# Existence of an optimal shape for the first eigenvalue of polyharmonic
operators

Roméo Leylekian 111Aix-Marseille Université, CNRS, I2M, Marseille, France -
email<EMAIL_ADDRESS>
###### Abstract

We prove the existence of an open set minimizing the first eigenvalue of the
Dirichlet polylaplacian of order $m\geq 1$ under volume constraint. Moreover,
the corresponding eigenfunction is shown to enjoy $C^{m-1,\alpha}$ Hölder
regularity. This is performed for dimension $2\leq d\leq 4m$. In particular,
our analysis answers the question of the existence of an optimal shape for the
clamped plate up to dimension $8$.

##  1 Introduction

The purpose of this document is to prove existence and partial regularity of
an optimal shape for the first eigenvalue of Dirichlet polylaplacians with
volume constraint. More precisely, for some $m\in\mathbb{N}^{*}$ (fixed from
now on), we consider the following eigenvalue problem.

$\left\\{\begin{array}[]{rcll}(-\Delta)^{m}u&=&\Gamma u&\text{in }\Omega,\\\
\partial_{n}^{i}u&=&0&\text{on }\partial\Omega,\text{ for all }0\leq
i<m,\end{array}\right.$ (1)

where $\Omega\subseteq\mathbb{R}^{d}$ is an open set of bounded volume
($d\in\mathbb{N}^{*}$ is fixed as well), $\partial_{n}=\nabla\cdot\vec{n}$ is
the outward normal derivative, $\Gamma$ is a real number, and
$u:\Omega\to\mathbb{R}$ is nontrivial and belongs to the Sobolev space
$H_{0}^{m}(\Omega)$. It is known that there exists an unbounded sequence of
positive such eigenvalues $\Gamma$. Here, we only focus on the lowest one,
that we denote $\Gamma(\Omega)$. Then, we are interested in solving

$\min\\{\Gamma(\Omega):\Omega\subseteq\mathbb{R}^{d}\text{ open set,
}|\Omega|=c\\},$ (2)

where $|\cdot|$ is the Lebesgue measure, and $c$ is a given positive real
number. For $m=1$, the Faber-Krahn inequality [Fab23, Kra25] shows that (2)
admits the ball as a solution. For $m=2$, in dimension $d=2,3$, it has been
proved in 1995 in [Nad95] and [AB95] that the ball also solves (2).
Unfortunately, the method employed does not give the result when $d\geq 4$.
For $m\geq 3$, it seems that nothing is known about the solution of (2) in any
dimension (except dimension $1$). In this context, it is a natural question to
ask whether there is at least one solution to (2). We will give a positive
answer to this question for a certain range of dimensions by proving the
following.

######  Theorem 1.

Let $2\leq d\leq 4m$. Problem (2) admits a solution $\Omega$. Moreover, any
corresponding eigenfunction extended by $0$ ouside $\Omega$ is
$C^{m-1,\alpha}(\mathbb{R}^{d})$ for all $0<\alpha<1$.

######  Remark 2.

  1. 1.

We do not claim anything on the boundedness of an optimal shape.

  2. 2.

Even if the eigenfunction is somewhat regular, we do not claim any regularity
on the optimal shape more than its openness.

  3. 3.

The regularity of the eigenfunction is not sharp in general (see [BMPV15,
Theorem 6.1]).

  4. 4.

In the case $m=1$, the restriction $d\leq 4m$ can be removed (see also Remark
6).

The proof of Theorem 1 will be divided into two steps, following the approach
of [Sto23]. First, we prove the existence of an optimal shape in a wider class
than the class of open sets. As we shall see, this part of the proof does not
require the restriction $2\leq d\leq 4m$. Then, we show that the generalised
optimal shape enjoys some regularity properties allowing to solve (2). Let us
begin with the existence of a generalised optimal shape.

###  1.1 Existence of an optimal shape

Several breakthroughs have been performed in the direction of exhibiting an
optimal shape for (2) or relaxed versions. We cite [AB03, Sto23], in which an
analogous problem is tackled for the buckling bilaplace eigenvalue problem,
using a concentration-compactness technique. Another interesting result is
[Buc05, Theorem 3.5], which establishes the existence of a quasi-open optimal
shape for a family of spectral optimization problems similar to (2). However,
the class of admissible shapes is restricted to some design region. Moreover,
we emphasize that the definition chosen in [Buc05] for the Sobolev space
$H_{0}^{m}(\Omega)$ does not match the classical one, unless $\Omega$ enjoys
some regularity (see [Buc05, p. 109] for details). As a consequence, even if
the optimal shape would be open, it would not necessarily solve (2). In the
same fashion, a similar result but with a different approach shall be found in
[Sto21]. There, the existence of an open optimal shape for problem (2) in
dimension $d\geq 4$ is evidenced, still among sets lying in a design region.
However, let us mention that it is unclear in the proof of [Sto21, Theorem 5]
whether the function $v$ is well-defined. Therefore, some of the claimed
results might need ancillary arguments.

As a first step towards a proof of Theorem 1, we would like to provide an
optimal shape for (2) in the wider class of quasi-open sets. Even if we need
preliminary notions to understand it correctly, let us give a precise
statement for our first main result.

######  Theorem 3.

Let $d\in\mathbb{N}^{*}$. The following problem admits a solution.

$\displaystyle\min\\{\Gamma(\Omega):\Omega\subseteq\mathbb{R}^{d}\text{,
}\Omega\text{ 1-quasi-open, }|\Omega|=c\\}$ (3)

######  Remark 4.

  1. 1.

Note that the optimal shape need not to be bounded, we only know that it has
bounded measure.

  2. 2.

At first glance it should be suprising that for $m$ arbitrary, the appropriate
class of quasi-open sets for getting existence is the class of $1$-quasi-open
sets instead of $m$-quasi-open sets. This will become clear as soon as we will
understand how the optimal shape is constructed.

The notion of quasi-open sets involved in the statement of Theorem 3 will be
investigated in section 2. Then, the proof of Theorem 3 is achieved in section
3.2. However, without entering into details, we shall briefly describe the
strategy of the proof: We follow [AB03], and apply the concentration-
compactness principle (see paragraph 3.1) directly to the first eigenfunctions
of a minimizing sequence of open sets. We show that only compactness shall
occur, hence we are able to extract a converging subsequence of first
eigenfunctions. The optimal shape is identified as the set of points where the
limit function or one of its derivatives up to order $m-1$ does not vanish.
Such a set is only $1$-quasi-open in general, which explains why we needed to
widen the class of admissible sets.

###  1.2 Regularity of the optimal shape

After providing an optimal shape among quasi-open sets, the next step in the
proof of Theorem 1 would be to show that this optimal set is actually open,
and to give some hint on its regularity. A wide literature have been produced
on this subject, especially in the case $m=1$ (see for instance [BMPV15,
BBV13] and the references given at the end of section 5). For the case $m=2$,
the work [Sto21], following [Sto22], gives partial answers, but again with the
assumption of a prescribed design region. This assumption was dropped
successfully only in the case of the buckling bilaplacian [Sto23]. Therefore,
as for the existence, the regularity of an optimal shape seems still open in
full generality. Indeed, gaining regularity from scratch is a quite difficult
task in general. As a consequence, we will not prove that any optimal shape
for (3) is a regular open set. Alternatively, in the wake of [BMPV15, Sto23],
we will obtain regularity for the first eigenfunction, and from this fact
build an open set competing with the optimal quasi-open set of Theorem 3. More
precisely, we will prove the following.

######  Theorem 5.

Assume that $2\leq d\leq 4m$ and that there exists a measurable set $\Omega$
solving

$\min\\{\Gamma(\omega):\omega\subset\mathbb{R}^{d}\text{ measurable,
}|\omega|=c\\}.$ (4)

Then, any first eigenfunction of $\Omega$ extended by $0$ outside $\Omega$ is
$C^{m-1,\alpha}(\mathbb{R}^{d})$ for all $0<\alpha<1$. As a consequence, there
exists an(other) optimal set $\tilde{\Omega}$ for (4) which is open.

######  Remark 6.

  1. 1.

The regularity which is obtained for the eigenfunction is not sharp in general
(see [BMPV15, Theorem 6.1]).

  2. 2.

The restriction $d\leq 4m$ comes from the fact that Theorem 18 is used with
$f=\Gamma(\Omega)u$ which needs to belong to $L_{loc}^{d/m}(\mathbb{R}^{d})$
(here $u$ is an eigenfunction in $\Omega$). When $d\leq 4m$, this holds by
Sobolev embeddings. In dimension greater than $4m$, such an integrability of
$u$ is known only when $\Omega$ is already very regular (for instance $C^{2m}$
by elliptic regularity [GGS10, Theorem 2.20] and bootstrap arguments). In the
special case $m=1$, this condition on $u$ is always automatically fulfilled
since the eigenfunctions are bounded [Dav90, Example 2.1.8]. For $m>1$, it
would be of interest to study the boundedness of the eigenfunctions in
measurable sets or even in open sets with regularity lower than $C^{2m}$.

The proof of Theorem 5 is given in section 4.2. The strategy for proving
$C^{m-1,\alpha}$ regularity for an eigenfunction $u$ of $\Omega$ is to resort
to Morrey’s Theorem (see [Gia83, Chapter III, Theorem 1.1]) which requires
some control on the growth of the $L^{2}$ norm of the derivatives of $u$ on
small balls. As shown in the proof of Theorem 18, such a control is precisely
a crucial property enjoyed by quasi-minimizers of Dirichlet energy functionals
(see paragraph 4.1 for the definition and first properties of quasi-
minimizers). As a consequence, one boils down to showing that $u$ is a quasi-
minimizer of some Dirichlet energy functional, which will come from the
optimality of $\Omega$. After proving the regularity of the eigenfunction, we
will define the competitor $\tilde{\Omega}$ as the set of points where $u$ or
one of its derivatives up to order $m-1$ does not vanish.

###  1.3 Structure of the paper

The document is structured as follows. Section 2 contains preliminaries. In
particular the notions of capacities, quasi-open sets, and related concepts
are recalled. Then, existence results are adressed in section 3. The method of
concentration-compactness adapted to our situation is presented in paragraph
3.1, after what the proof of Theorem 3 is achieved in paragraph 3.2. Section 4
deals with regularity results. Local quasi-minimizers are introduced and
briefly studied in paragraph 4.1 and the proof of Theorem 5 is finally
performed in paragraph 4.2. Section 5 concludes, providing the proof of
Theorem 1 and discussing some perspectives.

##  2 Preliminaries

We recall that if $\Omega$ is an open set of bounded measure, the eigenvalue
$\Gamma(\Omega)$ can be characterized variationally by the formula

$\Gamma(\Omega)=\min_{v\in
H_{0}^{m}(\Omega)}\frac{\int_{\Omega}|D^{m}v|^{2}}{\int_{\Omega}v^{2}}.$ (5)

Here, $D^{m}v$ denotes the differential of order $m$ of $v$. Its Euclidean
norm $|D^{m}v|^{2}=D^{m}v:D^{m}v$ is inherited from the Euclidean scalar
product denoted

$D^{m}v:D^{m}w=\sum_{\begin{subarray}{c}\gamma\in\mathbb{N}^{d},\\\
|\gamma|=m\end{subarray}}^{d}\partial^{\gamma}v\partial^{\gamma}w,$

where for $\gamma=(\gamma_{1},...,\gamma_{d})\in\mathbb{N}^{d}$,
$\partial^{\gamma}v=\partial_{1}^{\gamma_{1}}...\partial_{d}^{\gamma_{d}}v$.
It can be shown, thanks to the compact embedding
$H_{0}^{m}(\Omega)\hookrightarrow L^{2}(\Omega)$, that the minimum of (5) is
achieved for some function $u$. Then, $u$ is a first eigenfunction, that is
$(u,\Gamma(\Omega))$ is a distributional solution to (1). This last point is a
consequence of the integration by parts formula,

$\int_{\Omega}\left((-\Delta)^{m}u\right)v=\int_{\Omega}D^{m}u:D^{m}v,\qquad\forall
u,v\in C_{c}^{\infty}(\Omega).$ (6)

As explained in the introduction, we will need to define the first eigenvalue
on sets which are not necessarily open. For that, we will use the right-hand
side of formula (5) as a definition, hence we need to investigate the
functional space $H_{0}^{m}(\Omega)$ in the case where $\Omega$ is a general
set. This involves the notions of capacities of orders $1$ up to $m$. For the
sake of brevity, we will use the prefix $k$ when talking about capacity of
order $k$ and related notions.

