omega$.

######  Remark 2.3.

Let $\Omega=B_{R}(0)$ be an open ball in $\mathbb{R}^{n}$ centered at the
origin with a given radius $R>0$.

  1. (a)

It obviously holds $\Omega^{*}=\Omega$, and for any measurable function
$f:\Omega\to\mathbb{R}_{0}^{+}$, we have $f^{**}=f^{*}$.

  2. (b)

For any measurable function $f:\Omega\to\mathbb{R}_{0}^{+}$, its trivial
extension $f_{0}:\mathbb{R}^{n}\to\mathbb{R}_{0}^{+}$ with $f_{0}|_{\Omega}=f$
and $f_{0}|_{\mathbb{R}^{n}\backslash\Omega}=0$ is measurable and naturally
vanishes at infinity. In particular, we have $f_{0}^{*}|_{\Omega}=f^{*}$,
where the symmetric-decreasing rearrangement $f_{0}^{*}$ of the extension
$f_{0}$ is defined as in [25, Section 3.3].

Some important properties of the symmetric-decreasing rearrangement are
collected in the following lemma.

######  Lemma 2.4.

Let $\Omega=B_{R}(0)$ be an open ball in $\mathbb{R}^{n}$ centered at the
origin with a given radius $R>0$, and let $f,g:\Omega\to\mathbb{R}_{0}^{+}$ be
arbitrary measurable functions. Then the following statements hold:

  1. (a)

$f^{*}$ is measurable and symmetric-decreasing. Moreover, $f^{*}$ is defined
everywhere in $\Omega$. In particular, the condition (2.6) is satisfied
everywhere in $\Omega$.

  2. (b)

The level sets of $f^{*}$ are the rearrangements of the level sets of $f$,
meaning that

$\displaystyle\big{\\{}x\in\Omega:f^{*}(x)>t\big{\\}}=\big{\\{}x\in\Omega:f(x)>t\big{\\}}^{*}$

up to a Lebesgue null set in $\mathbb{R}^{n}$. In particular, if $f\in
L^{p}(\Omega)$ for some $p\in[1,\infty]$, it holds that $f^{*}\in
L^{p}(\Omega)$ with

$\displaystyle\left\|f^{*}\right\|_{L^{p}(\Omega)}=\left\|f\right\|_{L^{p}(\Omega)}.$

  3. (c)

Let $\Phi:\mathbb{R}_{0}^{+}\to\mathbb{R}_{0}^{+}$ be an non-decreasing, lower
semicontinuous function. Then it holds that

$\displaystyle(\Phi\circ f)^{*}=\Phi\circ
f^{*}\quad\text{a.e.\leavevmode\nobreak\ in $\Omega$}.$

  4. (d)

Let $\Psi\in C^{1}([0,1])$ with $\Psi(0)=0$. If $0\leq f\leq 1$ a.e. in
$\Omega$, it holds that

$\displaystyle\int_{\Omega}\Psi\circ
f^{*}\;\mathrm{d}\mathcal{L}^{n}=\int_{\Omega}\Psi\circ
f\;\mathrm{d}\mathcal{L}^{n}.$ (2.7)

  5. (e)

Hardy–Littlewood inequality: It holds that

$\displaystyle\int_{\Omega}f\,g\;\mathrm{d}\mathcal{L}^{n}\leq\int_{\Omega}f^{*}\,g^{*}\;\mathrm{d}\mathcal{L}^{n}$
(2.8)

with the convention that when the left-hand side is infinite, then also the
right-hand side is infinite.

  6. (f)

Nonexpansivity of the rearrangement: Let $F:\mathbb{R}\to\mathbb{R}_{0}^{+}$
be a convex function such that $F(0)=0$. Then

$\displaystyle\int_{\Omega}F\circ(f^{*}-g^{*})\;\mathrm{d}\mathcal{L}^{n}\leq\int_{\Omega}F\circ(f-g)\;\mathrm{d}\mathcal{L}^{n}.$

  7. (g)

Pólya–Szegő inequality: Suppose that $f\in
H^{1}_{0}(\Omega;\mathbb{R}_{0}^{+})$. Then, $f^{*}\in
H^{1}_{0}(\Omega;\mathbb{R}_{0}^{+})$ with

$\displaystyle\int_{\Omega}\left|\nabla
f^{*}\right|^{2}\;\mathrm{d}\mathcal{L}^{n}\leq\int_{\Omega}\left|\nabla
f\right|^{2}\;\mathrm{d}\mathcal{L}^{n}.$ (2.9)

Moreover, if $f>0$ almost everywhere in $\Omega$ and

$\displaystyle\mathcal{L}^{n}\big{(}\\{x\in\Omega\,|\,\nabla
f^{*}(x)=0\\}\big{)}=0,$ (2.10)

then equality in (2.9) holds if and only if $f=f^{*}$ almost everywhere in
$\Omega$.

The proof is deferred to Section 4.

######  Remark 2.5.

We point out that the condition that $f$ has a vanishing trace on
$\partial\Omega$ is actually a necessary assumption for the Pólya–Szegő
inequality (Lemma 2.4(g)). In general, as the following example shows, there
exist functions $f\in H^{1}(\Omega;\mathbb{R}_{0}^{+})$ such that $f^{*}$ does
not even belong to $H^{1}(\Omega;\mathbb{R}_{0}^{+})$.

Counterexample to the Pólya–Szegő inequality for functions in
$H^{1}(\Omega;\mathbb{R}_{0}^{+})$.
Let $\Omega=B_{1}(0)$ be the open unit ball in $\mathbb{R}^{2}$. We consider
the function

$\displaystyle f:\Omega\to\mathbb{R}_{0}^{+},\quad x\mapsto\left|x\right|\,,$

which obviously belongs to $H^{1}(\Omega;\mathbb{R}_{0}^{+})$ but not to
$H^{1}_{0}(\Omega;\mathbb{R}_{0}^{+})$. Its symmetric-decreasing rearrangement
$f^{*}$ is given by

$\displaystyle f^{*}:\Omega\to\mathbb{R}_{0}^{+},\quad
x\mapsto\sqrt{1-\left|x\right|^{2}}\,.$

Hence, $f^{*}$ is weakly differentiable with

$\displaystyle\nabla
f^{*}(x)=\frac{-x}{\sqrt{1-\left|x\right|^{2}}}\quad\text{for all
$x\in\Omega\backslash\\{0\\}$.}$

However, it is easy to see that the blow-up at $|x|=1$ causes

$\displaystyle\int_{\Omega}\left|\nabla
f^{*}\right|^{2}\;\mathrm{d}\mathcal{L}^{2}=+\infty.$

This means that $f^{*}\notin H^{1}(\Omega;\mathbb{R}_{0}^{+})$ and in
particular, the Pólya–Szegő inequality (2.9) does not hold.

###  2.4 Properties of functions of bounded variation

We recall some basic facts on functions of bounded variation and sets of
finite perimeter that will be used in the course of this paper. We refer to
[17, 26, 4] for more details.

The space of functions of bounded variation in $\Omega$ with values in
$\mathbb{R}$, also referred to as $BV$ functions, is defined as

$\displaystyle BV(\Omega)\coloneqq\left\\{u\in
L^{1}(\Omega)\;|\;V(u,\Omega)<\infty\right\\}.$

Here $V(u,\Omega)$ denotes the variation of a function $u\in
L^{1}_{\mathrm{loc}}(\Omega)$ defined as

$\displaystyle
V(u,\Omega)\coloneqq\sup\left\\{\int_{\Omega}u\,\operatorname{div}\boldsymbol{\xi}\;\mathrm{d}\mathcal{L}^{n}\;|\;\boldsymbol{\xi}\in
C_{0}^{1}(\Omega,\mathbb{R}^{n}),\,\left\|\boldsymbol{\xi}\right\|_{L^{\infty}(\Omega)}\leq
1\right\\}.$

Endowed with the norm

$\displaystyle\left\|u\right\|_{BV(\Omega)}\coloneqq\left\|u\right\|_{L^{1}(\Omega)}+V(u,\Omega),$

$BV(\Omega)$ is a Banach space. However, for practical purposes, the topology
induced by this norm is too strong. For this reason, the concept of strict
convergence is commonly used. We say that a sequence $u_{k}\in BV(\Omega)$
strictly converges to $u\in BV(\Omega)$ if

$\displaystyle u_{k}\to u\quad\text{\,in }L^{1}(\Omega)\quad\text{and}\quad
V(u_{k},\Omega)\to V(u,\Omega)$

as $k\to\infty$.

One of the fine properties of $BV$ functions is that they allow for a well
defined trace. Due to [4, Theorem 3.87] any $u\in BV(\Omega)$ possesses a
trace

$u_{|{\partial\Omega}}\in
L^{1}\big{(}(\partial\Omega,\mathcal{H}^{n-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\partial\Omega);\mathbb{R}\big{)},$

which is defined via the limit

$\displaystyle\underset{\rho\searrow 0}{\lim}\;\rho^{-n}\int_{\Omega\cap
B_{\rho}(x)}\left|u(y)-u_{|{\partial\Omega}}(x)\right|\;\mathrm{d}\mathcal{L}^{n}(y)=0.$
(2.11)

for $\mathcal{H}^{n-1}$ a.e. $x\in\partial\Omega$. Here,
$\mathcal{H}^{n-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\partial\Omega$ denotes the
restriction of the Hausdorff measure $\mathcal{H}^{n-1}$ to the boundary
$\partial\Omega$ and
$L^{1}\big{(}(\partial\Omega,\mathcal{H}^{n-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\partial\Omega);\mathbb{R}\big{)}$
is the space of $L^{1}$-functions on $\partial\Omega$ with respect to the
measure $\mathcal{H}^{n-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\partial\Omega$. In the
following, we will simply write $L^{1}(\partial\Omega)$ instead of
$L^{1}\big{(}(\partial\Omega,\mathcal{H}^{n-1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\partial\Omega);\mathbb{R}\big{)}$.
The corresponding norm on $L^{1}(\partial\Omega)$ is given by

$\displaystyle\left\|\cdot\right\|_{L^{1}(\partial\Omega)}=\int_{\partial\Omega}\left|\cdot\right|\;\mathrm{d}\mathcal{H}^{n-1}.$
(2.12)

Due to [4, Theorem 3.88], the operator

$\displaystyle BV(\Omega)$ $\displaystyle\to L^{1}(\partial\Omega),$
$\displaystyle u$ $\displaystyle\mapsto u_{|{\partial\Omega}},$

is continuous with respect to strict convergence in $BV(\Omega)$.

Concluding this section we give the definition of the relative perimeter. The
relative perimeter in $\Omega$ of a measurable set $E\subset\mathbb{R}^{n}$ is
defined as

$\displaystyle P_{\Omega}(E)\coloneqq V(\chi_{E},\Omega).$

##  3 Formulation of the problem and the main results

In the following, we consider the design domain
$\Omega=B_{R}(0)\subset\mathbb{R}^{n}$ with $n\geq 2$ and some finite radius
$R>0$.

For functions $\varphi\in L^{\infty}(\Omega)$, the eigenvalue problem
introduced in [21] reads as

$\displaystyle-\Delta w+b^{\varepsilon}(\varphi)w$ $\displaystyle=\lambda w$
$\displaystyle\qquad\text{in $\Omega$},$ (3.1a) $\displaystyle
w|_{\partial\Omega}$ $\displaystyle=0$ on $\partial\Omega$. (3.1b)

In view of the term $b^{\varepsilon}(\varphi)w$, this problem can be
understood as a phase-field approximation of the classical Dirichlet–Laplace
eigenvalue problem on the shape represented by the set
$\left\\{\varphi=1\right\\}$. For a detailed motivation and introduction of
this eigenvalue problem we refer to [21, Section 2]. To specify the notion of
weak solutions, eigenvalues and eigenfunctions, we recall the following
definition (cf. [21, Section 3.1]).

######  Definition 3.1.

Let $\varepsilon>0$ and $\varphi\in L^{\infty}(\Omega)$ be arbitrary.

  1. (a)

For any given $\lambda\in\mathbb{R}$, a function $w\in H^{1}_{0}(\Omega)$ is
called a weak solution of the system (3.1) if the weak formulation

$\displaystyle\int_{\Omega}\nabla
w\cdot\nabla\eta+b^{\varepsilon}(\varphi)\,w\eta\;\mathrm{d}\mathcal{L}^{n}=\lambda\int_{\Omega}w\eta\;\mathrm{d}\mathcal{L}^{n}$
(3.2)

is satisfied for all test functions $\eta\in H^{1}_{0}(\Omega)$.

  2. (b)

A real number $\lambda^{\varepsilon,\varphi}$ is called an eigenvalue
associated with $\varphi$, if there exists at least one nontrivial weak
solution $w^{\varepsilon,\varphi}\in H^{1}_{0}(\Omega)$ of the eigenvalue
problem (3.1) written for $\lambda=\lambda^{\varepsilon,\varphi}$.

In this case, $w^{\varepsilon,\varphi}$ is called an eigenfunction to the
eigenvalue $\lambda^{\varepsilon,\varphi}$.

We further recall some important properties of the eigenvalue problem (3.1).
The results can be found in [21], but are also accessible via the standard
literature (see, e.g., [1, 22]).

######  Proposition 3.2.

Let $\varepsilon>0$ and $\varphi\in L^{\infty}(\Omega)$ be arbitrary.

  1. (a)

The eigenvalue problem (3.1) has countably many eigenvalues and each of them
has a finite dimensional eigenspace. Repeating each eigenvalue according to
its multiplicity, we can write them as a sequence
$\big{(}\lambda_{k}^{\varepsilon,\varphi}\big{)}_{k\in\mathbb{N}}$ with

$\displaystyle
0<\lambda_{1}^{\varepsilon,\varphi}\leq\lambda_{2}^{\varepsilon,\varphi}\leq\lambda_{3}^{\varepsilon,\varphi}\leq...\quad\text{and}\quad\lambda_{k}^{\varepsilon,\varphi}\to\infty\;\;\text{as}\;\;k\to\infty.$

  2. (b)

There exists an orthonormal basis
$\big{(}w_{k}^{\varepsilon,\varphi}\big{)}_{k\in\mathbb{N}}$ of
$L^{2}(\Omega)$ where for every $k\in\mathbb{N}$,
$w_{k}^{\varepsilon,\varphi}$ is an eigenfunction to the eigenvalue
$\lambda_{k}^{\varepsilon,\varphi}$.

  3. (c)

The eigenvalue $\lambda_{1}^{\varepsilon,\varphi}$ is called the principal
eigenvalue. It can be represented via the Courant–Fischer characterization

$\displaystyle\lambda_{1}^{\varepsilon,\varphi}=\underset{w\in
H^{1}_{0}(\Omega)\backslash\\{0\\}}{\min}\frac{\int_{\Omega}\left|\nabla
w\right|^{2}+b^{\varepsilon}(\varphi)w^{2}\;\mathrm{d}\mathcal{L}^{n}}{\left\|w\right\|_{L^{2}(\Omega)}^{2}}\;.$
(3.3)

Any function $w\in H^{1}_{0}(\Omega)\backslash\\{0\\}$ at which this minimum
is attained is an eigenfunction to the eigenvalue
$\lambda_{1}^{\varepsilon,\varphi}$.

Moreover, the eigenspace of $\lambda_{1}^{\varepsilon,\varphi}$ is one-
dimensional, and there exists a unique eigenfunction $\overline{w}\in
H^{1}_{0}(\Omega)\backslash\\{0\\}$ corresponding to this eigenvalue which
fulfills

$\displaystyle\overline{w}>0\quad\text{a.e.\leavevmode\nobreak\ in
$\Omega$},\quad\text{and}\quad\left\|\overline{w}\right\|_{L^{2}(\Omega)}=1.$
(3.4)

We call $\overline{w}$ the positive-normalized eigenfunction. Without loss of
generality, as the choice the sign does not matter, we will always choose
$w_{1}^{\varepsilon,\varphi}=\overline{w}$ in the orthonormal basis given by
part (b).

Applying [22, Theorem 8.12] we directly infer the following statement.

######  Corollary 3.3.

Let $\varepsilon>0$ and $\varphi\in\Phi_{m}$ be arbitrary, and let $w$ be an
eigenfunction to the eigenvalue $\lambda_{1}^{\varepsilon,\varphi}$ in the
sense of Definition 3.1. Then it holds that $w\in H^{1}_{0}(\Omega)\cap
H^{2}(\Omega)$ and $w$ is a strong solution of the system (3.1), meaning that

$\displaystyle-\Delta w+b^{\varepsilon}(\varphi)w$
$\displaystyle=\lambda_{1}^{\varepsilon,\varphi}w$
$\displaystyle\qquad\text{a.e.\leavevmode\nobreak\ in $\Omega$},$
$\displaystyle w|_{\partial\Omega}$ $\displaystyle=0$ a.e. on
$\partial\Omega$.

Now we formulate the shape optimization problem for the principal eigenvalue.
This can be regarded as a special case of the framework in [21] by choosing
$\Psi(\lambda_{1})=\lambda_{1}$ there. Hence, we only briefly summarize the
main aspects concerning this optimization problem at this point.

Let $\psi$ be given as in assumption (A1). For any prescribed $m\in(0,1)$, we
define the set of admissible controls

$\displaystyle\Phi_{m}:=\left\\{\varphi\in
H^{1}_{0}(\Omega)\;|\;\begin{aligned} &0\leq\varphi(x)\leq 1\;\;\text{for a.e.
$x\in\Omega$},\\\
&\textstyle\fint_{\Omega}\varphi\;\mathrm{d}\mathcal{L}^{n}=m\end{aligned}\right\\}\subset
H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)\,.$

For $\varepsilon>0$ and $\varphi\in\Phi_{m}$, we now introduce the
Ginzburg–Landau energy

$\displaystyle
E^{\varepsilon}(\varphi):=\int_{\Omega}\frac{\varepsilon}{2}\left|\nabla\varphi\right|^{2}+\frac{1}{\varepsilon}\psi(\varphi)\;\mathrm{d}\mathcal{L}^{n}\,.$
(3.6)

This term regularizes the optimization problem in order for it to be well-
posed. We observe that the Ginzburg–Landau energy is decreasing with respect
to symmetric-decreasing rearrangement of its argument. This can be interpreted
as a phase-field version of the isoperimetric inequality.

######  Lemma 3.4 (Phase-field isoperimetric inequality).

Let $\varepsilon>0$ be arbitrary. Then, for all $\varphi\in
H_{0}^{1}(\Omega;[0,1])$, we have

$\displaystyle E^{\varepsilon}(\varphi^{*})\leq E^{\varepsilon}(\varphi).$
(3.7)

Furthermore we will prove the following phase-field version of the Faber–Krahn
inequality on the diffuse interface level.

######  Theorem 3.5 (Phase-field Faber–Krahn inequality).

Let $\varepsilon>0$ be arbitrary. Then, for all $\varphi\in
H_{0}^{1}(\Omega;[0,1])$, we have

$\displaystyle\lambda_{1}^{\varepsilon,\varphi^{*}}\leq\lambda_{1}^{\varepsilon,\varphi}.$

In order to recover the classical Faber–Krahn inequality in the sharp-
interface limit $\varepsilon\to 0$, we consider the following optimization
problem:

$\displaystyle\left\\{\begin{aligned}
&\text{Minimize}&&J^{\varepsilon}_{\gamma}(\varphi)=\lambda_{1}^{\varepsilon,\varphi}+\gamma
E^{\varepsilon}(\varphi)\\\ &\text{subject
to}&&\varphi\in\Phi_{m}\,.\end{aligned}\right.$
($\text{OP}^{\varepsilon}_{\gamma}$)

Here, $\lambda_{1}^{\varepsilon,\varphi}$ denotes the principal eigenvalue
corresponding to the function $\varphi$ as introduced in Proposition 3.2, and
$\gamma>0$ is the surface tension. Here, the additional summand $\gamma
E^{\varepsilon}(\varphi)$ acts as a regularization term which ensures well-
posedness of the optimization problem and is further used to gain additional
information about its minimizers. After passing to the sharp-interface limit
$\varepsilon\to 0$, we recover the classical Faber–Krahn inequality in the
framework of $BV$ functions by sending $\gamma\to 0$, which is possible as we
will see later that the minimizer on the sharp-interface level does not depend
on $\gamma$.

The existence of a minimizer $\varphi\in\Phi_{m}$ of the optimization problem
($\text{OP}^{\varepsilon}_{\gamma}$) was established in [21, Theorem 3.8].
This means that the following lemma holds.

######  Lemma 3.6.

Let $\varepsilon,\gamma>0$ be arbitrary. Then, the optimization problem
($\text{OP}^{\varepsilon}_{\gamma}$) possesses a minimizer
$\varphi\in\Phi_{m}$.

Now, based on Lemma 3.4 and Theorem 3.5, the following theorem shows that
minimizers of ($\text{OP}^{\varepsilon}_{\gamma}$) are necessarily symmetric-
decreasing. The same holds for the positive-normalized eigenfunction of the
corresponding principal eigenvalue.

######  Theorem 3.7.

Let $\varepsilon,\gamma>0$ be arbitrary, $m\in(0,1)$, and let
$\varphi\in\Phi_{m}$ be any minimizer of the optimization problem
($\text{OP}^{\varepsilon}_{\gamma}$). Then, $\varphi=\varphi^{*}$ almost
everywhere in $\Omega$, meaning that $\varphi$ is symmetric-decreasing, and
the positive-normalized eigenfunction $w_{1}^{\varepsilon,\varphi}$ to the
principal eigenvalue $\lambda_{1}^{\varepsilon,\varphi}$ also fulfills
$w_{1}^{\varepsilon,\varphi}=(w_{1}^{\varepsilon,\varphi})^{*}$ almost
everywhere in $\Omega$.

Furthermore, the following theorem, which is a direct consequence of the
boundedness of the Ginzburg–Landau energy along a sequence of minimizers for
$\varepsilon\to 0$, shows that the thickness of the interface up to an
infinitesimally small error is $\mathcal{O}(\varepsilon)$.

######  Theorem 3.8.

Let $\gamma>0$ and $m\in(0,1)$ be arbitrary. For any $\varepsilon>0$, let
$\varphi_{\varepsilon}$ be a minimizer of the optimization problem
($\text{OP}^{\varepsilon}_{\gamma}$). Then, there is a constant $C>0$ such
that for all $0<\delta<\frac{1}{2}$ it holds that

$\displaystyle\mathcal{L}^{n}\left(\left\\{\delta\leq\varphi_{\varepsilon}\leq
1-\delta\right\\}\right)\leq\frac{C\varepsilon}{\alpha_{\delta}\gamma}\quad\text{with}\quad\alpha_{\delta}:=\underset{[\delta,1-\delta]}{\min}\;\psi>0.$
(3.8)

The proofs of Theorem 3.7 and Theorem 3.8 are presented in Section 4.

Combining the preceding two results, we deduce that every minimizer
$\varphi_{\varepsilon}$ of ($\text{OP}^{\varepsilon}_{\gamma}$) is symmetric-
decreasing and exhibits the expected phase-field structure, i.e., for any
$0<\delta<\tfrac{1}{2}$, the width of the annulus on which
$\varphi_{\varepsilon}$ attains values between $\delta$ and $1-\delta$ is of
order $\varepsilon$. This behavior is illustrated in Figure 2.

Figure 2: Schematic sketch of a minimizer $\varphi_{\varepsilon}$ in radial
direction $r=|x|$.

Now, we investigate the limit $\varepsilon\to 0$. Therefore, let us fix a
sequence of minimizers $(\varphi_{\varepsilon})_{\varepsilon>0}$ of
($\text{OP}^{\varepsilon}_{\gamma}$). We intend to show that this sequence
converges to the characteristic function of the ball centered at the origin
with volume $m\left|\Omega\right|$ and that this ball is a minimizer of a
suitable limit cost functional (see Theorem 3.15). To this end, we recall the
most important aspects from [21].