We refer to [HP05, section 3.3] for the definition and properties of
$1$-capacity and $1$-quasi-open sets. We only recall that a $1$-quasi-
continuous function is a function for which the preimage of any open set is a
$1$-quasi-open set, and that any function in $H^{1}(\mathbb{R}^{d})$ has a
$1$-q.e. well-defined $1$-quasi-continuous representative. In the following,
we shall always identify $H^{1}$ functions with their $1$-quasi-continuous
representative. Then, Theorem 3.3.42 of [HP05] shows that, when $\Omega$ is
open,

$H_{0}^{1}(\Omega)=\\{u\in H^{1}(\mathbb{R}^{d}):u=0\text{ 1-q.e. in
}\Omega^{c}\\}$ (7)

In the same fashion, one might define an $m$-capacity and the corresponding
notions of $m$-quasi-open sets and $m$-quasi-continuous functions (see [AH96]
for a very general theory of capacities). Moreover, any $u\in H^{m}$ has an
$m$-q.e. well-defined $m$-quasi-continuous representative and we will always
identify $u$ with its representative. Then, when $\Omega$ is open, the space
$H_{0}^{m}(\Omega)$ shall be characterised thanks to the capacities up to
order $m$ in the following way (see for instance [GLM23, p.2]):

$H_{0}^{m}(\Omega)=\\{u\in H^{m}(\mathbb{R}^{d}):D^{k}u=0\text{
}(m-k)\text{-q.e. in }\Omega^{c}\text{, for }0\leq k<m\\}.$ (8)

Let us precise that in general $H_{0}^{m}(\Omega)$ does not coincide with the
set of $H^{m}(\mathbb{R}^{d})$ functions vanishing a.e. outside $\Omega$.
Indeed, such a property requires some regularity on $\Omega$ (see [GLM23]).
Observe that as the right-hand member of (8) makes sense regardless of the
regularity of $\Omega$, we shall use it to define $H_{0}^{m}(\Omega)$ even
when $\Omega$ is not open. Moreover, the compact injection

$H_{0}^{m}(\Omega)\hookrightarrow L^{2}(\Omega)$ (9)

holding when $\Omega$ was open, remains when $\Omega$ is only a measurable set
of finite measure, as explained below.

######  Definition 7.

Let $\Omega$ be a subset of $\mathbb{R}^{d}$. Then, $H_{0}^{m}(\Omega)$ is
defined by (8).

######  Lemma 8.

Let $\Omega$ be a measurable set of finite measure. Then, injection (9) is
compact.

###### Proof.

Let $(u_{n})_{n}$ be sequence in $H_{0}^{m}(\Omega)$. By outer regularity of
the Lebesgue measure, there exists an open set $\tilde{\Omega}\supseteq\Omega$
of measure $|\Omega|+1<+\infty$. Then,

$H_{0}^{m}(\Omega)\subseteq H_{0}^{m}(\tilde{\Omega})\hookrightarrow
L^{2}(\tilde{\Omega})$

is compact, so we shall extract from $(u_{n})_{n}$ a subsequence converging in
$L^{2}(\tilde{\Omega})$, hence in $L^{2}(\Omega)$. ∎

Besides, we recall that by [AH96, Theorem 10.1.1], for any set $\Omega$,
$H_{0}^{m}(\Omega)$ coincides with the closure of $C_{c}^{\infty}(\Omega)$
with respect to $H^{m}(\mathbb{R}^{d})$. Lastly, as a consequence of Lemma 8,
formula (5) generalises to any measurable set of finite measure, and the
minimum in (5) is also achieved in this setting. Before going further, let us
mention the following useful observation, which follows immediately from (8).

######  Lemma 9.

Let $u\in H^{m}(\mathbb{R}^{d})$. Then, $u\in
H_{0}^{m}(\bigcup_{k=0}^{m-1}\\{D^{k}u\neq 0\\})$.

######  Remark 10.

Although Lemma 9 looks obvious in view of the previous definition of
$H_{0}^{m}$, it has to be noted that the set
$\omega:=\bigcup_{k=0}^{m-1}\\{D^{k}u\neq 0\\}$ depends upon the choice of a
quasi-continuous representative not only for $u$ but also for its derivatives.
Consequently, one may find a set $\tilde{\omega}$ coinciding with $\omega$
only $1$-q.e. and such that $u\in H_{0}^{m}(\tilde{\omega})$. This might
appear paradoxical at first glance, since the assertion “$u\in
H_{0}^{m}(\tilde{\omega})$ for all $\tilde{\omega}=\omega$ $1$-q.e.” is false
when $m>1$. The nuance is that we only claim the existence of some
$\tilde{\omega}=\omega$ $1$-q.e. such that $u\in H_{0}^{m}(\tilde{\omega})$.
In other words, $\omega$ is defined more than just $1$-quasi everywhere.

To conclude this section, let us explain the relevance of our procedure of
relaxing the class of admissible domains for problem (2). This is justified by
the fact that minimizing $\Gamma$ among open sets, quasi-open sets, or even
measurable sets does not change the value of the infimum, as stated in the
next lemma. Its proof relies on the homogeneity of $\Gamma$, which reads, for
any set $\Omega$,

$\Gamma(R\Omega)=R^{-2m}\Gamma(\Omega),\qquad\forall R>0.$ (10)

######  Lemma 11.

For any $k\in\mathbb{N}^{*}$,

$\inf_{\begin{subarray}{c}\Omega\text{ open}\\\
|\Omega|=c\end{subarray}}\Gamma(\Omega)=\inf_{\begin{subarray}{c}\Omega\text{
k-quasi-open}\\\
|\Omega|=c\end{subarray}}\Gamma(\Omega)=\inf_{\begin{subarray}{c}\Omega\text{
measurable}\\\ |\Omega|=c\end{subarray}}\Gamma(\Omega)>0.$

###### Proof.

Clearly,

$\inf_{\begin{subarray}{c}\Omega\text{ open}\\\
|\Omega|=c\end{subarray}}\Gamma(\Omega)\geq\inf_{\begin{subarray}{c}\Omega\text{
k-quasi-open}\\\
|\Omega|=c\end{subarray}}\Gamma(\Omega)\geq\inf_{\begin{subarray}{c}\Omega\text{
measurable}\\\ |\Omega|=c\end{subarray}}\Gamma(\Omega),$

since any open set is also quasi-open, and any quasi-open set is measurable.
For the converse inequality, observe that if $\Omega$ is a measurable set of
volume $c$, by outer regularity of the Lebesgue measure there exists for every
$\epsilon>0$ an open set $\Omega_{\epsilon}$ of volume $c+\epsilon$ such that
$\Omega\subseteq\Omega_{\epsilon}$, and hence $H_{0}^{m}(\Omega)\subseteq
H_{0}^{m}(\Omega_{\epsilon})$. Then, by (10),

$\Gamma(\Omega)\geq\Gamma(\Omega_{\epsilon})\geq\left(\frac{c}{c+\epsilon}\right)^{\frac{2m}{d}}\inf_{\begin{subarray}{c}\Omega\text{
open}\\\ |\Omega|=c\end{subarray}}\Gamma(\Omega),$

and we get the result by taking $\epsilon\to 0$. In order to prove that the
infimum does not vanish, one may consider an open set $\Omega$ and combine the
classical Faber-Krahn inequality with the following one:

$\lambda(\Omega)^{m}\leq\Gamma(\Omega),$ (11)

where $\lambda(\Omega)$ is the first eigenvalue of the Laplacian, which is
defined by taking $m=1$ in (5). To prove (11), we let $u\in H_{0}^{m}(\Omega)$
be an eigenfunction (of the polylaplacian of order $m$) normalised in
$L^{2}(\Omega)$. After an integration by parts and two Cauchy-Schwarz
inequalities, we observe that for any $1\leq k<m$,

$\left(\int_{\Omega}|D^{k}u|^{2}\right)^{2}\leq\int_{\Omega}|D^{k-1}u|^{2}\int_{\Omega}|D^{k+1}u|^{2}.$

In other words, denoting $v_{k}=\int_{\Omega}|D^{k}u|^{2}$, we have

$v_{k}^{2}\leq v_{k-1}v_{k+1}.$

As a consequence, we prove by recursion that $(v_{k})_{k}$ obeys the rule

$v_{m-k}^{p_{k}}\leq v_{m-k-1}^{q_{k}}v_{m},\qquad\forall 1\leq k<m,$

where $(p_{k})_{k}$ and $(q_{k})_{k}$ are defined by $(p_{1},q_{1})=(2,1)$,
and for all $k\geq 1$,

$p_{k+1}=2p_{k}-q_{k},\qquad q_{k+1}=p_{k}.$

The couple $(p_{k},q_{k})_{k\geq 1}$ forms a constant-recursive sequence of
order $1$, the solution of which is $(p_{k},q_{k})=(k+1,k)$. Consequently,

$v_{1}^{m}\leq v_{0}^{m-1}v_{m},$ (12)

which gives the result since $u\in H_{0}^{1}(\Omega)$ is normalised in
$L^{2}$. ∎

##  3 Existence of an optimal shape for $\Gamma$

###  3.1 Concentration-Compactness principle

As mentioned in the introduction, the concentration-compactness principle is
the main tool used for proving Theorem 3. In this paragraph, we adapt to our
needs the standard concentration-compactness Lemma of Lions (see Lemma I.1 and
III.1 of [Lio84]). We will use the notation $\tau_{y}u$ to designate the
translation of a given function $u\in H^{m}(\mathbb{R}^{d})$ by a given point
$y\in\mathbb{R}^{d}$, that is the function defined by $\tau_{y}u(x)=u(x-y)$.

######  Lemma 12.

Let $(u_{n})_{n}$ be a bounded sequence of functions in
$H^{m}(\mathbb{R}^{d})$ such that $\|u_{n}\|_{L^{2}}^{2}\to\lambda$. Then, up
to extracting a subsequence, one of the following occurs:

  1. (i)

(compactness) there exists a sequence of points
$(y_{n})_{n}\subseteq\mathbb{R}^{d}$ such that

$\forall\epsilon>0,\exists R>0,\exists N\in\mathbb{N},\forall
n>N,\quad\|\tau_{y_{n}}u_{n}\|_{L^{2}(B_{R})}^{2}\geq\lambda-\epsilon.$

  2. (ii)

(vanishing)

$\forall
R>0,\quad\lim_{n\to\infty}\sup_{y\in\mathbb{R}^{d}}\|\tau_{y}u_{n}\|_{L^{2}(B_{R})}^{2}=0.$

  3. (iii)

(dichotomy) there exists $\alpha\in]0,\lambda[$ and functions $u_{n}^{1}$, and
$u_{n}^{2}$ in $H^{m}(\mathbb{R}^{d})$ such that

$\displaystyle\|u_{n}-(u_{n}^{1}+u_{n}^{2})\|_{L^{2}}\to 0,$
$\displaystyle\|u_{n}^{1}\|_{L^{2}}^{2}\to\alpha,\qquad\|u_{n}^{2}\|_{L^{2}}^{2}\to\lambda-\alpha,$
$\displaystyle\liminf\left[\|D^{m}u_{n}\|_{L^{2}}^{2}-\|D^{m}u_{n}^{1}\|_{L^{2}}^{2}-\|D^{m}u_{n}^{2}\|_{L^{2}}^{2}\right]\geq
0.$ Moreover, for any $0\leq k<m$, up to some sets of zero $(m-k)$-capacity,
$\displaystyle dist\left(\omega_{n,k}^{1},\omega_{n,k}^{2}\right)\to\infty,$
$\displaystyle\omega_{n,k}^{1},\omega_{n,k}^{2}\subseteq\omega_{n,k},$

where $\omega_{n,k}:=\bigcup_{j=0}^{k}\\{D^{j}u_{n}\neq 0\\}$ and
$\omega_{n,k}^{i}:=\bigcup_{j=0}^{k}\\{D^{j}u_{n}^{i}\neq 0\\}$ for $i=1,2$.

###### Proof.

We follow the proof of Lemma III.1 of [Lio84] and define

$Q_{n}(R):=\sup_{y\in\mathbb{R}^{d}}\|\tau_{y}u_{n}\|_{L^{2}(B_{R})}^{2}$

which induces a sequence of nonnegative nondecreasing uniformly bounded
functions on $\mathbb{R}_{+}$ satisfying, for each $n\in\mathbb{N}$,
$Q_{n}(R)\xrightarrow[R\to\infty]{}\lambda_{n}$, with $\lambda_{n}\to\lambda$.
Then, by Helly’s selection Theorem, up to extracting a subsequence, $Q_{n}$
pointwise converges to some function $Q$ on $\mathbb{R}_{+}$. Then $Q$, like
$Q_{n}$, enjoys the properties of being nonnegative nondecreasing and bounded
by $\lambda$. Now set

$\alpha:=\lim_{R\to\infty}\uparrow Q(R)\in[0,\lambda].$

If $\alpha=0$, we are in the vanishing case, because for any $R$, $0\leq
Q(R)\leq\alpha=0$, hence $Q(R)=0$ which is the result. If $\alpha=\lambda$, we
are in case of compactness. Indeed, let $\epsilon>0$ and choose $r$ such that
$\lambda-\epsilon<Q(r)$. Because $Q_{n}(r)\to Q(r)$, for all $n$ large enough,
$\lambda-\epsilon<Q_{n}(r)$. Then, there exists $z_{n}\in\mathbb{R}^{d}$ such
that $\lambda-\epsilon<\|\tau_{z_{n}}u_{n}\|_{L^{2}(B_{r})}^{2}$. Note that
the radius $r=r(\epsilon)$ and the points $z_{n}=z_{n}(\epsilon)$ depend on
$\epsilon$.

But as long as $\epsilon<\lambda/3$, the balls
$B(z_{n}(\epsilon),r(\epsilon))$ and $B(z_{n}(\lambda/3),r(\lambda/3))$
intersect each other for all $n$ large enough, otherwise we would have, up to
a subsequence,

$\|u_{n}\|_{L^{2}}^{2}\geq\|\tau_{z_{n}(\epsilon)}u_{n}\|_{L^{2}(B_{r(\epsilon)})}^{2}+\|\tau_{z_{n}(\lambda/3)}u_{n}\|_{L^{2}(B_{r(\lambda/3)})}^{2}>\lambda-\epsilon+\lambda-\lambda/3>4\lambda/3,$

which is impossible since $\|u_{n}\|_{L^{2}}^{2}\to\lambda$. As a consequence,
the distance between $z_{n}(\epsilon)$ and $z_{n}(\lambda/3)$ does not exceed
$r(\epsilon)+r(\lambda/3)$. Then, setting
$R(\epsilon):=r(\lambda/3)+2r(\epsilon)$, we get that
$B(z_{n}(\epsilon),r(\epsilon))$ is included in
$B(z_{n}(\lambda/3),R(\epsilon))$. Therefore, the sequence
$y_{n}=z_{n}(\lambda/3)$ satisfies that for any $0<\epsilon<\lambda/3$, there
exists $R:=R(\epsilon)$ such that for all $n$ large enough,

$\|\tau_{y_{n}}u_{n}\|_{L^{2}(B_{R})}^{2}\geq\|\tau_{z_{n}(\epsilon)}u_{n}\|_{L^{2}(B_{r(\epsilon)})}^{2}>\lambda-\epsilon.$

If now $\alpha\in]0,\lambda[$, let $\epsilon_{n}\to 0$ and $R_{n}\to\infty$
such that $\forall n\in\mathbb{N}$, $\alpha\geq Q(R_{n})>\alpha-\epsilon_{n}$.
Because $Q_{n}\to Q$ once again, we may extract a subsequence $(u_{n})_{n}$
and define a sequence $(y_{n})_{n}$ such that, $\forall n\in\mathbb{N}$,
$\alpha-\epsilon_{n}<\|\tau_{y_{n}}u_{n}\|_{L^{2}(B_{R_{n}})}^{2}$ and
$Q_{n}(kR_{n})<\alpha+\epsilon_{n}$ ($k=2,6$).

At this point, we use a smooth cut-off function over $B_{1}$ and with support
in $B_{2}$, say $\chi^{1}$, and set $\chi^{2}:=1-\chi^{1}$. Moreover, we
define the $m$-quasi-continuous functions

$\displaystyle u_{n}^{1}=u_{n}\tau_{-y_{n}}\delta_{R_{n}}\chi^{1},\qquad
u_{n}^{2}=u_{n}\tau_{-y_{n}}\delta_{3R_{n}}\chi^{2},$

where $\delta_{r}v$ is the dilation of a function $v$ by $r$, i.e.
$\delta_{r}v(x):=v(x/r)$. Note that $\delta_{r}\chi^{1}$ and
$\delta_{3r}\chi^{2}$, as well as their derivatives, are respectively
supported in $B_{2r}$ and in $\mathbb{R}^{d}\setminus B_{3r}$. As a
consequence, we have

$dist\left(\omega_{n,k}^{1},\omega_{n,k}^{2}\right)\geq R_{n}\to\infty.$

Moreover, for $i=1,2$, the inclusion $\omega_{n,k}^{i}\subseteq\omega_{n,k}$
follows from the definition of $u_{n}^{1}$ and $u_{n}^{2}$. Now let us check
that

$\displaystyle\left\|D^{m}u_{n}^{1}\right\|_{L^{2}}^{2}-\left\|(\tau_{-y_{n}}\delta_{R_{n}}\chi^{1})D^{m}u_{n}\right\|_{L^{2}}^{2}\underset{n\to\infty}{=}O(1/R_{n}),$
$\displaystyle\left\|D^{m}u_{n}^{2}\right\|_{L^{2}}^{2}-\left\|(\tau_{-y_{n}}\delta_{3R_{n}}\chi^{2})D^{m}u_{n}\right\|_{L^{2}}^{2}\underset{n\to\infty}{=}O(1/R_{n}).$

For that purpose, take $\gamma\in\mathbb{N}^{d}$ a multi-index of length $m$,
and recall that

$\partial^{\gamma}(fg)=\sum_{\beta\leq\gamma}\partial^{\beta}f\partial^{\gamma-\beta}g,$

from which we deduce

$\displaystyle\left\|\partial^{\gamma}u_{n}^{1}\right\|_{L^{2}}^{2}$
$\displaystyle=\left\|(\tau_{-y_{n}}\chi_{R_{n}}^{1})\partial^{\gamma}u_{n}\right\|_{L^{2}}^{2}+\sum_{\begin{subarray}{c}\beta\leq\gamma\\\
\beta\neq
0\end{subarray}}\frac{1}{R_{n}^{2|\beta|}}\left\|(\partial^{\gamma-\beta}u_{n})(\tau_{-y_{n}}\delta_{R_{n}}\partial^{\beta}\chi^{1})\right\|_{L^{2}}^{2}$
$\displaystyle+\sum_{\begin{subarray}{c}\beta,\beta^{\prime}\leq\gamma\\\
\beta\neq\beta^{\prime}\end{subarray}}\frac{1}{R_{n}^{|\beta|+|\beta^{\prime}|}}\left\langle(\partial^{\gamma-\beta}u_{n})(\tau_{-y_{n}}\delta_{R_{n}}\partial^{\beta}\chi^{1})\text{
,
}(\partial^{\gamma-\beta^{\prime}}u_{n})(\tau_{-y_{n}}\delta_{R_{n}}\partial^{\beta^{\prime}}\chi^{1})\right\rangle_{L^{2}}.$

Since $(u_{n})_{n}$ is bounded in $H^{m}(\mathbb{R}^{d})$, and since
$\chi^{1}\in W^{m,\infty}(\mathbb{R}^{d})$, the two sums above are
$O(1/R_{n})$, and we obtain as expected

$\left\|\partial^{\gamma}u_{n}^{1}\right\|_{L^{2}}^{2}\underset{n\to\infty}{=}\left\|(\tau_{-y_{n}}\chi_{R_{n}}^{1})\partial^{\gamma}u_{n}\right\|_{L^{2}}^{2}+O(1/R_{n}).$

An analogous discussion leads to an analogous result regarding
$\left\|\partial^{\gamma}u_{n}^{2}\right\|_{L^{2}}^{2}$. Now, using also the
inequality
$\left|\delta_{r}\chi^{1}\right|^{2}+\left|\delta_{3r}\chi^{2}\right|^{2}\leq
1$, the previous justifies that for all $n\in\mathbb{N}$,

$\displaystyle\|D^{m}u_{n}\|_{L^{2}}^{2}-\|D^{m}u_{n}^{1}\|_{L^{2}}^{2}-$
$\displaystyle\|D^{m}u_{n}^{2}\|_{L^{2}}^{2}$
$\displaystyle\geq\|D^{m}u_{n}\|_{L^{2}}^{2}-\|(\tau_{-y_{n}}\delta_{R_{n}}\chi^{1})D^{m}u_{n}\|_{L^{2}}^{2}-\|(\tau_{-y_{n}}\delta_{3R_{n}}\chi^{2})D^{m}u_{n}\|_{L^{2}}^{2}+O(1/R_{n})$
$\displaystyle\geq\|D^{m}u_{n}\|_{L^{2}}^{2}-\int_{\mathbb{R}^{d}}\tau_{-y_{n}}\left(\left|\delta_{R_{n}}\chi^{1}\right|^{2}+\left|\delta_{3R_{n}}\chi^{2}\right|^{2}\right)|D^{m}u_{n}|^{2}+O(1/R_{n})$
$\displaystyle\geq O(1/R_{n}).$

This means that
$\liminf_{n}\left[\|D^{m}u_{n}\|_{L^{2}}^{2}-\|D^{m}u_{n}^{1}\|_{L^{2}}^{2}-\|D^{m}u_{n}^{2}\|_{L^{2}}^{2}\right]\geq
0$. Moreover,

$\|u_{n}^{1}\|_{L^{2}}^{2}=\|\delta_{R_{n}}\chi^{1}\tau_{y_{n}}u_{n}\|_{L^{2}}^{2}\geq\|\tau_{y_{n}}u_{n}\|_{L^{2}(B_{R_{n}})}^{2}>\alpha-\epsilon_{n}\to\alpha.$

On the other hand, since $\delta_{R_{n}}\chi^{1}$ is supported in
$B_{2R_{n}}$,

$\|u_{n}^{1}\|_{L^{2}}^{2}=\|\delta_{R_{n}}\chi^{1}\tau_{y_{n}}u_{n}\|_{L^{2}}^{2}\leq\|\tau_{y_{n}}u_{n}\|_{L^{2}(B_{2R_{n}})}^{2}\leq
Q_{n}(2R_{n})<\alpha+\epsilon_{n}\to\alpha.$