First of all, we recall the limit eigenvalue problem, i.e., the eigenvalue
problem corresponding to (3.2) on the sharp interface level. For details we
refer again to [21]. For any given $\varphi\in BV(\Omega;\left\\{\pm
1\right\\})$, we want to solve

$\displaystyle-\Delta w$ $\displaystyle=\lambda w$
$\displaystyle\qquad\text{in $E^{\varphi}$},$ (3.9a) $\displaystyle
w|_{\partial E^{\varphi}}$ $\displaystyle=0$ on $\partial E^{\varphi}$, (3.9b)

where

$\displaystyle E^{\varphi}$
$\displaystyle=\left\\{x\in\Omega\;|\;\varphi(x)=1\right\\}.$

Note that, in general, $E^{\varphi}$ is only a set of finite perimeter and
therefore, it merely enjoys a very weak regularity. However, the following
definition turns out to be the suitable notion of weak solution as it is
compatible with the sharp-interface limit $\varepsilon\to 0$ (see Proposition
3.12).

######  Definition 3.9.

Let

$\displaystyle\varphi\in\Phi_{m}^{0}\coloneqq\left\\{\varphi\in
BV(\Omega;\left\\{0,1\right\\})\;|\;\textstyle\fint_{\Omega}\varphi\;\mathrm{d}\mathcal{L}^{n}=m\right\\}$

be arbitrary.

  1. (a)

For any given $\lambda\in\mathbb{R}$, a function $w\in V^{\varphi}$ is called
a weak solution of the system (3.9) if the weak formulation

$\displaystyle\int_{\Omega}\nabla
w\cdot\nabla\eta\;\mathrm{d}\mathcal{L}^{n}=\lambda\int_{\Omega}w\eta\;\mathrm{d}\mathcal{L}^{n}$
(3.10)

is satisfied for all test functions $\eta\in V^{\varphi}$, where

$\displaystyle V^{\varphi}$ $\displaystyle=\left\\{\eta\in
H_{0}^{1}(\Omega)\;|\;\eta=0\text{ a.e. in }\Omega\backslash
E^{\varphi}\right\\}.$

  2. (b)

A real number $\lambda^{0,\varphi}$ is called an eigenvalue associated with
$\varphi$, if there exists at least one nontrivial weak solution
$w^{0,\varphi}\in H^{1}_{0}(\Omega)$ of the eigenvalue problem (3.9) written
for $\lambda=\lambda^{0,\varphi}$.

In this case, $w^{0,\varphi}$ is called an eigenfunction to the eigenvalue
$\lambda^{0,\varphi}$.

We recall the following proposition which is a direct consequence of [21,
Theorem 4.2].

######  Proposition 3.10.

Suppose that $\varphi\in\Phi_{m}^{0}$ with
$V^{\varphi}\neq\left\\{0\right\\}$.

  1. (a)

The minimum in

$\displaystyle\min\left\\{\left.\frac{\int_{\Omega}\left|\nabla
v\right|^{2}\;\mathrm{d}\mathcal{L}^{n}}{\int_{\Omega}\left|v\right|^{2}\;\mathrm{d}\mathcal{L}^{n}}\right|v\in
V^{\varphi}\backslash\\{0\\}\right\\}=:\lambda_{1}^{0,\varphi}.$ (3.11)

is attained and any minimizer $w\in V^{\varphi}\backslash\left\\{0\right\\}$
is an eigenfunction of the limit problem (3.9) to the eigenvalue
$\lambda_{1}^{0,\varphi}$ in the sense of Definition 3.9(b).

  2. (b)

$\lambda_{1}^{0,\varphi}>0$ is the smallest eigenvalue of the limit problem
(3.9) in the sense of Definition 3.9(b).

######  Remark 3.11.

Note that in [21, Theorem 4.2], we are in the situation that $V^{\varphi}$ is
an infinite dimensional vector space. In the present paper, we only assume
that $V^{\varphi}$ is non-trivial, but the above proposition can be
established analogously using classical spectral theory. If
$V^{\varphi}=\left\\{0\right\\}$, we set $\lambda_{1}^{0,\varphi}=+\infty$,
which is consistent with the above proposition.

The following continuity result for the principal eigenvalues in the limit
$\varepsilon\to 0$ was established in [21, Lemma 4.4].

######  Proposition 3.12.

Let $\left(\varphi_{\varepsilon}\right)_{\varepsilon>0}\subset L^{1}(\Omega)$
with $\varphi_{\varepsilon}\in[0,1]$ almost everywhere in $\Omega$ and suppose
that $\varphi\in BV(\Omega,\left\\{0,1\right\\})$ with
$V^{\varphi}\neq\left\\{0\right\\}$ such that

$\displaystyle\underset{\varepsilon\searrow
0}{\lim\,}\left\|\varphi_{\varepsilon}-\varphi\right\|_{L^{1}(\Omega)}=0,$

Moreover, we demand the additional convergence rate

$\displaystyle\underset{\varepsilon\searrow
0}{\lim\,}\left\|\varphi_{\varepsilon}-\varphi\right\|_{L^{1}\big{(}E^{\varphi}\cap\left\\{\varphi_{\varepsilon}<\frac{1}{2}\right\\}\big{)}}=\mathcal{O}(\varepsilon).$

Then, there exists an eigenfunction $u\in V^{\varphi}$ of the limit problem
(3.9) to the eigenvalue $\lambda_{1}^{0,\varphi}$ such that

$\displaystyle\underset{\varepsilon\searrow
0}{\lim\,}\int_{\Omega}b^{\varepsilon}(\varphi_{\varepsilon})\big{|}w_{1}^{\varepsilon,\varphi_{\varepsilon}}\big{|}^{2}\;\mathrm{d}\mathcal{L}^{n}=\int_{\Omega}b^{0}(\varphi)\left|u\right|^{2}\;\mathrm{d}\mathcal{L}^{n}=0,$

as well as

$\displaystyle\underset{\varepsilon\searrow
0}{\lim\,}\left\|w_{1}^{\varepsilon,\varphi_{\varepsilon}}-u\right\|_{H^{1}(\Omega)}=0\quad\text{and}\quad\underset{\varepsilon\searrow
0}{\lim\,}\lambda_{1}^{\varepsilon,\varphi_{\varepsilon}}=\lambda_{1}^{0,\varphi}.$

We point out that in [21], the above result was established for any eigenvalue
in the case where $V^{\varphi}$ is an infinite dimensional vector space.
However, as we only consider the principal eigenvalue, the Rayleigh quotient
merely needs to be minimized over the set $V^{\varphi}\backslash\\{0\\}$ (cf.
(3.11)). It is thus clear that the proof of [21] also works under the weaker
assumption $V^{\varphi}\neq\\{0\\}$.

Since $\lambda_{1}^{0,\varphi}=+\infty$ if $V^{\varphi}=\\{0\\}$, the
following corollary is a direct consequence.

######  Corollary 3.13.

Let the assumptions of the previous theorem be fulfilled, but allow for the
case $V^{\varphi}=\left\\{0\right\\}$. Then, it still holds

$\displaystyle\underset{\varepsilon\searrow
0}{\limsup\,}\lambda_{1}^{\varepsilon,\varphi_{\varepsilon}}\leq\lambda_{1}^{0,\varphi}.$

Finally, we consider the limit cost functional

$\displaystyle J^{0}_{\gamma}(\varphi)\coloneqq\begin{cases}\begin{aligned}
&\lambda_{1}^{0,\varphi}+\gamma
c_{0}\left(P_{\Omega}(E^{\varphi})+\int_{\partial\Omega}\varphi_{|{\partial\Omega}}\text{\,d}\mathcal{H}^{n-1}\right)&&\quad\text{if
}\varphi\in\Phi_{m}^{0},\\\ &+\infty&&\quad\text{if }\varphi\in
L^{1}(\Omega)\backslash\Phi_{m}^{0},\end{aligned}\end{cases}$ (3.12)

where $c_{0}=\int_{0}^{1}\sqrt{2\psi(t)}\text{\,d}t$ and
$\varphi_{|\partial\Omega}\in L^{1}(\partial\Omega)$ denotes the trace of the
$BV$ function $\varphi$ (see Section 2.4).

######  Remark 3.14.

We note that in [29], where the phase-fields are subject to a more complex
inhomogeneous space dependent Dirichlet boundary condition, the corresponding
term in the limit cost functional resulting from the Ginzburg–Laundau energy
is written (transferred to our notation) as

$\displaystyle\int_{\Omega}\left|\nabla\Psi(\varphi)\right|\;\mathrm{d}\mathcal{L}^{n}+\int_{\partial\Omega}\left|\Psi(\varphi_{|\partial\Omega})\right|\;\mathrm{d}\mathcal{H}^{n-1}.$

Here, the function $\Psi$ is defined by

$\displaystyle\Psi(s)\coloneqq\int_{0}^{s}\sqrt{2\psi(t)}\text{\,d}t,$ (3.13)

and $\int_{\Omega}\left|\nabla\Psi(\varphi)\right|\;\mathrm{d}\mathcal{L}^{n}$
denotes the variation of $\Psi(\varphi)\in L^{1}(\Omega)$ as given in Section
2.4. As obviously $\Psi(\varphi)=\Psi(1)\chi_{E^{\varphi}}$ in
$BV(\Omega,\\{0,1\\})$ we obtain

$\displaystyle\int_{\Omega}\left|\nabla\Psi(\varphi)\right|\;\mathrm{d}\mathcal{L}^{n}=c_{0}P_{\Omega}(E^{\varphi}).$

Furthermore, due to the definition of the trace in (2.11) and $\varphi\in
BV(\Omega,\left\\{0,1\right\\})$, we see that also $\varphi_{|\partial\Omega}$
only attains the values $0$ and $1$. Hence, we have
$\Psi(\varphi_{|\partial\Omega})=\Psi(1)\varphi_{|\partial\Omega}$ in
$L^{1}(\partial\Omega)$, which yields

$\displaystyle\int_{\partial\Omega}\left|\Psi(\varphi_{|\partial\Omega})\right|\;\mathrm{d}\mathcal{H}^{n-1}=c_{0}\int_{\partial\Omega}\varphi_{|\partial\Omega}\;\mathrm{d}\mathcal{H}^{n-1}.$

Note that, as we are imposing a homogeneous Dirichlet boundary condition, we
do not need to rely on the very technical construction of a recovery sequence
presented in [29] as we can simply perform a cut-off procedure as in [8]. The
idea in [8] is to approximate any finite perimeter set by truncated sets that
are compactly contained within $\Omega$. For these truncated sets, we then
perform a diffuse interface approximation in the spirit of [27, 33]. Using
this approach, the need of the additional boundary integral in the limit cost
functional can be clearly seen: In the course of this approximation, the
boundaries of the truncated sets are getting closer and closer to the boundary
of $\Omega$. Therefore, the whole boundary of the limit set has to be
perceived by the limit energy. For more details, we refer to the proof of
Theorem 3.17 given in Section 4.

The previous discussion allows us to state the desired theorem which states
the convergence of minimizers as $\varepsilon$ tends to zero.

######  Theorem 3.15.

Let $\varphi_{0}\in\Phi^{0}_{m}$ be the characteristic function of the ball
centered at the origin with volume $m\left|\Omega\right|$ and let
$\left(\varphi_{\varepsilon}\right)_{\varepsilon>0}$ be a sequence of
minimizers of ($\text{OP}^{\varepsilon}_{\gamma}$).
Then

$\displaystyle\underset{\varepsilon\searrow
0}{\lim\,}\left\|\varphi_{\varepsilon}-\varphi_{0}\right\|_{L^{1}(\Omega)}=0,\quad\underset{\varepsilon\searrow
0}{\lim\,}J^{\varepsilon}_{\gamma}(\varphi_{\varepsilon})=J^{0}_{\gamma}(\varphi_{0}),$

and $\varphi_{0}$ is a minimizer of $J^{0}_{\gamma}$.

As a direct consequence we finally obtain the classical Faber–Krahn theorem in
our framework by sending the surface tension parameter $\gamma$ to zero.

######  Corollary 3.16 (Faber–Krahn theorem for $BV$ functions).

Let $\varphi_{0}\in\Phi^{0}_{m}$ be the characteristic function of the ball
centered at the origin with volume $m\left|\Omega\right|$. Then, it holds that

$\displaystyle\lambda_{1}^{0,\varphi_{0}}=\min\left\\{\lambda_{1}^{0,\varphi}\;|\;\varphi\in\Phi^{0}_{m}\right\\}.$

We point out that this result even extends the classical Faber–Krahn theorem
which merely states that an (open) ball is a minimizer among all open sets of
the same volume. In this regard, Corollary 3.16 is more general as it even
allows for the larger class of finite perimeter sets. Alternatively, in [12],
it was shown that the Faber–Krahn theorem remains correct if the minimization
problem is formulated for the class of quasi-open sets.

The proof of Theorem 3.15 can be found in Section 4. In this proof, the key
step is to show that
$J^{\varepsilon}_{\gamma}\overset{\Gamma}{\to}J^{0}_{\gamma}$ as
$\varepsilon\to 0$, i.e., $J^{\varepsilon}_{\gamma}$ converges to
$J^{0}_{\gamma}$ in the sense of $\Gamma$-convergence. Our strategy will be
similar as in [21].

The first step is to prove the $\Gamma$-convergence for slightly modified
functionals $F_{\varepsilon}^{\gamma}$ where the corresponding set of
admissible phase-fields does not contain a volume constraint. In the proof, we
need to revisit the construction of the recovery sequence in [33] as we allow
for more general potentials $\psi$. In order to tackle the Dirichlet boundary
constraint hidden in $H_{0}^{1}(\Omega)$, we apply the idea of [8, Theorem
3.1]. Namely, to construct a recovery sequence for any given $\varphi\in
BV(\Omega;\left\\{0,1\right\\})$, we approximate the corresponding set
$\\{\varphi=1\\}$ by truncated sets which are compactly contained in $\Omega$.
The $\Gamma$-convergence result is stated by the following theorem.

######  Theorem 3.17.

For any $\varepsilon,\gamma>0$, let the functions
$F_{\varepsilon}^{\gamma},F_{0}^{\gamma}:L^{1}(\Omega)\to\mathbb{R}\cup\left\\{+\infty\right\\}$
be defined as

$\displaystyle
F_{\varepsilon}^{\gamma}(\varphi)=\begin{cases}\lambda_{1}^{\varepsilon,\varphi_{\varepsilon}}+\gamma\int_{\Omega}\frac{\varepsilon}{2}\left|\nabla\varphi\right|^{2}+\frac{1}{\varepsilon}\psi(\varphi)\;\mathrm{d}\mathcal{L}^{n}&\quad\textup{if
}\varphi\in H_{0}^{1}(\Omega;[0,1]),\\\
+\infty&\quad\textup{else},\end{cases}$

and

$\displaystyle
F_{0}^{\gamma}(\varphi)=\begin{cases}\lambda_{1}^{0,\varphi}+c_{0}\gamma\left(P_{\Omega}(E^{\varphi})+\int_{\partial\Omega}\varphi_{|\partial\Omega}\;\mathrm{d}\mathcal{H}^{n-1}\right)&\quad\textup{if
}\varphi\in BV(\Omega;\left\\{0,1\right\\}),\\\
+\infty&\quad\textup{else}.\end{cases}$

Then $F_{\varepsilon}^{\gamma}\overset{\Gamma}{\to}F_{0}^{\gamma}$.

The second step is to modify the recovery sequence obtained by Theorem 3.17,
as in [21, Theorem 4.8], via suitable $C^{1}$-diffeomorphisms such that the
modified sequence is actually a recovery sequence for
$J^{\varepsilon}_{\gamma}$ satisfying the volume constraint included in
$\Phi_{m}$. This is done in the following theorem.

######  Theorem 3.18.

For any $\varepsilon,\gamma>0$, let the functions
$J^{\varepsilon}_{\gamma},J^{0}_{\gamma}:L^{1}(\Omega)\to\mathbb{R}\cup\left\\{+\infty\right\\}$
be defined as

$\displaystyle
J^{\varepsilon}_{\gamma}(\varphi)=\begin{cases}\lambda_{1}^{\varepsilon,\varphi}+\gamma\int_{\Omega}\frac{\varepsilon}{2}\left|\nabla\varphi\right|^{2}+\frac{1}{\varepsilon}\psi(\varphi)\;\mathrm{d}\mathcal{L}^{n}&\quad\textup{if
}\varphi\in\Phi_{m},\\\ +\infty&\quad\textup{else},\end{cases}$

and

$\displaystyle
J^{0}_{\gamma}(\varphi)=\begin{cases}\lambda_{1}^{0,\varphi}+c_{0}\gamma\left(P_{\Omega}(E^{\varphi})+\int_{\partial\Omega}\varphi_{|\partial\Omega}\;\mathrm{d}\mathcal{H}^{n-1}\right)&\quad\textup{if
}\varphi\in\Phi_{m}^{0},\\\ +\infty&\quad\textup{else}.\end{cases}$

Then $J^{\varepsilon}_{\gamma}\overset{\Gamma}{\to}J^{0}_{\gamma}$.

The proofs of Theorem 3.17 and Theorem 3.18 are presented in Section 4.

######  Remark 3.19.

Although for our purposes we have fixed $\Omega=B_{R}(0)$ the results
presented in Theorem 3.17 and Theorem 3.18 hold true for any bounded, open set
$\Omega\subset\mathbb{R}^{n}$ with Lipschitz boundary.

##  4 Proofs

We first assure that the basic properties of radially symmetric-decreasing
rearrangements in $\mathbb{R}^{n}$ carry over to our local case.

  * Proof of Lemma 2.4.

In view of Remark 2.3(b), the statements (a)–(c), (e) and (f) are direct
consequences of the results in [25, Sections 3.3–3.5].

To prove (d), we use the decomposition
$\Psi^{\prime}=\Psi^{\prime}_{+}-\Psi^{\prime}_{-}$, where
$\Psi^{\prime}_{+}:=\max(\Psi^{\prime},0)$ and
$\Psi^{\prime}_{-}:=-\min(\Psi^{\prime},0)$ denote the positive part and the
negative part of $\Psi^{\prime}$, respectively. Now, we define

$\displaystyle\Psi_{1}(t):=\int_{0}^{t}\Psi_{+}^{\prime}(s)\,\mathrm{d}s\quad\text{and}\quad\Psi_{2}(t):=\int_{0}^{t}\Psi_{-}^{\prime}(s)\,\mathrm{d}s\quad\text{for
all $t\in[0,1]$.}$

Recalling $\Psi(0)=0$, we apply the fundamental theorem of calculus to derive
the decomposition ${\Psi=\Psi_{1}-\Psi_{2}}$. As the functions $\Psi_{1}$ and
$\Psi_{2}$ are non-decreasing, (2.7) follows directly from [25, Section
3.3(iv)].

To prove (g), let $f\in H^{1}_{0}(\Omega;\mathbb{R}_{0}^{+})$ be any function,
and let $f_{0}:\mathbb{R}^{n}\to\mathbb{R}_{0}^{+}$ denote its trivial
extension as in Remark 2.3(b). This means that $f_{0}\in
H^{1}(\mathbb{R}^{n})$ is a non-negative function with compact support. We
further define

$\displaystyle A:[0,\infty)\to[0,\infty),\,x\mapsto x^{2}.$

Hence $A\in C^{2}([0,\infty)$ is strictly increasing, $A(0)=0$ and
$A^{\frac{1}{2}}$ is convex. Thus, as all conditions are fulfilled, we can
apply the first part of [10, Theorem 1.1] and obtain

$\displaystyle\int_{\mathbb{R}^{n}}\big{|}\nabla
f_{0}^{\ast}\big{|}^{2}\;\mathrm{d}\mathcal{L}^{n}\leq\int_{\mathbb{R}^{n}}\big{|}\nabla
f_{0}\big{|}^{2}\;\mathrm{d}\mathcal{L}^{n},$ (4.1)

which directly implies (2.9) since $\nabla f_{0}=0$ and $\nabla f^{*}_{0}=0$
almost everywhere on $\mathbb{R}^{n}\backslash\Omega$.

In addition, let us now assume that condition (2.10) holds true and that $f>0$
almost everywhere in $\Omega$. Since $f^{*}_{0}=0$ on
$\mathbb{R}^{n}\backslash\Omega$, we have
$\left\\{f_{0}^{\ast}>0\right\\}\subset\Omega$ and thus,

$\displaystyle\mathcal{L}^{n}\Big{(}\\{x\in\mathbb{R}^{n}\,|\,\nabla
f_{0}^{*}(x)=0\\}\cap\big{(}f_{0}^{*}\big{)}^{-1}\big{(}(0,\infty)\big{)}\Big{)}\leq\mathcal{L}^{n}\big{(}\\{x\in\Omega\,|\,\nabla
f^{*}(x)=0\\}\big{)}=0.$

Therefore, [10, Theorem 1.1] states that equality in (4.1) holds if and only
if $f_{0}$ is a translate of $f_{0}^{*}$. This directly entails that equality
in (2.9) holds if and only if $f$ is a translate of $f^{*}$. However, since
$f\in H^{1}_{0}(\Omega)$ with $f>0$ almost everywhere in $\Omega=B_{R}(0)$,
this is possible if and only if $f=f^{*}$ almost everywhere in $\Omega$, which
proves the claim. ∎

Now we are in the position to present the proofs of our main results.

  * Proof of Theorem 3.4.

In view of the definition of $E^{\varepsilon}$ given in (3.6), the assertion
follows directly by using the Pólya–Szegő inequality (Lemma 2.4(g)) to
estimate the gradient term, and by applying Lemma 2.4(d) to the potential
term. ∎

  * Proof of Theorem 3.5.

Let $\varphi\in H_{0}^{1}(\Omega;[0,1])$ be arbitrary.

First of all, we derive some general inequalities. Therefore, let $w\in
H^{1}_{0}(\Omega)\backslash\\{0\\}$ with $w\geq 0$ a.e. in $\Omega$ be
arbitrary. In view of (A3), the coefficient $b^{\varepsilon}$ is continuous,
decreasing and $b^{\varepsilon}(0)=\beta^{\varepsilon}$, we infer that the
function

$\displaystyle B^{\varepsilon}:[0,1]\to[0,\beta^{\varepsilon}],\quad s\mapsto
b^{\varepsilon}(0)-b^{\varepsilon}(s)$ (4.2)

is continuous and increasing. Hence, according to Lemma 2.4(c), we have

$\displaystyle\big{(}B^{\varepsilon}(\varphi)\big{)}^{*}=B^{\varepsilon}(\varphi^{*})\quad\text{and}\quad\big{(}w^{2}\big{)}^{*}=\big{(}w^{*}\big{)}^{2}$

almost everywhere in $\Omega$. Applying Lemma 2.4(b) and (e), we thus obtain

$\displaystyle\int_{\Omega}b^{\varepsilon}(\varphi)w^{2}\;\mathrm{d}\mathcal{L}^{n}$
$\displaystyle=b^{\varepsilon}(0)\int_{\Omega}w^{2}\;\mathrm{d}\mathcal{L}^{n}-\int_{\Omega}B^{\varepsilon}(\varphi)w^{2}\;\mathrm{d}\mathcal{L}^{n}$
$\displaystyle\geq
b^{\varepsilon}(0)\int_{\Omega}\big{(}w^{*}\big{)}^{2}\;\mathrm{d}\mathcal{L}^{n}-\int_{\Omega}\big{(}B^{\varepsilon}(\varphi)\big{)}^{*}\;\big{(}w^{2}\big{)}^{*}\;\mathrm{d}\mathcal{L}^{n}$
$\displaystyle=b^{\varepsilon}(0)\int_{\Omega}\big{(}w^{*}\big{)}^{2}\;\mathrm{d}\mathcal{L}^{n}-\int_{\Omega}B^{\varepsilon}(\varphi^{*})\,\big{(}w^{*}\big{)}^{2}\;\mathrm{d}\mathcal{L}^{n}$
$\displaystyle=\int_{\Omega}b^{\varepsilon}(\varphi^{*})\,\big{(}w^{*}\big{)}^{2}\;\mathrm{d}\mathcal{L}^{n}.$
(4.3)