Hence $\|u_{n}^{1}\|_{L^{2}}^{2}\to\alpha$. In the same spirit, regarding
$u_{n}^{2}$,

$\|u_{n}^{2}\|_{L^{2}}^{2}=\|\delta_{3R_{n}}\chi^{2}\tau_{y_{n}}u_{n}\|_{L^{2}}^{2}\leq\|u_{n}\|_{L^{2}}^{2}-\|\tau_{y_{n}}u_{n}\|_{L^{2}(B_{3R_{n}})}^{2}\leq\|u_{n}\|_{L^{2}}^{2}-\alpha+\epsilon_{n}\to\lambda-\alpha,$

and

$\|u_{n}^{2}\|_{L^{2}}^{2}\geq\|u_{n}\|_{L^{2}}^{2}-Q_{n}(6R_{n})>\|u_{n}\|_{L^{2}}^{2}-\alpha-\epsilon_{n}\to\lambda-\alpha,$

showing that $\|u_{n}^{2}\|_{L^{2}}^{2}\to\lambda-\alpha$. To conclude,

$\displaystyle\|u_{n}-(u_{n}^{1}+u_{n}^{2})\|_{L^{2}}^{2}$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\delta_{3R_{n}}\chi^{1}-\delta_{R_{n}}\chi^{1}\right)^{2}|\tau_{y_{n}}u_{n}|^{2}\to
0,$

by dominated convergence. ∎

###  3.2 Proof of Theorem 3

Equipped with the concentration-compactness principle, we can now turn to the
proof of Theorem 3, which claims the existence of a $1$-quasi-open set
minimizing $\Gamma$ under volume constraint. Following [AB03], the strategy is
to apply the concentration-compactness procedure to the eigenfunctions of a
minimizing sequence of sets, and to show that only compactness occurs. As a
consequence, the eigenfunctions are shown to converge in $L^{2}$ towards a
limit function. The final step is to identify the optimal set from the limit
function, which is performed in the next proposition.

######  Proposition 13.

Let $\Omega_{n}$ be a sequence of measurable sets of volume $c$, and $u_{n}\in
H_{0}^{m}(\Omega_{n})$ the corresponding eigenfunctions converging to some $u$
in $H^{m}(\mathbb{R}^{d})$ weakly and in $L^{2}(\mathbb{R}^{d})$ strongly.
Then there exists a $1$-quasi-open set $\Omega$ of volume $|\Omega|\leq c$
such that

$\liminf_{n}\Gamma(\Omega_{n})\geq\Gamma(\Omega).$

###### Proof.

Thanks to weak $H^{m}$ and strong $L^{2}$ convergence of $(u_{n})_{n}$, we
immediately have

$\liminf_{n}\Gamma(\Omega_{n})\geq\frac{\int_{\mathbb{R}^{d}}|D^{m}u|^{2}}{\int_{\mathbb{R}^{d}}u^{2}}.$

That’s why one would like to define the optimal shape as the domain where $u$
or one of its derivatives does not vanish. Fortunately, the set
$\Omega:=\cup_{k=0}^{m-1}\\{D^{k}u\neq 0\\}$ is $1$-quasi-open, and $u\in
H_{0}^{m}(\Omega)$ by Lemma 9. Therefore,

$\liminf_{n}\Gamma(\Omega_{n})\geq\Gamma(\Omega).$

It remains only to show that $|\Omega|\leq c$. For that purpose, observe that
not only $(u_{n})_{n}$ converges in $L^{2}(\mathbb{R}^{d})$, but also that
$(D^{k}u_{n})_{n}$ converges in $L^{2}(\mathbb{R}^{d})$ by interpolation, for
all $0\leq k<m$ (recall Gagliardo-Nirenberg inequalities). Moreover, up to a
subsequence, we shall assume that the convergence holds almost everywhere.
Then, from the $1$-q.p. inclusion

$\Omega\setminus\Omega_{n}\subseteq\bigcup_{0\leq k<m}\\{D^{k}u\neq
0\\}\setminus\\{D^{k}u_{n}\neq 0\\},$

we get $|\Omega\setminus\Omega_{n}|\to 0$, which gives as desired
$|\Omega|\leq\lim|\Omega_{n}|=c$. ∎

Let us now apply the concentration-compactness principle and proceed to the
proof of Theorem 3.

###### Proof of Theorem 3.

Let us consider a minimizing sequence $\Omega_{n}$. Actually, thanks to Lemma
11, we can take the same minimizing sequence for problems (3) and (2), and
hence choose $\Omega_{n}$ to be open. Moreover, we have
$\inf_{n}\Gamma(\Omega_{n})=:I>0$. We note $u_{n}\in H_{0}^{m}(\Omega_{n})$
the eigenfunction on $\Omega_{n}$, and assume without loss of generality that
$u_{n}$ is normalised in $L^{2}$. Then, by Lemma 12, we extract a subsequence
from $(u_{n})_{n}$ such that three situations shall occur: compactness,
vanishing, or dichotomy.

If we are in the vanishing situation, for any $0<R<\infty$,

$\lim_{n}\sup_{y\in\mathbb{R}^{d}}\int_{B_{R}}|\tau_{y}u_{n}|^{2}=0.$

If we assumed that $(u_{n})_{n}$ were bounded in $H^{1}$, then thanks to
[AB03, Lemma 3.6], there would exist a sequence of points
$(y_{n})_{n}\subseteq\mathbb{R}^{d}$ such that $\tau_{y_{n}}u_{n}$ admit no
subsequence weakly converging to $0$ in $H^{1}$. Consequently, there would
exist $\epsilon>0$ and a test function $\varphi$ with support in a ball of
radius say $R>0$, such that, for $n$ large enough,

$\epsilon\leq\left(\int\tau_{y_{n}}u_{n}\varphi\right)^{2}\leq\sup_{y\in\mathbb{R}^{d}}\int_{B_{R}}|\tau_{y}u_{n}|^{2}\int_{B_{R}}\varphi^{2}.$

This would contradict the fact that $(u_{n})_{n}$ vanishes. Therefore, we can
extract a subsequence such that $\int_{\Omega_{n}}|\nabla
u_{n}|^{2}\to\infty$. But thanks to inequality (12), we obtain that
$\int_{\Omega_{n}}|D^{m}u_{n}|^{2}\to\infty$, showing in turn that
$\Gamma(\Omega_{n})\to\infty$. To conclude, the vanishing situation cannot
occur.

If the dichotomy occurs, there exists $\alpha\in]0,1[$ and sequences
$(u_{n}^{1})_{n}$, $(u_{n}^{2})_{n}$ in $H^{m}(\mathbb{R}^{d})$ with

$\displaystyle\|u_{n}-(u_{n}^{1}+u_{n}^{2})\|_{L^{2}}\to 0,$
$\displaystyle\|u_{n}^{1}\|_{L^{2}}^{2}\to\alpha,\qquad\|u_{n}^{2}\|_{L^{2}}^{2}\to\lambda-\alpha,$
$\displaystyle\liminf\left[\|D^{m}u_{n}\|_{L^{2}}^{2}-\|D^{m}u_{n}^{1}\|_{L^{2}}^{2}-\|D^{m}u_{n}^{2}\|_{L^{2}}^{2}\right]\geq
0,$ $\displaystyle
dist\left(\omega_{n,k}^{1},\omega_{n,k}^{2}\right)\to\infty,$
$\displaystyle\text{up to sets of null }(m-k)\text{-capacity},$
$\displaystyle\omega_{n,k}^{1},\omega_{n,k}^{2}\subseteq\omega_{n,k}\subseteq\Omega_{n},$
$\displaystyle\text{up to sets of null }(m-k)\text{-capacity},$

where $\omega_{n,k}:=\bigcup_{j=0}^{k}\\{D^{j}u_{n}\neq 0\\}$ and
$\omega_{n,k}^{i}:=\bigcup_{j=0}^{k}\\{D^{j}u_{n}^{i}\neq 0\\}$ for $i=1,2$.
Then, thanks to the disjunction of $\\{u_{n}^{1}\neq 0\\}$ and
$\\{u_{n}^{2}\neq 0\\}$, we have that for any $\epsilon>0$, for any $n$ large
enough,

$\displaystyle\int_{\Omega_{n}}|u_{n}|^{2}\leq\int_{\Omega_{n}}|u_{n}^{1}|^{2}+\int_{\Omega_{n}}|u_{n}^{2}|^{2}+2\epsilon,$
$\displaystyle\int_{\Omega_{n}}|D^{m}u_{n}|^{2}\geq\int_{\Omega_{n}}|D^{m}u_{n}^{1}|^{2}+\int_{\Omega_{n}}|D^{m}u_{n}^{2}|^{2}-2\epsilon.$

Therefore,

$\frac{\int_{\Omega_{n}}|D^{m}u_{n}|^{2}}{\int_{\Omega_{n}}u_{n}^{2}}\geq\frac{\int_{\Omega_{n}}|D^{m}u_{n}^{1}|^{2}+\int_{\Omega_{n}}|D^{m}u_{n}^{2}|^{2}-2\epsilon}{\int_{\Omega_{n}}|u_{n}^{1}|^{2}+\int_{\Omega_{n}}|u_{n}^{2}|^{2}+2\epsilon}.$

Note that $u_{n}^{1},u_{n}^{2}\in H_{0}^{m}(\Omega_{n})$, hence we have,

$\int_{\Omega_{n}}|u_{n}^{i}|^{2}\leq\frac{1}{\Gamma(\Omega_{n})}\int_{\Omega_{n}}|D^{m}u_{n}^{i}|^{2}\leq
I^{-1}\int_{\Omega_{n}}|D^{m}u_{n}^{i}|^{2},\qquad i=1,2.$

As a consequence, both $\liminf_{n}\int_{\Omega_{n}}|D^{m}u_{n}^{1}|^{2}>0$
and $\liminf_{n}\int_{\Omega_{n}}|D^{m}u_{n}^{2}|^{2}>0$, so we may choose
$\epsilon$ small enough for having
$\int_{\Omega_{n}}|D^{m}u_{n}^{i}|^{2}-\epsilon>0$ for $i=1,2$ and any large
$n$. Then, using the algebraic property
$\frac{a+b}{c+d}\geq\min\left(\frac{a}{c},\frac{b}{d}\right)$ for $a,b,c,d>0$,
and passing to the limit $n\to\infty$, we obtain that

$\liminf_{n}\frac{\int_{\Omega_{n}}|D^{m}u_{n}|^{2}}{\int_{\Omega_{n}}u_{n}^{2}}\geq\min\left(\liminf_{n}\frac{\int_{\Omega_{n}}|D^{m}u_{n}^{1}|^{2}-\epsilon}{\int_{\Omega_{n}}|u_{n}^{1}|^{2}+\epsilon},\liminf_{n}\frac{\int_{\Omega_{n}}|D^{m}u_{n}^{2}|^{2}-\epsilon}{\int_{\Omega_{n}}|u_{n}^{2}|^{2}+\epsilon}\right).$

Shrinking $\epsilon\to 0$ and assuming that the minimum is achieved for the
quotient involving $u_{n}^{1}$, we get

$\liminf_{n}\Gamma(\Omega_{n})\geq\liminf_{n}\frac{\int_{\Omega_{n}}|D^{m}u_{n}^{1}|^{2}}{\int_{\Omega_{n}}|u_{n}^{1}|^{2}}.$

Now, for $i=1,2$, define $\omega_{n}^{i}:=\omega_{n,m-1}^{i}=\bigcup_{0\leq
k<m}\\{D^{k}u_{n}^{i}\neq 0\\}$. Observe that $\omega_{n}^{i}$ is $1$-quasi-
open, and that, by Lemma 9, $u_{n}^{i}\in H_{0}^{m}(\omega_{n}^{i})$ for
$i=1,2$. In particular,

$\liminf_{n}\Gamma(\Omega_{n})\geq\liminf_{n}\Gamma(\omega_{n}^{1}).$

Moreover, since $\omega_{n}^{1}$ and $\omega_{n}^{2}$ are $1$-q.e. disjoint
(recall that $dist(\omega_{n}^{1},\omega_{n}^{2})\to\infty$) subsets of
$\Omega_{n}$, we have

$|\omega_{n}^{1}|\leq|\Omega_{n}|-|\omega_{n}^{2}|.$

But $\liminf_{n}|\omega_{n}^{2}|>0$. Indeed, since $u_{n}^{2}$ belongs to
$H_{0}^{m}(\omega_{n}^{2})$, we have thanks to relation (10) and to Lemma 11,

$|\omega_{n}^{2}|^{\frac{2m}{d}}\frac{\int\left|D^{m}u_{n}^{2}\right|^{2}}{\int\left|u_{n}^{2}\right|^{2}}\geq|\omega_{n}^{2}|^{\frac{2m}{d}}\Gamma(\omega_{n}^{2})\geq
c^{\frac{2m}{d}}\inf\\{\Gamma(\Omega):\Omega\text{ 1-quasi-open,
}|\Omega|=c\\}=c^{\frac{2m}{d}}I>0.$

Now because $u_{n}^{2}$ is uniformly bounded in $H^{m}(\mathbb{R}^{d})$, and
because $\|u_{n}^{2}\|_{L^{2}}^{2}\to 1-\alpha\neq 0$, we end up with

$\liminf_{n}|\omega_{n}^{2}|>0.$

As a result, we have $\limsup_{n}|\omega_{n}^{1}|=:c^{\prime}<c$.
Consequently,
$\Omega_{n}^{\prime}:=\left(c/|\omega_{n}^{1}|\right)^{\frac{1}{d}}\omega_{n}^{1}$
is an admissible sequence made of $1$-quasi-open sets of volume $c$, hence the
following contradiction

$\liminf_{n}\Gamma(\omega_{n}^{1})\leq\liminf_{n}\Gamma(\Omega_{n})\leq\liminf_{n}\Gamma(\Omega_{n}^{\prime})=\left(c^{\prime}/c\right)^{\frac{2m}{d}}\liminf_{n}\Gamma(\omega_{n}^{1}).$

We have shown that neither vanishing nor dichotomy can occur. The only
remaining possibility is compactness. This means that there exists a sequence
of points $(y_{n})_{n}\subset\mathbb{R}^{d}$ such that for all $\epsilon>0$,
there exists some radius $R>0$ such that, for any $n$ large enough,

$\|\tau_{y_{n}}u_{n}\|_{L^{2}(B_{R})}\geq 1-\epsilon.$

First, for readability, considering $\Omega_{n}+y_{n}$ instead of
$\Omega_{n}$, it is possible to assume that $y_{n}=0$. Now, note that
$(u_{n})_{n}$ is bounded in $L^{2}(\mathbb{R}^{d})$, hence converges weakly to
some $u\in L^{2}(\mathbb{R}^{d})$. We will see that the convergence is also
$L^{2}$ strong. Indeed, let $(R_{k})_{k}$ be an increasing sequence of
radiuses such that, for each $k\in\mathbb{N}^{*}$, for each large
$n\in\mathbb{N},\|u_{n}\|_{L^{2}(B_{R_{k}})}\geq 1-1/k$. Note that for each
$k\in\mathbb{N}^{*}$, the sequence $(u_{n})_{n}$ is bounded in
$H^{m}(B_{R_{k}})$, hence we may extract diagonaly a subsequence converging
strongly in each $L^{2}(B_{R_{k}})$ to $u$. As a consequence,
$\|u\|_{L^{2}}\geq 1$ since for any $k\in\mathbb{N}^{*}$,
$\|u\|_{L^{2}}\geq\|u\|_{L^{2}(B_{R_{k}})}=\lim_{n}\|u_{n}\|_{L^{2}(B_{R_{k}})}\geq
1-1/k$. On the other hand, by weak convergence $\|u\|_{L^{2}}\leq 1$.
Therefore, in $L^{2}(\mathbb{R}^{d})$, the convergence holds simultaneously
weakly and in norm, hence strongly.

Weak $H^{m}$ and strong $L^{2}$ convergence of $(u_{n})_{n}$ towards $u$,
together with Proposition 13 show that it is possible to build a $1$-quasi-
open set $\Omega$, of volume lower or equal than $c$ such that
$\liminf_{n}\Gamma(\Omega_{n})\geq\Gamma(\Omega)$. Therefore, $\Omega$, or
more precisely its dilation fitting the volume constraint, solves (3). ∎

##  4 Regularity of an optimal shape

###  4.1 Quasi-minimizers

In optimization, a common strategy for getting regularity is to resort to the
notion of local quasi-minimizers222We mention that as far as we know, there is
no direct link between capacity and quasi-minimizers. In particular, the
prefix “quasi” of quasi-minimizers does not refer to any capacity. as defined
in [BMPV15], based on the ideas of [BHP05, AC81]. In this paragraph, we recall
the notion of local quasi-minimizer of a functional defined on
$H^{m}(\mathbb{R}^{d})$.

######  Definition 14.

Let $J:H^{m}(\mathbb{R}^{d})\to\mathbb{R}$ be a functional and $u\in
H^{m}(\mathbb{R}^{d})$. We say that $u$ is a local quasi-minimizer for $J$ if
there exists $C,\alpha,r_{0}>0$ such that for all $x_{0}\in\mathbb{R}^{d}$ and
$0<r<r_{0}$,

$J(u)\leq J(v)+Cr^{d},$ (13)

for all $v\in\mathcal{A}_{r,\alpha}:=\\{v\in H^{m}(\mathbb{R}^{d}):u-v\in
H_{0}^{m}(B_{r}(x_{0})),\int_{B_{r}(x_{0})}|D^{m}(u-v)|^{2}\leq\alpha\\}$.

######  Remark 15.

The constants $C$, $\alpha$, and $r_{0}$ may depend not only on $J$, but also
on $u$.

For the archetypal case of Dirichlet energy functionals, a central property of
quasi-minimizers is given in the next lemma.

######  Lemma 16.

Let $f\in L^{2}(\mathbb{R}^{d})$ and
$J_{f}:H^{m}(\mathbb{R}^{d})\to\mathbb{R}$ defined by

$J_{f}(v)=\frac{1}{2}\int_{\mathbb{R}^{d}}|D^{m}v|^{2}-\int_{\mathbb{R}^{d}}fv.$

Let $u\in H^{m}(\mathbb{R}^{d})$ be a local quasi-minimizer for $J_{f}$. Then,
there exists $C,r_{0}>0$ (depending upon $d$,$m$,$f$ and $u$) such that for
all $x_{0}\in\mathbb{R}^{d}$ and $0<r<r_{0}$,

$\langle(-\Delta)^{m}u-f,\varphi\rangle_{H^{-m},H_{0}^{m}}\leq
Cr^{d/2}\left(\int_{B_{r}(x_{0})}|D^{m}\varphi|^{2}\right)^{1/2},\qquad\forall\varphi\in
H_{0}^{m}(B_{r}(x_{0})).$ (14)

######  Remark 17.

Observe that, in the setting of Lemma 16, (14) is actually equivalent to (13)
(see Remark 3.3 and Remark 3.4 of [BMPV15]).

###### Proof.

Let $\psi\in H_{0}^{m}(B_{r}(x_{0}))$, such that
$\int_{B_{r}(x_{0})}|D^{m}\psi|^{2}\leq\alpha$. Because $\psi\in
H_{0}^{m}(B_{r}(x_{0}))$, we may integrate by parts (recall (6)) to obtain

$\langle(-\Delta)^{m}u-f,\psi\rangle_{H^{-m},H_{0}^{m}}=\int_{\mathbb{R}^{d}}D^{m}u:D^{m}\psi-\int_{\mathbb{R}^{d}}f\psi.$

On the other hand, $v:=u-\psi\in\mathcal{A}_{r,\alpha}$, hence from the quasi-
minimality of $u$ we get

$J_{f}(u)\leq
J_{f}(v)+Cr^{d}=J_{f}(u)+J_{f}(-\psi)-\int_{\mathbb{R}^{d}}D^{m}u:D^{m}\psi+Cr^{d},$

from which we deduce

$\int_{\mathbb{R}^{d}}D^{m}u:D^{m}\psi-\int_{\mathbb{R}^{d}}f\psi\leq\frac{1}{2}\int_{\mathbb{R}^{d}}|D^{m}\psi|^{2}+Cr^{d}.$

Now if $\varphi\in H_{0}^{m}(B_{r}(x_{0}))$, set
$\psi=\alpha^{1/2}\left(\frac{r}{r_{0}}\right)^{d/2}\|D^{m}\varphi\|_{L^{2}(B_{r}(x_{0}))}^{-1}\varphi$
so that $\psi\in H_{0}^{m}(B_{r}(x_{0}))$ and
$\int_{B_{r}(x_{0})}|D^{m}\psi|^{2}\leq\alpha$. The previous discussion shows
that

$\alpha^{1/2}\left(\frac{r}{r_{0}}\right)^{d/2}\|D^{m}\varphi\|_{L^{2}(B_{r}(x_{0}))}^{-1}\langle(-\Delta)^{m}u-f,\varphi\rangle_{H^{-m},H_{0}^{m}}\leq\frac{\alpha}{2}\left(\frac{r}{r_{0}}\right)^{d}+Cr^{d}.$

In other words, there exists $C^{\prime}>0$ such that for all $r<r_{0}$,

$\langle(-\Delta)^{m}u-f,\varphi\rangle_{H^{-m},H_{0}^{m}}\leq
C^{\prime}r^{d/2}\left(\int_{B_{r}(x_{0})}|D^{m}\varphi|^{2}\right)^{1/2},$

which concludes. ∎

###  4.2 Proof of Theorem 5

To prove Theorem 5, the first step is to gain regularity on an optimal
eigenfunction. For that, the concept of local quasi-minimizer provides an
approriate framework. Indeed, the following result gives the regularity of
local quasi-minimizers of Dirichlet energy functionals. The regularity
obtained is not sharp in general, as one shall see in [BMPV15, Theorem 3.6].

######  Theorem 18.

Assume $d\geq 2$. Let $f\in L^{2}\cap L_{loc}^{p}(\mathbb{R}^{d})$,
$p\geq\frac{d}{m}$ and $J_{f}:H^{m}(\mathbb{R}^{d})\to\mathbb{R}$ defined by

$J_{f}(v)=\frac{1}{2}\int_{\mathbb{R}^{d}}|D^{m}v|^{2}-\int_{\mathbb{R}^{d}}fv.$
(15)

Let $u\in H^{m}(\mathbb{R}^{d})$ be a local quasi-minimizer for $J_{f}$. Then,
$u\in C^{m-1,\alpha}(\mathbb{R}^{d})$ for all $0<\alpha<1$.