In particular, since $\varphi^{**}=\varphi^{*}$, this already entails that

$\displaystyle\int_{\Omega}b^{\varepsilon}(\varphi^{*})w^{2}\;\mathrm{d}\mathcal{L}^{n}\geq\int_{\Omega}b^{\varepsilon}(\varphi^{*})\,\big{(}w^{*}\big{)}^{2}\;\mathrm{d}\mathcal{L}^{n}.$
(4.4)

These general estimates can now be used to prove the assertion
$\lambda_{1}^{\varepsilon,\varphi^{*}}\leq\lambda_{1}^{\varepsilon,\varphi}$.
Therefore, we define the functional

$\displaystyle\mathcal{R}:H^{1}_{0}(\Omega)\backslash\\{0\\}\to\mathbb{R},\quad
w\mapsto\frac{\int_{\Omega}\left|\nabla
w\right|^{2}+b^{\varepsilon}(\varphi^{*})\,w^{2}\;\mathrm{d}\mathcal{L}^{n}}{\left\|w\right\|_{L^{2}(\Omega)}^{2}}\;.$
(4.5)

We now consider the positive-normalized eigenfunction
$w_{1}^{\varepsilon,\varphi^{*}}$ associated with $\varphi$, which is
obviously a minimizer of $\mathcal{R}$. Using (4.4) along with the Pólya–Szegő
inequality (Lemma 2.4(g)) and Lemma 2.4(b), we find that

$\displaystyle\mathcal{R}\Big{(}\big{(}w_{1}^{\varepsilon,\varphi^{*}}\big{)}^{*}\Big{)}\leq\mathcal{R}\Big{(}w_{1}^{\varepsilon,\varphi^{*}}\Big{)}.$

Thus, $\big{(}w_{1}^{\varepsilon,\varphi^{*}}\big{)}^{*}$ is also a minimizer
of $\mathcal{R}$ and thus, due to Proposition 3.2(c), it is an eigenfunction
to the eigenvalue $\lambda_{1}^{\varepsilon,\varphi^{*}}$. As
$\big{(}w_{1}^{\varepsilon,\varphi^{*}}\big{)}^{*}$ is non-negative and
$L^{2}$-normalized, this is enough to deduce

$\displaystyle\big{(}w_{1}^{\varepsilon,\varphi^{*}}\big{)}=\big{(}w_{1}^{\varepsilon,\varphi^{*}}\big{)}^{*}\quad\text{a.e.
in }\Omega,$

as the eigenspace to $\lambda_{1}^{\varepsilon,\varphi^{*}}$ is one-
dimensional. On the other hand, the Courant–Fischer characterization (3.3)
yields that for any $w\in H_{0}^{1}(\Omega)\backslash\left\\{0\right\\}$ with
$w\geq 0$ a.e. in $\Omega$, we have

$\displaystyle\lambda_{1}^{\varepsilon,\varphi^{*}}\leq\frac{\int_{\Omega}\left|\nabla
w^{*}\right|^{2}+b^{\varepsilon}(\varphi^{*})\,(w^{*})^{2}\;\mathrm{d}\mathcal{L}^{n}}{\left\|w^{*}\right\|_{L^{2}(\Omega)}^{2}}\leq\frac{\int_{\Omega}\left|\nabla
w\right|^{2}+b^{\varepsilon}(\varphi)\,(w)^{2}\;\mathrm{d}\mathcal{L}^{n}}{\left\|w\right\|_{L^{2}(\Omega)}^{2}}.$
(4.6)

Here, we applied the Pólya–Szegő inequality (Lemma 2.4(g)), estimate (Proof of
Theorem 3.5.) and Lemma 2.4(b). Hence, choosing
$w=w_{1}^{\varepsilon,\varphi}$ we use Proposition 3.2(c) to conclude

$\displaystyle\lambda_{1}^{\varepsilon,\varphi^{*}}\leq\lambda_{1}^{\varepsilon,\varphi}$
(4.7)

and thus, the proof is complete. ∎

  * Proof of Theorem 3.7.

Let $\varphi\in\Phi_{m}$ be any minimizer of the optimization problem
($\text{OP}^{\varepsilon}_{\gamma}$). Combining the estimates (3.7) and (4.7),
we deduce that $J^{\varepsilon}_{\gamma}(\varphi^{*})\leq
J^{\varepsilon}_{\gamma}(\varphi)$.Since $\varphi$ is a minimizer, this
implies that

$\displaystyle
J^{\varepsilon}_{\gamma}(\varphi^{*})=J^{\varepsilon}_{\gamma}(\varphi).$
(4.8)

This proves that the symmetric-decreasing rearrangement $\varphi^{*}$ is also
a minimizer of the optimization problem ($\text{OP}^{\varepsilon}_{\gamma}$).

Therefore, it remains to prove that the eigenfunction
$w_{1}^{\varepsilon,\varphi}$ and the minimizer $\varphi$ are symmetric-
decreasing, meaning that $\varphi=\varphi^{*}$ and
$w_{1}^{\varepsilon,\varphi}=(w_{1}^{\varepsilon,\varphi})^{*}$ a.e. in
$\Omega$.

First of all, using (3.7) and (4.8), we obtain the estimate

$\displaystyle\lambda_{1}^{\varepsilon,\varphi}=J^{\varepsilon}_{\gamma}(\varphi)-\gamma
E^{\varepsilon}(\varphi)\leq J^{\varepsilon}_{\gamma}(\varphi^{*})-\gamma
E^{\varepsilon}(\varphi^{*})=\lambda_{1}^{\varepsilon,\varphi^{*}}.$

Hence, in combination with (4.7), we conclude that

$\displaystyle\lambda:=\lambda_{1}^{\varepsilon,\varphi}=\lambda_{1}^{\varepsilon,\varphi^{*}}.$
(4.9)

As above let $w=w_{1}^{\varepsilon,\varphi}$ be the positive-normalized
eigenfunction corresponding to the principal eigenvalue $\lambda$ associated
with the minimizer $\varphi$. Combining (4.6) with (4.9) we arrive at

$\displaystyle\lambda_{1}^{\varepsilon,\varphi^{*}}=\frac{\int_{\Omega}\left|\nabla
w^{*}\right|^{2}+b^{\varepsilon}(\varphi^{*})\,(w^{*})^{2}\;\mathrm{d}\mathcal{L}^{n}}{\left\|w^{*}\right\|_{L^{2}(\Omega)}^{2}}=\frac{\int_{\Omega}\left|\nabla
w\right|^{2}+b^{\varepsilon}(\varphi)\,(w)^{2}\;\mathrm{d}\mathcal{L}^{n}}{\left\|w\right\|_{L^{2}(\Omega)}^{2}}=\lambda_{1}^{\varepsilon,\varphi}.$
(4.10)

Since, according to Proposition 3.2(c), the eigenspace to the eigenvalue
$\lambda$ is one-dimensional, we conclude

$\displaystyle
w^{*}=\big{(}w_{1}^{\varepsilon,\varphi}\big{)}^{*}=w_{1}^{\varepsilon,\varphi^{*}}\quad\text{a.e.
in }\Omega.$ (4.11)

Moreover, Proposition 3.2(c) further yields $w^{*}>0$ a.e. in $\Omega$. As
$w^{*}$ is a symmetric-decreasing rearrangement, it follows from Lemma 2.4(a)
that $w^{*}>0$ actually holds everywhere in $\Omega$, which will be crucial in
the following. Since
$\left\|w^{*}\right\|_{L^{2}(\Omega)}=\left\|w\right\|_{L^{2}(\Omega)}=1$,
(4.10) entails that

$\displaystyle\int_{\Omega}\left|\nabla
w\right|^{2}+b^{\varepsilon}(\varphi)w^{2}\;\mathrm{d}\mathcal{L}^{n}=\int_{\Omega}\left|\nabla
w^{*}\right|^{2}+b^{\varepsilon}(\varphi^{*})(w^{*})^{2}\;\mathrm{d}\mathcal{L}^{n}.$
(4.12)

Invoking (Proof of Theorem 3.5.), we thus obtain

$\displaystyle\int_{\Omega}\left|\nabla
w\right|^{2}\;\mathrm{d}\mathcal{L}^{n}-\int_{\Omega}\left|\nabla
w^{*}\right|^{2}\;\mathrm{d}\mathcal{L}^{n}=\int_{\Omega}b^{\varepsilon}(\varphi^{*})(w^{*})^{2}\;\mathrm{d}\mathcal{L}^{n}-\int_{\Omega}b^{\varepsilon}(\varphi)w^{2}\;\mathrm{d}\mathcal{L}^{n}\leq
0.$ (4.13)

Hence, we have equality in the Pólya–Szegő inequality (Lemma 2.4(g)):

$\displaystyle\int_{\Omega}\left|\nabla
w\right|^{2}\;\mathrm{d}\mathcal{L}^{n}=\int_{\Omega}\left|\nabla
w^{*}\right|^{2}\;\mathrm{d}\mathcal{L}^{n}.$ (4.14)

In order to prove $w=w^{*}$ a.e. in $\Omega$, we now intend to show that
$w^{*}$ is even strictly symmetric-decreasing. Therefore, we argue by
contradiction and assume that this is not the case. This means that there
exists a direction $x\in\mathbb{R}^{n}$ with $\left|x\right|=1$ as well as
$0\leq s<t\leq R$ such that $w^{*}(sx)=w^{*}(tx)=:c$. As the function $w^{*}$
is positive in $\Omega$, we deduce that $c>0$. Moreover, since $w^{*}$ is non-
increasing in radial direction, we infer that $w^{*}(\tau x)=c$ for all
$\tau\in[s,t]$. Because of spherical symmetry, this already implies

$\displaystyle
w^{*}=c\quad\text{in}\;\big{\\{}x\in\Omega\,\big{|}\,s\leq\left|x\right|\leq
t\big{\\}}.$ (4.15)

We further know from Corollary 3.3 that $w^{*}\in H^{2}(\Omega)$ is a strong
solution of the eigenvalue problem (3.1). Hence, we have

$\displaystyle 0=\Delta
w^{*}=\big{(}b^{\varepsilon}(\varphi^{*})-\lambda\big{)}w^{*}\quad\text{a.e.\leavevmode\nobreak\
in}\;\big{\\{}x\in\Omega\,\big{|}\,s<\left|x\right|<t\big{\\}}.$

As $w^{*}>0$ in $\Omega$ and since $b^{\varepsilon}(\varphi^{*})-\lambda$ is
non-decreasing and defined everywhere in $\Omega$, we infer that

$\displaystyle b^{\varepsilon}(\varphi^{*})-\lambda$
$\displaystyle=0\quad\text{in}\;\big{\\{}x\in\Omega\,\big{|}\,s<\left|x\right|<t\big{\\}},$
$\displaystyle b^{\varepsilon}(\varphi^{*})-\lambda$ $\displaystyle\geq
0\quad\text{in}\;\;A\coloneqq\big{\\{}x\in\Omega\,\big{|}\,s<\left|x\right|<R\big{\\}},$

which in turn implies

$\displaystyle\Delta
w^{*}=\big{(}b^{\varepsilon}(\varphi^{*})-\lambda\big{)}w^{*}\geq
0\quad\text{a.e.\leavevmode\nobreak\ in}\;A.$ (4.16)

Due to (4.15), we have

$\displaystyle\underset{A}{\sup}\,w^{*}=c,$

since $w^{*}$ is symmetric-decreasing. Applying the strong maximum principle
for the Laplace operator (see, e.g., [22, Theorem 8.19] with $L=\Delta$), we
infer that

$\displaystyle w^{*}=c\quad\text{in}\;A.$

However, since $c>0$, this is a contradiction to the zero-trace condition
hidden in $w^{*}\in H^{1}_{0}(\Omega)$. We have thus proven that $w^{*}$ is
strictly symmetric-decreasing.

As a consequence, we have $\nabla w^{*}\neq 0$ almost everywhere in $\Omega$,
meaning that

$\displaystyle\mathcal{L}^{n}\big{(}\big{\\{}x\in\Omega\,\big{|}\,\nabla
w^{*}=0\big{\\}}\big{)}=0.$

Recalling that $w^{*}>0$ in $\Omega$, we use Lemma 2.4(g) along with (4.14) to
conclude

$\displaystyle w=w^{*}\quad\text{a.e. in }\Omega,$

meaning that $w$ is symmetric-decreasing. Plugging this into (4.12) we arrive
at

$\displaystyle\int_{\Omega}(b^{\varepsilon}(\varphi^{*})-b^{\varepsilon}(\varphi))\big{(}w^{*}\big{)}^{2}\;\mathrm{d}\mathcal{L}^{n}=0.$

Since $w^{*}>0$ in $\Omega$, we infer

$\displaystyle
b^{\varepsilon}(\varphi^{*})=b^{\varepsilon}(\varphi)\quad\text{a.e. in
}\Omega.$

This directly implies $\varphi=\varphi^{*}$ a.e. in $\Omega$ as
$b^{\varepsilon}$ is strictly decreasing (and thus injective). This proves
that $\varphi$ is symmetric-decreasing and hence, the proof is complete. ∎

  * Proof of Theorem 3.17.

The $\liminf$ inequality of the Ginzburg–Landau part directly carries over
from [29, Lemma 1] as in our case, we only need to consider trivial extensions
of $\varphi_{\varepsilon},\varphi$ instead of the more complicated boundary
value function $h^{\varepsilon}$ discussed there. We point out that (A1) is
sufficient for the proof to work, as only the continuity in $[0,1]$ and the
non-negativity of the potential $\psi$ is needed to ensure that the function
$\Psi$ defined in (3.13) is well-defined and differentiable. Note that we
actually need to include the factor $\sqrt{2}$ in the definition of $\Psi$ in
order for the Modica–Mortola trick

$\displaystyle\int_{\Omega}\big{|}\nabla\Psi(\varphi_{\varepsilon})\big{|}\;\mathrm{d}\mathcal{L}^{n}=\int_{\Omega}\sqrt{2\psi(\varphi_{\varepsilon})}\big{|}\nabla\varphi_{\varepsilon}\big{|}\;\mathrm{d}\mathcal{L}^{n}\leq\int_{\Omega}\frac{\varepsilon}{2}\big{|}\nabla\varphi_{\varepsilon}\big{|}^{2}+\frac{1}{\varepsilon}\psi(\varphi_{\varepsilon})\;\mathrm{d}\mathcal{L}^{n}=E^{\varepsilon}(\varphi_{\varepsilon})$

in the proof of the $\liminf$ inequality to work, see e.g., also [5, Formula
(3.61)]. In [29], the factor $2$ is used which is due to the fact that there
the gradient term in the energy is not scaled with $\frac{1}{2}$.

To verify the $\liminf$ inequality for the eigenvalue term, we proceed as in
[21, Theorem 4.12]. However, we need to be careful with the constraints for
the limit cost functional. In [21], the additional constraint
$\varphi\in\mathcal{U}$ was imposed which fixes a non-trivial open set $S_{1}$
such that $S_{1}\subset\left\\{\varphi=1\right\\}$. This guarantees that all
the eigenvalues are finite. In our framework, we now additionally need to
consider the case $\lambda_{1}^{\varphi}=+\infty$. Therefore we consider
$\varphi_{\varepsilon}\to\varphi$ in $L^{1}(\Omega)$ such that

$\displaystyle\underset{\varepsilon\searrow
0}{\liminf\,}F_{\varepsilon}^{\gamma}(\varphi_{\varepsilon})<+\infty.$

Applying Fatou’s Lemma to the potential term as in [27, Proposition 1] (which
only requires the continuity of $\psi$ demanded in (A1)), we obtain that
$\varphi\in BV(\Omega;\left\\{0,1\right\\})$ and, up to subsequence
extraction, that the sequence of eigenvalues
$(\lambda^{\varepsilon,\varphi_{\varepsilon}})_{\varepsilon>0}$ is bounded.
Hence, as in the proof of [21, Theorem 4.10], the sequence of minimizers
$(v_{\varepsilon})_{\varepsilon>0}$ of the problem

$\displaystyle\min\left\\{\left.\int_{\Omega}\left|\nabla
v\right|^{2}\;\mathrm{d}\mathcal{L}^{n}+\int_{\Omega}b_{\varepsilon}(\varphi_{\varepsilon})\left|v\right|^{2}\;\mathrm{d}\mathcal{L}^{n}\right|\begin{aligned}
&v\in H^{1}_{0}(\Omega),\\\
&\left\|v\right\|_{L^{2}(\Omega)}=1\end{aligned}\right\\}$

fulfills

$\displaystyle\begin{aligned}
v_{\varepsilon}&\rightharpoonup\overline{v}\quad\text{in
}H^{1}_{0}(\Omega),\qquad v_{\varepsilon}\to\overline{v}\quad\text{in
}L^{2}(\Omega),\qquad v_{\varepsilon}\to\overline{v}\quad\text{a.e. in
}\Omega,\end{aligned}$

after another subsequence extraction. Now due to the boundedness of the
sequence
$\int_{\Omega}b_{\varepsilon}(\varphi_{\varepsilon})\left|v_{\varepsilon}\right|^{2}\;\mathrm{d}\mathcal{L}^{n}$
we proceed as in [21] and use Fatou’s lemma to infer $\overline{v}\in
V^{\varphi}$. Consequently, $V^{\varphi}$ is non-trivial and
$\lambda_{1}^{0,\varphi}<\infty$. This means that the case
$\lambda_{1}^{0,\varphi}=+\infty$ can only occur if also

$\displaystyle\underset{\varepsilon\searrow
0}{\liminf\,}F_{\varepsilon}^{\gamma}(\varphi_{\varepsilon})=+\infty.$

Therefore, the $\liminf$ inequality is established as in [21, Theorem 4.10].

It remains to prove the $\limsup$ inequality. First of all, as already
mentioned in Remark 2.1, we want to show that the proof given in [33, Theorem
1] for the smooth double well potential carries over to general potentials
satisfying assumptions (A1) and (A2). The key step is to consider the ordinary
differential equation

$\displaystyle\left\\{\begin{aligned}
\eta^{\prime}(t)&=\sqrt{2\psi(\eta(t))},\\\
\eta(0)&=\tfrac{1}{2}.\end{aligned}\right.$ (4.17)

As the right hand side is locally Lipschitz away from $\eta(t)=0$ and
$\eta(t)=1$, the Picard–Lindelöf theorem provides the existence of a unique
maximal solution $\eta$. Moreover, since the right-hand side is non-negative,
we know that $\eta$ is non-decreasing. Now, depending on the choice of the
potential $\psi$, the values $0$ and/or $1$ can either be reached for finite
$t$ or the solution tends to those values asymptotically, meaning that

$\displaystyle\underset{t\to\infty}{\lim}\eta(t)=1\quad\text{and/or}\quad\underset{t\to-\infty}{\lim}\eta(t)=0.$
(4.18)

If the solution $\eta$ satisfies $\eta(t_{0})=0$ for some $t_{0}<0$ (or
$\eta(t_{1})=1$ for some $t_{1}>0$), it holds that $\eta(t)=0$ for all $t\leq
t_{0}$ (or $\eta(t)=1$ for all $t\geq t_{1}$). In particular, in any case, the
solution exists for all $t\in\mathbb{R}$. These properties follow from
classical ODE theory, exploiting that, due to (A1), the right-hand side of
(4.17) is strictly positive whenever $\eta(t)\in(0,1)$.

As in [33, (1.22)], we construct the profile function
$\rho_{\varepsilon}^{P}:\mathbb{R}\to[0,1]$ by defining

$\displaystyle\rho_{\varepsilon}^{P}(t)\coloneqq\begin{cases}1&\quad\text{for
}t>2\sqrt{\varepsilon},\\\
1+\left(1-\eta\left(\varepsilon^{-\frac{1}{2}}\right)\right)\left(\frac{t-2\sqrt{\varepsilon}}{\sqrt{\varepsilon}}\right),&\quad\text{for
}\sqrt{\varepsilon}\leq t\leq 2\sqrt{\varepsilon},\\\
\eta\left(\frac{t}{\varepsilon}\right),&\quad\text{for
}\left|t\right|\leq\sqrt{\varepsilon},\\\
\eta\left(-\varepsilon^{-\frac{1}{2}}\right)\left(\frac{t+2\sqrt{\varepsilon}}{\sqrt{\varepsilon}}\right),&\quad\text{for
}-2\sqrt{\varepsilon}\leq t\leq-\sqrt{\varepsilon},\\\ 0,&\quad\text{for
}t<-2\sqrt{\varepsilon}.\end{cases}$ (4.19)

The idea behind these profiles is to use the solution of (4.17) and possibly
linearly interpolate the values where $\eta$ is close to $0$ or $1$. This
interpolation is necessary to obtain a transition from $0$ to $1$ on a finite
interval scaling suitably with $\varepsilon$, even though the solution of
(4.17) does possibly not reach these values in finite time.

In case $\eta$ actually reaches the values $0$ and/or $1$, the interpolations
in the second and fourth line of this definition become trivial and therefore
negligible, provided that $\varepsilon>0$ is sufficiently small.

In case the values $0$ and/or $1$ are only reached asymptotically, we still
need the following exponential convergence rates in order for the proof in the
spirit of [33] to work out:

    * –

If the value $1$ is only reached asymptotically, there exists $T_{1}>0$ as
well as constants $C_{1},a_{1}>0$ such that

$\displaystyle\left|1-\eta(t)\right|\leq C_{1}\exp(-a_{1}t)\quad\text{for all
$t\geq T_{1}$.}$ (4.20)

    * –

If the value $0$ is only reached asymptotically, there exists $T_{0}>0$ as
well as constants $C_{0},a_{0}>0$ such that

$\displaystyle\left|\eta(t)\right|\leq C_{0}\exp(-a_{0}t)\quad\text{for all
$t\leq-T_{0}$.}$ (4.21)

We will only prove estimate (4.20) since estimate (4.21) can be established
completely analogously. Therefore, we assume that $\eta$ reaches the value $1$
only asymptotically. Due to the monotonicity of $\eta$, we have
$\eta(t)\in\big{[}\tfrac{1}{2},1\big{)}$ and thus
$\psi\big{(}\eta(t)\big{)}>0$ for all $t\in[0,\infty)$. Hence, $\eta$ is twice
continuously differentiable with

$\displaystyle\eta^{\prime\prime}(t)=\frac{1}{2\sqrt{2\psi(\eta(t))}}2\psi^{\prime}(\eta(t))\eta^{\prime}(t)=\psi^{\prime}(\eta(t))\quad\text{for
all }t\in[0,\infty).$

Combining (4.18) and (4.17), we further deduce

$\displaystyle\underset{t\to\infty}{\lim}\eta^{\prime}(t)=0.$

Hence, for any $\xi\in[0,\infty)$, we have

$\displaystyle\int_{\xi}^{\infty}\psi^{\prime}(\eta(t))\text{\,d}t=-\eta^{\prime}(\xi).$
(4.22)

Let us now assume that $\psi^{\prime}(1)\neq 0$. Since
$\psi\colon[0,1]\to\mathbb{R}$ possesses a local minimum at $1$, this already
entails $\psi^{\prime}(1)<0$. Hence, due to the continuity of $\psi^{\prime}$,
we have

$\displaystyle\psi^{\prime}(s)<\frac{1}{2}\psi^{\prime}(1)<0$

for all $s\in(0,1)$ in a suitably small neighborhood around $1$. However, this
is an obvious contradiction to the finiteness of the integral in (4.22). We
thus conclude

$\displaystyle\psi^{\prime}(1)=0.$ (4.23)

This equality will now be the crucial ingredient in applying the comparison
principle for ODEs. Recalling that $\psi\in C^{2}([0,1])$, for any
$s\in(0,1]$, we consider the Taylor expansion

$\displaystyle\psi(s)=\psi(1)+\psi^{\prime}(1)(s-1)+\frac{1}{2}\psi^{\prime\prime}(\xi_{s})(s-1)^{2}=\frac{1}{2}\psi^{\prime\prime}(\xi_{s})(s-1)^{2},$
(4.24)

for a $\xi_{s}$ between $1$ and $s$. In the light of (4.23) and (A2), there
exist $c,\delta>0$ such that