###### Proof.

We proceed in a similar way than [Sto22, Theorem 3 to Theorem 4]. More
precisely, our goal is to prove that for all $\alpha\in]0,1[$, there exists
$R_{0},C>0$ (depending on $d$,$m$,$p$,$f$,$u$, and $\alpha$) such that, for
all $x_{0}\in\mathbb{R}^{d}$ and $0<r<R_{0}$,

$\int_{B_{r}(x_{0})}|D^{m}u|^{2}\leq Cr^{d-2+2\alpha}.$ (16)

Indeed, due to Morrey’s Theorem (see [Gia83, Chapter III, Theorem 1.1]), this
yields that $D^{m-1}u\in C^{0,\alpha}$ for all $\alpha\in]0,1[$. In order to
show (16), we let $x_{0}\in\mathbb{R}^{d}$, $0<R$ and define $v_{x_{0},R}$
such that $u-v_{x_{0},R}\in H_{0}^{m}(B_{R}(x_{0}))$ and
$(-\Delta)^{m}v_{x_{0},R}=f$ in $B_{R}(x_{0})$. Such a function might be
constructed by minimizing $J_{f}$ over the set $u-H_{0}^{m}(B_{R}(x_{0}))$.
Now we will estimate the two terms in the right-hand side of

$\int_{B_{r}(x_{0})}|D^{m}u|^{2}\leq
2\int_{B_{r}(x_{0})}|D^{m}(u-v_{x_{0},R})|^{2}+2\int_{B_{r}(x_{0})}|D^{m}v_{x_{0},R}|^{2}.$
(17)

We will control the first term in the larger ball $B_{R}(x_{0})$ proceeding as
in the proof of [BMPV15, inequality (A.8)]. Indeed, integrating by parts, and
then using the definition of $v_{x_{0},R}$, we get

$\displaystyle\int_{B_{R}(x_{0})}|D^{m}(u-v_{x_{0},R})|^{2}$
$\displaystyle=\langle(-\Delta)^{m}(u-v_{x_{0},R}),(u-v_{x_{0},R})\rangle_{H^{-m},H_{0}^{m}}$
$\displaystyle=\langle(-\Delta)^{m}u-f,(u-v_{x_{0},R})\rangle_{H^{-m},H_{0}^{m}}.$
(18)

The quasi-minimality of $u$ and Lemma 16 show that there exists $C,R_{0}>0$
such that, as long as $R<R_{0}$,

$\langle(-\Delta)^{m}u-f,(u-v_{x_{0},R})\rangle_{H^{-m},H_{0}^{m}}\leq
CR^{d/2}\left(\int_{B_{R}(x_{0})}|D^{m}(u-v_{x_{0},R})|^{2}\right)^{1/2}.$
(19)

Finally, combining (4.2) and (19), we obtain (still for all $R<R_{0}$),

$\int_{B_{R}(x_{0})}|D^{m}(u-v_{x_{0},R})|^{2}\leq CR^{d}.$ (20)

For the second term in (17), let us decompose $v_{x_{0},R}$ into $y_{x_{0},R}$
and $z_{x_{0},R}$ , which are defined in the following way: $y_{x_{0},R}$ is
the function in $H_{0}^{m}(B_{R}(x_{0}))$ such that
$(-\Delta)^{m}y_{x_{0},R}=f$; and $z_{x_{0},R}$ is the $m$-polyharmonic
function such that $v_{x_{0},R}-z_{x_{0},R}\in H_{0}^{m}(B_{R}(x_{0}))$.
Notice that, using the definition of $y_{x_{0},R}$, Cauchy-Schwarz inequality,
and relation (10), we find

$\begin{split}\int_{B_{R}(x_{0})}|D^{m}y_{x_{0},R}|^{2}=\int_{B_{R}(x_{0})}fy_{x_{0},R}&\leq\|f\|_{L^{2}(B_{R}(x_{0}))}\|y_{x_{0},R}\|_{L^{2}(B_{R}(x_{0}))}\\\
&\leq
CR^{m}\|f\|_{L^{2}(B_{R}(x_{0}))}\|D^{m}y_{x_{0},R}\|_{L^{2}(B_{R}(x_{0}))}.\end{split}$

In particular, using Hölder inequality (with $q=p/2$) and the fact that $f\in
L^{p}(B_{R_{0}}(x_{0}))$,

$\int_{B_{R}(x_{0})}|D^{m}y_{x_{0},R}|^{2}\leq
CR^{2m}\int_{B_{R}(x_{0})}f^{2}\leq CR^{2m+d(1-2/p)}\leq CR^{d},$ (21)

the last inequality coming from $m-d/p\geq 0$ and $R\leq R_{0}$. On the other
hand, using inequality (3.2) of [GM79] (see also [Sto22, Lemma 2]), we obtain
the existence of $C>0$ such that, for all $0<r<R<R_{0}$,

$\int_{B_{r}(x_{0})}|D^{m}z_{x_{0},R}|^{2}\leq
C\left(\frac{r}{R}\right)^{d}\int_{B_{R}(x_{0})}|D^{m}z_{x_{0},R}|^{2}.$

Because $z_{x_{0},R}$ minimizes $J_{0}$ amongst functions $z$ such that
$v_{x_{0},R}-z\in H_{0}^{m}(B_{R}(x_{0}))$, the previous turns into the next

$\int_{B_{r}(x_{0})}|D^{m}z_{x_{0},R}|^{2}\leq
C\left(\frac{r}{R}\right)^{d}\int_{B_{R}(x_{0})}|D^{m}u|^{2}.$ (22)

Combining (21) and (22), we find some $C>0$ such that for all $0<r<R<R_{0}$,

$\int_{B_{r}(x_{0})}|D^{m}v_{x_{0},R}|^{2}\leq
C\left(R^{d}+\left(\frac{r}{R}\right)^{d}\int_{B_{R}(x_{0})}|D^{m}u|^{2}\right).$
(23)

Thanks to (20) and (23), we can now conclude that for some $C>0$ and for all
$0<r<R<R_{0}$,

$\int_{B_{r}(x_{0})}|D^{m}u|^{2}\leq
C\left[R^{d}+\left(\frac{r}{R}\right)^{d}\int_{B_{R}(x_{0})}|D^{m}u|^{2}\right].$
(24)

In view of [Gia83, Chapter III, Lemma 2.1, p.86] applied to
$\phi(r)=\int_{B_{r}(x_{0})}|D^{m}u|^{2}$ with $\alpha=d$ and $\beta=d-\gamma$
for some $0<\gamma<d$, we conclude that, for all $0<r<R<R_{0}$,

$\int_{B_{r}(x_{0})}|D^{m}u|^{2}\leq
C\left(\frac{r}{R}\right)^{d-\gamma}\left[\int_{B_{R}(x_{0})}|D^{m}u|^{2}+R^{d-\gamma}\right],$

where $C$ depends as usual on $d$,$m$,$p$,$f$,$u$, but now also on $\gamma$.
Fixing for instance $R=R_{0}/2$, we end up with

$\int_{B_{r}(x_{0})}|D^{m}u|^{2}\leq Cr^{d-\gamma},$

for all $r<R_{0}/4$, which is exactly (16) as long as we set
$\gamma=2-2\alpha<2\leq d$ where $0<\alpha<1$. ∎

Thanks to Theorem 18, it remains only to show that an optimal eigenfunction is
a local quasi-minimizer, which is performed below and allows to conclude the
proof of Theorem 5.

###### Proof of Theorem 5.

To begin with, we explain the last part of Theorem 5, which asserts that, as
long as an eigenfunction $u$ associated with $\Gamma(\Omega)$ is
$C^{m-1,\alpha}(\mathbb{R}^{d})$, we shall construct an open set
$\tilde{\Omega}$ which is optimal. Indeed, we proceed in a similar fashion
than in the proof of [BMPV15, Theorem 6.1], and define

$\tilde{\Omega}:=\bigcup_{0\leq k<m}\\{D^{k}u\neq 0\\}.$

Then, $\tilde{\Omega}$ is open and, by Lemma 9, $u\in H_{0}^{m}(\Omega)$. In
particular, $\Gamma(\tilde{\Omega})\leq\Gamma(\Omega)$. It remains to check
that $\tilde{\Omega}$ satisfies the volume constraint. But since $u\in
H_{0}^{m}(\Omega)$, we know that, for each $k<m$, $\\{D^{k}u\neq 0\\}$ is
included in $\Omega$ up to a set of null $(m-k)$-capacity, and hence
$\tilde{\Omega}\subseteq\Omega$ up to a set of null $1$-capacity. As a
consequence, $|\tilde{\Omega}|\leq|\Omega|$, and $\tilde{\Omega}$ satisfies
the volume constraint up to a dilation, which concludes.

Let us now turn to the proof of the fact that $u$ is
$C^{m-1,\alpha}(\mathbb{R}^{d})$. In view of Theorem 18, it is enough to show
that $u$ is a local quasi-minimizer of the functional $J_{f}$ defined in (15)
for some $f\in L^{d/m}(\mathbb{R}^{d})$. For that purpose, we introduce the
concept of quasi-minimizer of a shape functional: we say that $\Omega$ is a
local quasi-minimizer of the functional $\Gamma$ if there exists $C,r_{0}>0$
such that, for all $x_{0}\in\mathbb{R}^{d}$ and $0<r<r_{0}$, and for any
measurable set $\Omega^{\prime}$,

$\Omega^{\prime}\Delta\Omega\subseteq
B_{r}(x_{0})\quad\Longrightarrow\quad\Gamma(\Omega)\leq\Gamma(\Omega^{\prime})+Cr^{d},$

where $\Omega^{\prime}\Delta\Omega$ is the symmetric difference between
$\Omega$ and $\Omega^{\prime}$. As shown in [BMPV15, Lemma 4.6], the local
quasi-minimality of $u$ with respect to $J_{f}$ shall be deduced from the
local quasi-minimality of $\Omega$ with respect to $\Gamma$. Indeed, without
loss of generality, we assume $u$ to be $L^{2}$ normalised. Then, as long as
$\Omega$ is a local quasi-minimizer, for any $v\in H^{m}(\mathbb{R}^{d})$ with
$v-u\in H_{0}^{m}(B_{r}(x_{0}))$ and $r\in]0,r_{0}[$, we have that $v\in
H_{0}^{m}(\Omega\cup B_{r}(x_{0}))$, hence, using the variational definition
of $\Gamma(\Omega\cup B_{r}(x_{0}))$,

$\Gamma(\Omega)\leq\Gamma(\Omega\cup
B_{r}(x_{0}))+Cr^{d}\leq\frac{\int_{\mathbb{R}^{d}}|D^{m}v|^{2}}{\int_{\mathbb{R}^{d}}v^{2}}+Cr^{d},$
(25)

Now let $\alpha>0$ and take an arbitrary $v\in\mathcal{A}_{r,\alpha}$ (recall
Definition 14). Setting $\varphi:=v-u$, because $u$ is normalised, $r<r_{0}$
and $\int_{B_{r}(x_{0})}|D^{m}\varphi|^{2}<\alpha$ we have

$\int_{\mathbb{R}^{d}}v^{2}\leq
2\int_{\mathbb{R}^{d}}u^{2}+\frac{2\alpha}{\Gamma(B_{r_{0}}(x_{0}))}\leq C,$

hence after multiplying (25) by $\int_{\mathbb{R}^{d}}v^{2}$ we obtain

$\Gamma(\Omega)\int_{\mathbb{R}^{d}}(u+\varphi)^{2}\leq\int_{\mathbb{R}^{d}}|D^{m}(u+\varphi)|^{2}+Cr^{d}.$

Expand both sides and find

$\Gamma(\Omega)\int_{\mathbb{R}^{d}}u\varphi-\frac{1}{2}\int_{\mathbb{R}^{d}}|D^{m}\varphi|^{2}-\int_{\mathbb{R}^{d}}D^{m}u:D^{m}\varphi\leq
Cr^{d}.$