$\displaystyle\left|\psi^{\prime\prime}(x)\right|\geq c>0,\quad\text{for all
}x\in B_{\delta}(1)\cap[0,1].$

Hence, defining $a_{1}\coloneqq\sqrt{c}$, we obtain

$\displaystyle\sqrt{2\psi(s)}\geq a_{1}(1-s)$ (4.25)

for all for any $s\in B_{\delta}(1)\cap[0,1]$. Due to (4.18), there exists
$t_{1}\in\mathbb{R}$ such that $s_{1}\coloneqq\eta(t_{1})\in
B_{\delta}(1)\cap(0,1)$. We now consider the initial value problem

$\displaystyle\begin{aligned} \mu^{\prime}(t)&=a_{1}(1-\mu(t)),\\\
\mu(t_{1})&=s_{1},\end{aligned}$ (4.26)

which possesses the unique global solution

$\displaystyle\mu:\mathbb{R}\to\mathbb{R},\quad\mu(t)=(s_{1}-1)\exp(-a_{1}(t-t_{1}))+1.$

Due to (4.25) and the fact that $\eta$ solves (4.26) with the right-hand side
being replaced by $\sqrt{2\psi(\eta(t))}$, the comparison principle for ODEs
implies

$\displaystyle\eta(t)\geq\mu(t)\quad\text{for all }t\geq t_{1}.$

Hence, we conclude

$\displaystyle\left|\eta(t)-1\right|=1-\eta(t)\leq(1-s_{1})\exp(-a_{1}(t-t_{1}))\quad\text{for
all }t\geq t_{1},$ (4.27)

which proves (4.20) with $\bar{C}_{1}:=(1-s_{1})\exp(a_{1}t_{1})>0$ and
$T_{1}:=t_{1}$.

Now, the estimates (4.20) and (4.21) allow us to continue as in [33].
Proceeding as in the proof of [33, Theorem 1], we conclude that for any
smooth, bounded, open set $E\subset\mathbb{R}^{n}$ having finite perimeter and
satisfying the transversality condition

$\displaystyle\mathcal{H}^{n-1}(\partial E\cap\partial\Omega)=0,$

there exists a recovery sequence
$(\varphi_{\varepsilon})_{\varepsilon>0}\subset H^{1}(\Omega;[0,1])$ which
satisfies

$\displaystyle\left\\{\begin{aligned} \underset{\varepsilon\searrow
0}{\limsup\,}\int_{\Omega}\frac{\varepsilon}{2}\left|\nabla{\varphi}_{\varepsilon}\right|^{2}+\frac{1}{\varepsilon}\psi({\varphi}_{\varepsilon})\;\mathrm{d}\mathcal{L}^{n}&\leq
c_{0}\text{Per}_{\Omega}(E),\\\
\left\|\varphi_{\varepsilon}-\chi_{E}\right\|_{L^{1}(\Omega)}&=\mathcal{O}(\varepsilon).\end{aligned}\right.$
(4.28)

We point out that the above $L^{1}$-convergence rate can be obtained by
following the line of argument in Step 1 of [33, Theorem 1]. The key fact is
that, due to (4.20) and (4.21), the profile $\rho_{\varepsilon}^{P}$ converges
in the interpolation parts exponentially to $0$ and $1$, respectively. In the
middle part, $\rho_{\varepsilon}^{P}$ is scaled with $\varepsilon$ such that,
using the coarea formula and a change of variables, we obtain the desired
rate.

Obviously, this recovery sequence is not yet admissible as (in general) it
does not fulfill the homogeneous Dirichlet boundary condition hidden in
$H_{0}^{1}(\Omega)$. Following the idea in [8, Theorem 3.1], we make the
following observation: If the finite perimeter set $E$ is compactly contained
in $\Omega$, then $\left(\varphi_{\varepsilon}\right)_{\varepsilon>0}\subset
H_{0}^{1}(\Omega)$ is guaranteed provided that $\varepsilon>0$ is sufficiently
small. This directly follows from the construction of $\varphi_{\varepsilon}$
via the optimal profile $\rho_{\varepsilon}^{P}$ which vanishes in all points
$t<-2\sqrt{\varepsilon}$. This means that outside of a small tubular
neighborhood around the boundary of $E$ we indeed have
$\varphi_{\varepsilon}=0$. Therefore, we now approximate any finite perimeter
set $E\subset\Omega$ by smooth, open finite perimeter sets which are compactly
contained in $\Omega$. Although the line of argument is outlined in the proof
of [8, Theorem 3.1], we highlight the key steps in order to present a
comprehensive proof.

Let now $\varphi\in BV(\Omega;\left\\{0,1\right\\})$ be arbitrary and
$E:=\\{\varphi=1\\}$. In the following, we use the notation

$\displaystyle
E^{\Omega}:=E\cap\Omega,\quad\lambda_{1}^{0}(E)\coloneqq\lambda_{1}^{0,\chi_{E^{\Omega}}}\quad\text{and}\quad
F_{0}^{\gamma}(E)\coloneqq F_{0}^{\gamma}(\chi_{E^{\Omega}}).$

In order to construct a recovery sequence in $H_{0}^{1}(\Omega;[0,1])$, we
first approximate the finite perimeter set $E$ by a sequence
$(E_{k})_{k\in\mathbb{N}}$ of bounded, smooth, open sets
$E_{k}\subset\mathbb{R}^{n}$ fulfilling

$\displaystyle\left\\{\begin{aligned} \mathcal{H}^{n-1}(\partial
E_{k}\cap\partial\Omega)&=0,\\\
\text{Per}_{\Omega}(E_{k})&\to\text{Per}_{\Omega}(E)\quad\text{for
}k\to\infty,\\\ \chi_{E_{k}^{\Omega}}&\to\chi_{E}\quad\text{in
}L^{1}(\Omega)\text{ for }k\to\infty,\\\
\underset{k\to\infty}{\limsup}\,\lambda_{1}^{0}(E_{k})&\leq\lambda_{1}^{0}(E).\end{aligned}\right.$
(4.29)

Such a sequence is constructed in the proof of [21, Theorem 4.12] relying on
ideas of [7, 32, 27]. Note that the second and the third property of (4.29)
mean that

$\displaystyle\chi_{E_{k}^{\Omega}}\to\chi_{E}\text{ {strictly} in
}BV(\Omega)$ (4.30)

as $k\to\infty$, see Subsection 2.4. Due to the continuity of the trace
operator with respect to strict $BV$-convergence, we further know

$\displaystyle\big{\|}\chi_{E_{k}^{\Omega}}\big{\|}_{L^{1}(\partial\Omega)}\to\left\|\chi_{E}\right\|_{L^{1}(\partial\Omega)}=\int_{\partial\Omega}\varphi_{|\partial\Omega}\;\mathrm{d}\mathcal{H}^{n-1},$

for $k\to\infty$.

Now, the crucial idea of [8, Theorem 3.1] is to perform a further
approximation by cutting off $E_{k}$ in a tubular neighborhood around the
boundary of $\Omega$ such that the truncated set is compactly contained in
$\Omega$. As we fix $k\in\mathbb{N}$ in the following, we omit this index for
a cleaner presentation.
For any $\delta>0$, the truncated set is defined as $E^{\delta}\coloneqq E\cap
B_{\delta}$ with

$B_{\delta}\coloneqq\left\\{x\in\Omega\;|\;\mathrm{dist}(x,\partial\Omega)>\delta\right\\}.$

Obviously, $E^{\delta}$ is compactly contained in $\Omega$ and it also is a
set of finite perimeter. We now intend to show that

$\displaystyle\left\\{\begin{aligned} &\;\underset{\delta\to
0}{\limsup}\;F_{0}^{\gamma}(E^{\delta})\leq F_{0}^{\gamma}(E),\\\
&\;\underset{\delta\to
0}{\lim}\;\big{\|}\chi_{E^{\delta}}-\chi_{E}\big{\|}_{L^{1}(\Omega)}=0.\end{aligned}\right.$
(4.31)

Then, applying a diagonal sequence argument will yield the desired $\limsup$
inequality, see below.

The $L^{1}$ convergence in (4.31) is clear by construction. To establish the
first line of (4.31), we consider the eigenvalue term and the perimeter term
separately. For the eigenvalue term, it suffices to prove the Mosco condition
(M1), see [23, Theorem 2.3.10 and Remark 2.3.12]. This means, we need to show
that for any $v\in H_{0}^{1}(E)$, there exists a sequence
$(v_{\delta})_{\delta>0}$ with $v_{\delta}\in H_{0}^{1}(E^{\delta})$ such that

$\displaystyle\underset{\delta\to
0}{\lim}\left\|v_{\delta}-v\right\|_{H^{1}(\Omega)}=0.$

Since $C_{0}^{\infty}(E)$ is dense in $H_{0}^{1}(E)$, there exists a sequence
$(v_{\delta})_{\delta>0}$ in $C_{0}^{\infty}(E)$ with $v_{\delta}\to v$
strongly in $H_{0}^{1}(E)$ as $\delta\to 0$. As the support of $v_{\delta}$ is
compactly contained in $E$, we can assume without loss of generality (after
possibly relabeling the index $\delta$) that the support of $v_{\delta}$ is
also compactly contained in $E_{\delta}$. Hence, we have $v_{\delta}\in
H_{0}^{1}(E^{\delta})$ and thus, the Mosco condition (M1) in [23] is verified.
In view of [23, Remark 2.3.12], we thus infer

$\displaystyle\underset{\delta\to
0}{\limsup}\;\lambda_{1}^{0}(E^{\delta})\leq\lambda_{1}^{0}(E).$ (4.32)

For the perimeter term, we obtain

$\displaystyle P_{\Omega}(E^{\delta})=P_{\Omega}(E\cap
B_{\delta})\leq\mathcal{H}^{n-1}(\partial(E\cap B_{\delta}))$ (4.33)

due to [4, Proposition 3.62]. Applying [4, Proposition 2.95], we deduce that

$\displaystyle\mathcal{H}^{n-1}(\partial E\cap\partial
B_{\delta})=\mathcal{H}^{n-1}\big{(}\partial
E\cap\big{\\{}\mathrm{dist}(\cdot,\partial\Omega)=\delta\big{\\}}\big{)}=0.$

for almost every $\delta>0$. Using the simple fact

$\displaystyle\partial(E\cap B_{\delta})\subset(\partial
E\cap\overline{B_{\delta}})\cup(\partial B_{\delta}\cap\overline{E}),$

we arrive at

$\displaystyle\mathcal{H}^{n-1}(\partial(E\cap B_{\delta}))$
$\displaystyle\leq\mathcal{H}^{n-1}(\partial E\cap
B_{\delta})+\mathcal{H}^{n-1}(E\cap\partial B_{\delta})$
$\displaystyle=P_{B_{\delta}}(E)+\mathcal{H}^{n-1}\big{(}E\cap\big{\\{}\mathrm{dist}(\cdot,\partial\Omega)=\delta\big{\\}}\big{)}.$
(4.34)

Here, for the equality in the second line, the smoothness of $E$ is crucial.
The first summand in the second line of (Proof of Theorem 3.17.) side can be
expressed as $P_{B_{\delta}}(E)=\left|D\chi_{E}\right|(B_{\delta}),$ where
$\left|D\chi_{E}\right|$ denotes the total variation of the Radon-measure
$D\chi_{E}$ associated with $\chi_{E}\in BV(\Omega)$ (see, e.g., [26, Chapter
12]). From to the $\sigma$-additivity of $\left|D\chi_{E}\right|$, we directly
infer

$\displaystyle\underset{\delta\to
0}{\lim}\;P_{B_{\delta}}(E)=\underset{\delta\to
0}{\lim}\;\left|D\chi_{E}\right|(B_{\delta})=\left|D\chi_{E}\right|(\Omega)=P_{\Omega}(E).$
(4.35)

For the second summand in the second line of (Proof of Theorem 3.17.), we use
the transversality condition $\mathcal{H}^{n-1}(\partial
E\cap\partial\Omega)=0$ in order to apply [32, Lemma 13.9]. This yields

$\displaystyle\underset{\delta\to
0}{\lim}\;\mathcal{H}^{n-1}\big{(}E\cap\big{\\{}\mathrm{dist}(\cdot,\partial\Omega)=\delta\big{\\}}\big{)}=\mathcal{H}^{n-1}(E\cap\partial\Omega)=\int_{\partial\Omega}\varphi_{|\partial\Omega}\;\mathrm{d}\mathcal{H}^{n-1}.$
(4.36)

Noticing that the trace of $\varphi_{\delta}$ vanishes on $\partial\Omega$ in
the sense of Section 2.4, we combine (4.33)–(4.36) to obtain

$\displaystyle\underset{\delta\to 0}{\lim\sup}\;P_{\Omega}(E^{\delta})\leq
P_{\Omega}(E).$ (4.37)

Combining (4.32) and (4.37), we eventually conclude (4.31).