Observe that the left-hand member corresponds exactly to $J_{f}(u)-J_{f}(v)$
for $f=\Gamma(\Omega)u$. As a result, $u$ is a local quasi-minimizer of
$J_{f}$ with $f=\Gamma(\Omega)u$. At this point, remark that thanks to Sobolev
injections, $u\in L^{p}(\Omega)$ for $p=2d/(d-2m)$. Consequently, the
restriction $d\leq 4m$ yields $p\geq d/m$. Thanks to Theorem 18, $u$ is then
$C^{m-1,\alpha}(\mathbb{R}^{d})$. Thus, it remains only to show the local
quasi-minimality of $\Omega$ with respect to $\Gamma$. As mentioned in the
second point of Remark 5.2 in [BMPV15], this will follow if we show that
$\Omega$ is a local shape supersolution for $\Gamma+\Lambda|\cdot|$ for some
$\Lambda>0$, or, in other words, that there is some $r_{0}>0$ such that for
all $x_{0}\in\mathbb{R}^{d}$ and all measurable set
$\Omega\subseteq\Omega^{\prime}\subseteq\Omega\cup B_{r_{0}}(x_{0})$,

$\Gamma(\Omega)+\Lambda|\Omega|\leq\Gamma(\Omega^{\prime})+\Lambda|\Omega^{\prime}|.$

Indeed, if $\Omega$ is such a shape super-solution, for any
$x_{0}\in\mathbb{R}^{d}$, any $0<r<r_{0}$, and any measurable set
$\Omega^{\prime}$ with $\Omega^{\prime}\Delta\Omega\subseteq B_{r}(x_{0})$, we
have that $\Omega\subseteq\Omega^{\prime}\cup\Omega\subseteq\Omega\cup
B_{r_{0}}(x_{0})$, hence

$\Gamma(\Omega^{\prime})\geq\Gamma(\Omega^{\prime}\cup\Omega)\geq\Gamma(\Omega)+\Lambda|\Omega|-\Lambda|\Omega^{\prime}\cup\Omega|\geq\Gamma(\Omega)-\Lambda|\Omega^{\prime}\setminus\Omega|,$

which yields as expected

$\Gamma(\Omega)\leq\Gamma(\Omega^{\prime})+\Lambda|B_{r}(x_{0})|.$

To conclude, let us show that $\Omega$ is a shape super-solution. This is a
consequence of the optimality of $\Omega$, as explained in the proof of
[BMPV15, Theorem 6.1], since for any
$\Omega\subseteq\Omega^{\prime}\subseteq\Omega\cup B_{r_{0}}(x_{0})$, setting
$t=\left(|\Omega^{\prime}|/|\Omega|\right)^{\frac{1}{d}}$ and recalling (10),

$\Gamma(\Omega)\leq\Gamma(\Omega^{\prime}/t)\leq
t^{2m}\Gamma(\Omega^{\prime})\leq\Gamma(\Omega^{\prime})+(t^{2m}-1)\Gamma(\Omega^{\prime})\leq\Gamma(\Omega^{\prime})+C_{r_{0}}(t^{d}-1)\Gamma(\Omega)=\Gamma(\Omega^{\prime})+\Lambda(|\Omega^{\prime}|-|\Omega|),$

with $\Lambda=C_{r_{0}}\Gamma(\Omega)/|\Omega|$. Note that, besides
L’Hôpital’s rule, we used the fact that $t$ is bounded irrespective of
$\Omega^{\prime}\subseteq\Omega\cup B_{r_{0}}(x_{0})$ in order to get the
bound $t^{2m}-1\leq C_{r_{0}}(t^{d}-1)$. ∎

##  5 Conclusion

Fortified by Theorem 3 and Theorem 5, we shall propose a proof for Theorem 1.

###### Proof of Theorem 1.

Let $2\leq d\leq 4m$. Due to Theorem 3, there exists a $1$-quasi-open set
solving (3). From Lemma 11 we deduce that this set also solves (4). Then, as
desired, Theorem 5 produces an open set $\Omega$ solving (2). Moreover,
$\Omega$ also solves (4) by Lemma 11. Applying again Theorem 5, we conclude
that any first eigenfunction on $\Omega$ is $C^{m-1,\alpha}(\mathbb{R}^{d})$.
∎

Theorem 1 is of course an advance in the context of shape optimization.
However, as stated in Remark 2, it has several limits besides the dimensional
restriction. Indeed, in optimization, existence results are often used as a
preliminary step in the derivation of optimality conditions in view of
identifying an optimizer. Nevertheless, in shape optimization, some regularity
is frequently needed on the optimal shape to apply the theory of shape
derivatives in order to derive optimality conditions. See [HP05, Chapter 5]
for a general theory on shape derivatives. See for instance [BL13, BL15, OZ00,
Moh75], [ABF19, section 5], and [Ley23, section 3] for results on shape
derivatives involving fourth and higher order operators. Finally, see [AB03]
for an example where shape derivatives are performed without regularity
assumption on the optimal shape.

Unfortunately, as mentioned in Remark 2, the optimal shape of Theorem 1 enjoys
only little regularity. A first step towards upgrading it would be to prove
boundedness. Some results are known in this direction, but only in the case of
second order operators, up to our knowledge (see [Buc12, MP13]). After
boundedness, the question of finer regularity of the optimal set shall be
tackled. See [LP17] for a review on this subject. The main results known are
covered by some works of Lamboley et al. [BL09, DLPV18, LP23]. However, as far
as we know, those results hold only in special situations (for instance $m=1$
with [BL09]), or when ancillary constraints allow to gain regularity (see
[DLPV18, LP23]). To conclude let us mention that [Sto21] also contains some
hints on this topic in the case $m=2$ (see for instance [Sto21, Lemma 9]).

## Acknowledgements

Let me thank Dorin Bucur for suggestions about building an open optimal set.
Thank you to Michel Pierre and Antoine Lemenant for interesting discussions
and references about capacities of higher order. Lastly, I would like to thank
Enea Parini and François Hamel for their help and their useful remarks.

## References

  * [AH96] David R. Adams and Lars I. Hedberg  “Function spaces and potential theory” 314, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]  Springer-Verlag, Berlin, 1996, pp. xii+366  DOI: 10.1007/978-3-662-03282-4
  * [AC81] Hans W. Alt and Luis A. Caffarelli  “Existence and regularity for a minimum problem with free boundary”  In _J. Reine Angew. Math._ 325, 1981, pp. 105–144
  * [ABF19] Pedro R.. Antunes, Davide Buoso and Pedro Freitas  “On the behavior of clamped plates under large compression”  In _SIAM J. Appl. Math._ 79.5, 2019, pp. 1872–1891  DOI: 10.1137/19M1249606
  * [AB95] Mark S. Ashbaugh and Rafael D. Benguria  “On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions”  In _Duke Math. J._ 78.1, 1995, pp. 1–17  DOI: 10.1215/S0012-7094-95-07801-6
  * [AB03] Mark S. Ashbaugh and Dorin Bucur  “On the isoperimetric inequality for the buckling of a clamped plate” Special issue dedicated to Lawrence E. Payne  In _Z. Angew. Math. Phys._ 54.5, 2003, pp. 756–770  DOI: 10.1007/s00033-003-3204-3
  * [BHP05] Tanguy Briançon, Mohammed Hayouni and Michel Pierre  “Lipschitz continuity of state functions in some optimal shaping”  In _Calc. Var. Partial Differential Equations_ 23.1, 2005, pp. 13–32  DOI: 10.1007/s00526-004-0286-5
  * [BL09] Tanguy Briançon and Jimmy Lamboley  “Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints”  In _Ann. Inst. H. Poincaré C Anal. Non Linéaire_ 26.4, 2009, pp. 1149–1163  DOI: 10.1016/j.anihpc.2008.07.003
  * [Buc05] Dorin Bucur  “How to prove existence in shape optimization”  In _Control Cybernet._ 34.1, 2005, pp. 103–116
  * [Buc12] Dorin Bucur  “Minimization of the $k$-th eigenvalue of the Dirichlet Laplacian”  In _Arch. Ration. Mech. Anal._ 206.3, 2012, pp. 1073–1083  DOI: 10.1007/s00205-012-0561-0
  * [BBV13] Dorin Bucur, Giuseppe Buttazzo and Bozhidar Velichkov  “Spectral optimization problems with internal constraint”  In _Ann. Inst. H. Poincaré C Anal. Non Linéaire_ 30.3, 2013, pp. 477–495  DOI: 10.1016/j.anihpc.2012.10.002
  * [BMPV15] Dorin Bucur, Dario Mazzoleni, Aldo Pratelli and Bozhidar Velichkov  “Lipschitz regularity of the eigenfunctions on optimal domains”  In _Arch. Ration. Mech. Anal._ 216.1, 2015, pp. 117–151  DOI: 10.1007/s00205-014-0801-6
  * [BL13] Davide Buoso and Pier D. Lamberti  “Eigenvalues of polyharmonic operators on variable domains”  In _ESAIM Control Optim. Calc. Var._ 19.4, 2013, pp. 1225–1235  DOI: 10.1051/cocv/2013054
  * [BL15] Davide Buoso and Pier D. Lamberti  “On a classical spectral optimization problem in linear elasticity”  In _New trends in shape optimization_ 166, Internat. Ser. Numer. Math.  Birkhäuser/Springer, Cham, 2015, pp. 43–55  DOI: 10.1007/978-3-319-17563-8\\_3
  * [Dav90] Edward B. Davies  “Heat kernels and spectral theory” 92, Cambridge Tracts in Mathematics  Cambridge University Press, Cambridge, 1990, pp. x+197
  * [DLPV18] Guido De Philippis, Jimmy Lamboley, Michel Pierre and Bozhidar Velichkov  “Regularity of minimizers of shape optimization problems involving perimeter”  In _J. Math. Pures Appl. (9)_ 109, 2018, pp. 147–181  DOI: 10.1016/j.matpur.2017.05.021
  * [Fab23] Georg Faber  “Beweis, daß unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt.”, Münch. Ber. 1923, 169-172 (1923)., 1923
  * [GGS10] Filippo Gazzola, Hans-Christoph Grunau and Guido Sweers  “Polyharmonic boundary value problems” Positivity preserving and nonlinear higher order elliptic equations in bounded domains 1991, Lecture Notes in Mathematics  Springer-Verlag, Berlin, 2010, pp. xviii+423  DOI: 10.1007/978-3-642-12245-3
  * [Gia83] Mariano Giaquinta  “Multiple integrals in the calculus of variations and nonlinear elliptic systems” 105, Annals of Mathematics Studies  Princeton University Press, Princeton, NJ, 1983, pp. vii+297
  * [GM79] Mariano Giaquinta and Giuseppe Modica  “Regularity results for some classes of higher order nonlinear elliptic systems”  In _J. Reine Angew. Math._ 311/312, 1979, pp. 145–169
  * [GLM23] Jean-François Grosjean, Antoine Lemenant and Rémy Mougenot  “Stable domains for higher order elliptic operators” Preprint at https://arxiv.org/abs/2307.07217 arXiv, 2023  arXiv:2307.07217
  * [HP05] Antoine Henrot and Michel Pierre  “Variation et optimisation de formes” Une analyse géométrique. 48, Mathématiques & Applications (Berlin)  Springer, Berlin, 2005, pp. xii+334  DOI: 10.1007/3-540-37689-5
  * [Kra25] Edgar Krahn  “Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises”  In _Math. Ann._ 94.1, 1925, pp. 97–100  DOI: 10.1007/BF01208645
  * [LP17] Jimmy Lamboley and Michel Pierre  “Regularity of optimal spectral domains”  In _Shape optimization and spectral theory_ De Gruyter Open, Warsaw, 2017, pp. 29–77  DOI: 10.1515/9783110550887-003
  * [LP23] Jimmy Lamboley and Raphaël Prunier  “Regularity in shape optimization under convexity constraint”  In _Calc. Var. Partial Differential Equations_ 62.3, 2023, pp. Paper No. 10144  DOI: 10.1007/s00526-023-02440-7
  * [Ley23] Roméo Leylekian  “Sufficient conditions yielding the Rayleigh Conjecture for the clamped plate” Preprint at https://arxiv.org/abs/2302.06313 arXiv, 2023  arXiv:2302.06313
  * [Lio84] Pierre-Louis Lions  “The concentration-compactness principle in the calculus of variations. The locally compact case. I”  In _Ann. Inst. H. Poincaré Anal. Non Linéaire_ 1.2, 1984, pp. 109–145  URL: http://www.numdam.org/item?id=AIHPC_1984__1_2_109_0
  * [MP13] Dario Mazzoleni and Aldo Pratelli  “Existence of minimizers for spectral problems”  In _J. Math. Pures Appl. (9)_ 100.3, 2013, pp. 433–453  DOI: 10.1016/j.matpur.2013.01.008
  * [Moh75] Ernst Mohr  “Über die Rayleighsche Vermutung: unter allen Platten von gegebener Fläche und konstanter Dichte und Elastizität hat die kreisförmige den tiefsten Grundton”  In _Ann. Mat. Pura Appl. (4)_ 104, 1975, pp. 85–122  DOI: 10.1007/BF02417012
  * [Nad95] Nikolai S. Nadirashvili  “Rayleigh’s conjecture on the principal frequency of the clamped plate”  In _Arch. Rational Mech. Anal._ 129.1, 1995, pp. 1–10  DOI: 10.1007/BF00375124
  * [OZ00] Jaime H. Ortega and Enrique Zuazua  “Generic simplicity of the spectrum and stabilization for a plate equation”  In _SIAM J. Control Optim._ 39.5, 2000, pp. 1585–1614  DOI: 10.1137/S0363012900358483
  * [Sto21] Kathrin Stollenwerk  “Existence of an optimal domain for minimizing the fundamental tone of a clamped plate of prescribed volume in arbitrary dimension” Preprint at https://arxiv.org/abs/2109.01455 arXiv, 2021  arXiv:2109.01455
  * [Sto22] Kathrin Stollenwerk  “Existence of an optimal domain for the buckling load of a clamped plate with prescribed volume”  In _Ann. Mat. Pura Appl. (4)_ 201.4, 2022, pp. 1677–1704  DOI: 10.1007/s10231-021-01175-6
  * [Sto23] Kathrin Stollenwerk  “On the optimal domain for minimizing the buckling load of a clamped plate”  In _Z. Angew. Math. Phys._ 74.1, 2023, pp. Paper No. 1216  DOI: 10.1007/s00033-022-01898-3