Now, we close the proof by means of a final diagonal sequence argument.
Therefore, we reinstate the index $k$. Note that, without loss of generality,
we can assume $E_{k}^{\delta}$, which is compactly contained in $\Omega$, is
smooth by performing again the approximation of (4.29). We point out that by
performing this approximation of $E_{k}^{\delta}$, the approximation sets are
still compactly contained in $\Omega$. This is because the corresponding proof
in [21, Theorem 4.10] is based on classical convolution with mollifiers and
thus, the set $E_{k}^{\delta}$ is only modified up to a small tubular
neighborhood.

As we now take for granted that the sets $E_{k}^{\delta}$ are smooth, there
exists a recovery sequence $\varphi_{\varepsilon}^{k,\delta}\subset
H_{0}^{1}(\Omega;[0,1])$ fulfilling (4.28) with $E$ replaced by
$E_{k}^{\delta}$. Using the convergence rate and the upper semicontinuity of
eigenvalues provided by Corollary 3.13, we further know for any for fixed
$k\in\mathbb{N}$ and $\delta>0$,

$\displaystyle\underset{\varepsilon\searrow
0}{\limsup\,}\lambda_{1}^{\varepsilon,\varphi_{\varepsilon}^{k,\delta}}\leq\lambda_{1}(E_{k}^{\delta})$

and consequently,

$\displaystyle\underset{\varepsilon\searrow
0}{\limsup\,}F_{\varepsilon}^{\gamma}(\varphi_{\varepsilon}^{k,\delta})\leq
F_{0}^{\gamma}(E_{k}^{\delta}).$

Now, according to (4.29) and (4.31), for every $k\in\mathbb{N}$ we can find a
sufficiently small $\delta_{k}>0$ such that

$\displaystyle\underset{k\to\infty}{\limsup}F_{0}^{\gamma}(E_{k}^{\delta_{k}})$
$\displaystyle\leq F_{0}^{\gamma}(E)\quad\text{and}$
$\displaystyle\underset{k\to\infty}{\lim}\big{\|}\chi_{E_{k}^{\delta_{k}}}-\chi_{E}\big{\|}_{L^{1}(\Omega)}$
$\displaystyle=0.$

This in turn allows us now to choose also $\varepsilon_{k}>0$ small enough
such that finally

$\displaystyle\underset{k\to\infty}{\limsup}F_{\varepsilon_{k}}(\varphi_{\varepsilon_{k}}^{k,\delta_{k}})$
$\displaystyle\leq F_{0}^{\gamma}(E)\quad\text{and}$
$\displaystyle\underset{k\to\infty}{\lim}\big{\|}\varphi_{\varepsilon_{k}}^{k,\delta_{k}}-\chi_{E}\big{\|}_{L^{1}(\Omega)}$
$\displaystyle=0.$

Thus, the proof is complete. ∎

  * Proof of Theorem 3.18.

We have $\Phi_{m}\subset H_{0}^{1}(\Omega;[0,1])$ and $\Phi_{m}^{0}\subset
BV(\Omega;{0,1})$ and we know that the volume constraint is preserved under
$L^{1}$ convergence. Hence, the $\liminf$ inequality is a direct consequence
of Theorem 3.17 as now there are less admissible sequences.

It remains to prove the $\limsup$ inequality, namely that for every
$\varphi\in\Phi_{m}^{0}$ there exists a sequence
$(\tilde{\varphi}_{\varepsilon})_{\varepsilon>0}\subset\Phi_{m}$ fulfilling

$\displaystyle\underset{\varepsilon\searrow
0}{\lim\,}\left\|\tilde{\varphi}_{\varepsilon}-\varphi\right\|_{L^{1}(\Omega)}$
$\displaystyle=0,$ (4.38) $\displaystyle\underset{\varepsilon\searrow
0}{\limsup\,}J^{\varepsilon}_{\gamma}(\tilde{\varphi}_{\varepsilon})$
$\displaystyle\leq J^{0}_{\gamma}(\varphi).$ (4.39)

For any $\varphi\in\Phi_{m}^{0}$, a recovery sequence
$(\varphi_{\varepsilon})_{\varepsilon>0}\subset H^{1}_{0}(\Omega;[0,1])$ for
the functional $F_{\varepsilon}^{\gamma}$ was constructed in Theorem 3.17. It
was shown that this recovery sequence converges in $L^{1}(\Omega)$ and
fulfills the $\limsup$ inequality for $F_{\varepsilon}^{\gamma}$. Now, our
goal is to carefully modify this recovery sequence such that it preserves
these properties but additionally fulfills the mean value constraint. For this
modification, we proceed completely analogously as in the proof of [21,
Theorem 4.8]. Therefore, we only briefly sketch the most important steps.
Since $\varphi\in\Phi_{m}^{0}$, it is non-constant. Hence, we can find a
function $\boldsymbol{\xi}\in C_{0}^{1}(\Omega,\mathbb{R}^{n})$ such that

$\displaystyle\int_{\Omega}\varphi\nabla\cdot\boldsymbol{\xi}\;\mathrm{d}\mathcal{L}^{n}>0.$

For any $s\in\mathbb{R}$, we define the function

$\displaystyle T_{s}:\mathbb{R}^{n}$ $\displaystyle\to\mathbb{R}^{n},$
$\displaystyle x$ $\displaystyle\mapsto x+s\boldsymbol{\xi}(x),$

which is a $C^{1}$-diffeomorphism if $s$ is sufficiently small. By means of
the implicit function theorem, we deduce that for any sufficiently small
$\varepsilon>0$, there exists $s(\varepsilon)\in\mathbb{R}$ such that

$\displaystyle\tilde{\varphi}_{\varepsilon}\coloneqq\varphi_{\varepsilon}\circ
T_{s(\varepsilon)}^{-1}\in\Phi_{m},$

and $s(\varepsilon)\to 0$ as $\varepsilon\to 0$. In particular, the property
$\tilde{\varphi}_{\varepsilon}\in H_{0}^{1}(\Omega)$ holds since for $x$ close
to $\partial\Omega$, we have $T_{s(\varepsilon)}(x)=x$ due to the fact that
$\boldsymbol{\xi}$ has compact support in $\Omega$. Eventually, we show that
the sequence $(\tilde{\varphi}_{\varepsilon})_{\varepsilon>0}$ satisfies the
properties (4.38) and (4.39) and thus, it is a suitable recovery sequence. ∎

  * Proof of Theorem 3.8.

In the proof of Theorem 3.18, for any admissible $\varphi\in\Phi_{m}^{0}$, we
have constructed a recovery sequence
$(\tilde{\varphi}_{\varepsilon})_{\varepsilon>0}$. In particular, this implies
that the cost functional $J_{\gamma}^{\varepsilon}$ is bounded uniformly in
$\varepsilon$ along any sequence $(\varphi_{\varepsilon})_{\varepsilon>0}$ of
minimizers to the optimization problem ($\text{OP}^{\varepsilon}_{\gamma}$).
Consequently, there exists a constant $C>0$ independent of $\varepsilon$ and
$\gamma$ such that

$\displaystyle\int_{\Omega}\psi(\varphi_{\varepsilon})\;\mathrm{d}\mathcal{L}^{n}\leq\varepsilon
E^{\varepsilon}(\varphi_{\varepsilon})\leq\frac{\varepsilon}{\gamma}J_{\gamma}^{\varepsilon}(\varphi_{\varepsilon})\leq\frac{C\varepsilon}{\gamma},$
(4.40)

for all $\varepsilon>0$. Recalling that $\psi\in C^{2}([0,1])$ is non-
negative, we thus have

$\displaystyle\Big{(}\,\underset{[\delta,1-\delta]}{\min}\;\psi\,\Big{)}\;\mathcal{L}^{n}\left(\left\\{\delta\leq\varphi_{\varepsilon}\leq
1-\delta\right\\}\right)\leq\int_{\left\\{\delta\leq\varphi_{\varepsilon}\leq
1-\delta\right\\}}\psi(\varphi_{\varepsilon})\;\mathrm{d}\mathcal{L}^{n}$
$\displaystyle\quad\leq\int_{\Omega}\psi(\varphi_{\varepsilon})\;\mathrm{d}\mathcal{L}^{n}\leq\frac{C\varepsilon}{\gamma},.$
(4.41)

Since $\psi>0$ on $[\delta,1-\delta]$ due to (A1), the assertion directly
follows. ∎

  * Proof of Theorem 3.15.

We apply the compactness of the Ginzburg Landau energy of [27, Proposition
3(a)] which only relies on the fact that

$\displaystyle\Psi:[0,1]\to[0,c_{0}],\quad
s\mapsto\int_{0}^{s}\sqrt{2\psi(t)}\text{\,d}t,$

is invertible. This is true since due to (A1), we have $\psi>0$ in $(0,1)$.
Consequently, there exists a function $\varphi_{0}\in
BV(\Omega,\left\\{0,1\right\\})$ such that

$\displaystyle\underset{\varepsilon\searrow
0}{\lim\,}\left\|\varphi_{\varepsilon}-\varphi_{0}\right\|_{L^{1}(\Omega)}=0$
(4.42)

along a non-relabeled subsequence of
$(\varphi_{\varepsilon})_{\varepsilon>0}$. We further recall that
$\varphi_{\varepsilon}=\varphi_{\varepsilon}^{\ast}$ according to Theorem 3.7.
Hence, the non-expansivity of the rearrangement (see Lemma 2.4(f)) yields

$\displaystyle\int_{\Omega}\left|\varphi_{0}-\varphi_{0}^{\ast}\right|\;\mathrm{d}\mathcal{L}^{n}$
$\displaystyle\leq\int_{\Omega}\left|\varphi_{0}-\varphi_{\varepsilon}\right|\;\mathrm{d}\mathcal{L}^{n}+\int_{\Omega}\left|\varphi_{\varepsilon}^{\ast}-\varphi_{0}^{\ast}\right|\;\mathrm{d}\mathcal{L}^{n}$
$\displaystyle\leq
2\int_{\Omega}\left|\varphi_{\varepsilon}-\varphi_{0}\right|\;\mathrm{d}\mathcal{L}^{n}.$

Hence, we infer $\varphi_{0}=\varphi_{0}^{\ast}$ almost everywhere in
$\Omega$. As the mean value is preserved under $L^{1}$ convergence this is
already enough to deduce that $\varphi_{0}$ is the characteristic function of
the ball centered at the origin with volume $m\left|\Omega\right|$. Obviously
the limit $\varphi_{0}$ does not depend on the choice of the subsequence of
$(\varphi_{\varepsilon})_{\varepsilon>0}$. Hence, the convergence (4.42) even
holds for the whole sequence.

Eventually, using Theorem 3.18 which states that
$J^{\varepsilon}\overset{\Gamma}{\to}J^{0}$, we conclude that $\varphi_{0}$ is
a minimizer of $J^{0}_{\gamma}$ as $\Gamma$-convergence implies the
convergence of minimizers. ∎

## Acknowledgments

Paul Hüttl and Patrik Knopf gratefully acknowledge the support by the
Graduiertenkolleg 2339 IntComSin of the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – Project-ID 321821685. Tim Laux has received
funding from the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813.

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