††thanks: ORCID 0000-0001-9980-0363††thanks: ORCID
0000-0001-9896-1018††thanks: ORCID 0000-0003-1168-7883

# Heider balance on Archimedean lattices

Krzysztof Malarz  Maciej Wołoszyn<EMAIL_ADDRESS>Krzysztof Kułakowski
AGH University, Faculty of Physics and Applied Computer Science, al.
Mickiewicza 30, 30-059 Kraków, Poland

###### Abstract

The phenomenon of Heider (structural) balance is known for a long time (P.
Bonacich and P. Lu, Introduction to Mathematical Sociology, Princeton UP,
2012). Yet it attracts attention of numerous computational scholars, as it is
an example of a macroscopic ordering which emerges as a consequence of local
interactions. In this paper, we investigate the thermal evolution (driven by
thermal noise level $T$) of the work function $U(T)$ for Heider balance on
several Archimedean lattices that contain separated triangles, pairs of
triangles, chains of triangles and complex structures of triangles. To that
end, the heat-bath algorithm is applied. Two schemes of link values updating
are considered: synchronous and asynchronous. In the latter case, the
analytical formula $U(T)=-\tanh(1/T)$ based on the partition function is
provided. The Archimedean lattices are encoded with adjacency matrices, and
Fortran procedures for their construction are provided. Finally, we present
the mathematical proof that for any two-dimensional lattice, perfect
structural (Heider) balance is unreachable at $T>0$.

##  I Introduction

The sociophysical problem of structural balance [1, 2, 3, 4, 5, 6, 7].
governed by Heider’s algebra [1]:

  * •

a friend of my friend is my friend;

  * •

a friend of my enemy is my enemy;

  * •

an enemy of my friend is my enemy;

  * •

an enemy of my enemy is my friend

brought attention of physicists (see Refs. 8, 9 for recent reviews) as a three
body problem with Hamiltonian

$\mathcal{H}=-\sum_{i}\sum_{j>i}\sum_{k>j}a_{ij}x_{ij}\cdot a_{jk}x_{jk}\cdot
a_{ki}x_{ki},$ (1)

where $x_{ij}=\pm 1$ represent friendly and hostile relations between actors
$i$ and $j$ and $a_{ij}\in\\{0,1\\}$ define the network/lattice topology.

(a)

(b)

(c)

(d)

Figure 1: Heider’s triads corresponding to balanced (1(a) and 1(c)) and
imbalanced (1(b) and 1(d)) states

When in every triangle these relations obey Heider’s rules mentioned above,
the system is termed as balanced in Heider’s sense—which means that only
triangles presented in Figures 1(a) and 1(c) are observed in the system.

The earlier studies have been devoted to triangular [10], diluted [11] and
densified triangulations [12]; chains of nodes [13]; classical random graphs
[14]; and complete graphs [15, 16].

Recently, reaching structural balance has been investigated with thermal noise
modeled either with heat-bath [17, 15, 16] or Glauber dynamics [18, 19]. A
more refined dynamics based on the indirect reciprocity mechanism on a
complete graph has been studied in Reference [20].

In this paper, we continue studies of the influence of thermal noise on the
possibility of reaching structural balance on Archimedean lattices containing
triangles (see Figure 2). There are eleven Archimedean lattices [21], which
contain only regular polygons allowing for covering plane with tiles and
keeping the translational symmetry of the lattice. The systematic names of
these lattices come from the number of edges in tiles attached to every
lattice node sorted in the possibly lowest lexicographic order—for example, we
call lattice presented in Figure 2(b) $(3^{2},4,3,4)$ and not $(4,3^{2},4,3)$
or $(4,3,4,3^{2})$. In this way, the square lattice is called $(4^{4})$, the
triangular lattice is called $(3^{6})$, the honeycomb lattice is called
$(6^{3})$ and kagomé lattice is called $(3,6,3,6)$.

(a)

(b)

(c)

(d)

(e)

Figure 2: Shapes of the Archimedean lattices: (2(a)) $(3^{6})$; (2(b))
$(3^{2},4,3,4)$; (2(c)) $(3^{3},4^{2})$; (2(d)) $(3^{4},6)$; (2(e))
$(3,6,3,6)$

The Archimedean lattices investigated here are:

  * •

separated triangles: kagomé—$(3,6,3,6)$ lattice, Figure 2(e);

  * •

separated pairs of triangles: $(3^{2},4,3,4)$ lattice, Figure 2(b);

  * •

chains of triangles: $(3^{3},4^{2})$ lattice, Figure 2(c);

  * •

and segments of chains of triangles aligned on honeycomb lattice as a back-
bone: maple leaf—$(3^{4},6)$ lattice, Figure 2(d).

Additionally, we inspect the system evolution for triangular $(3^{6})$ lattice
(Figure 2(a)), diluted triangulations, and classical random graphs with mean
node degree equal to $k=4$ and $k=5$, which also characterize the above-
mentioned lattices. In Table 1 characteristics of the Archimedean lattices
discussed in this work are collected. The social noise is modeled by the
temperature parameter $T$ in the heat-bath scheme.

Our calculations are performed both with asynchronous and synchronous scheme.
The former is known to reproduce properly the thermodynamic properties of
systems in thermal equilibrium. The latter is justified if we intend to mimic
the dynamics of opinions gathered periodically in polls. In this case, a
respondent is not aware of the answers of other respondents during the same
round. Such procedure of data collection has been applied in [22], to give an
example.

The paper is organized as follows: in Section II we describe the methodology
of studies (including the creation of adjacency matrices for the Archimedean
lattices mentioned above, the heat-bath algorithm and calculation of the work
function), Section III contains results of Monte Carlo simulations and Section
IV is devoted to discussion of the obtained results and presentation of
mathematical proof that structurally ordered phase is limited to $T=0$ on any
two-dimensional lattice.

The paper is finalized with Appendix A showing lattices and adjacency matrices
shapes and Appendix B with Fortran95 procedures for creating adjacency
matrices for the lattices considered here.

##  II Methods

We keep the simulation methodology as presented for diluted/densified
triangulations and classical random graphs [11, 12, 14]. In this approach (1),
every link $x_{ij}$ represents friendly ($x_{ij}=+1$) or hostile ($x_{ij}=-1$)
attitudes between actors $i$ and $j$, while (binary and symmetric) adjacency
matrix $\mathbf{A}=[a_{ij}]$ elements

$a_{ij}=\begin{cases}1&\iff i\text{ and }j\text{ are connected},\\\
0&\iff\text{otherwise},\end{cases}$ (2)

define geometry of the lattice.

The Heider balance can be easily identified by checking the system work
function [23, 24] per number of triangles $\Delta$ on lattice

$U/\Delta\equiv-\frac{\operatorname{\mathrm{Tr}}[(\mathbf{A}\circ\mathbf{X})^{3}]}{\operatorname{\mathrm{Tr}}(\mathbf{A}^{3})},$
(3)

where $\circ$ stands for the Hadamard product of the matrices and the matrix
$\mathbf{X}=[x_{ij}]$. The system work function per triangle $U/\Delta$ is
equal to $-1$ if and only if all triangles in the system are balanced.

###  II.1 Adjacency matrix representation of Archimedean lattices

In Appendix A in Figures 13(a), 10(a), 11(a), 12(a) and 9(a) the equivalents
of the lattice shapes presented in Figure 2 are mapped onto distorted
triangular lattices embedded in square lattices. The lattice edges are printed
in thick solid black lines, while the underlying triangular lattice is
presented in thin solid blue lines. The ‘unit cell’ is marked with a green
dashed line. The periodic boundary conditions are assumed, and replicas of
‘elementary cell’ are marked with dotted cyan lines.

The Fortran procedures providing $\mathbf{A}$ for considered lattices are
presented in Appendix B. Independently of the considered lattice, the module
‘par’ (see LABEL:lst:module-par in Appendix B) is required to define the
linear size of the system $W$ and the lattice connectivity $k$.

Table 1: Characteristics of some Archimedean lattices. The second column indicates the lattice connectivity $k$. References to figures showing lattices’ shapes (and their shapes embedded in a distorted triangular lattice) are given in the third and the fourth columns. Figures showing the shapes of the adjacency matrix $\mathbf{A}$ are indicated in the fifth column. In the sixth column the labels of listings supplying $\mathbf{A}$ are listed lattice | $k$ | shape | shape in Appendix A | matrix $\mathbf{A}$ in Appendix A | subroutine in Appendix B
---|---|---|---|---|---
$(3^{6})$ | 6 | Figure 2(a) | Figure 9(a) | Figure 9(b) | LABEL:lst:code-triangular
$(3^{2},4,3,4)$ | 5 | Figure 2(b) | Figure 10(a) | Figure 10(b) | LABEL:lst:code-tri2al33434
$(3^{3},4^{2})$ | 5 | Figure 2(c) | Figure 11(a) | Figure 11(b) | LABEL:lst:code-al33344
$(3^{4},6)$ | 5 | Figure 2(d) | Figure 12(a) | Figure 12(b) | LABEL:lst:code-tri2mapleleaf
$(3,6,3,6)$ | 4 | Figure 2(e) | Figure 13(a) | Figure 13(b) | LABEL:lst:code-tri2kagome

###  II.2 Heat-bath

The social noise level is mimicked by the temperature $T$ introduced in the
probabilities of setting a hostile or friendly relation between actors. If
$a_{ij}\neq 0$ then evolution of the link between nodes $i$ and $j$ of value
$x_{ij}$ is given by

$x_{ij}(t+1)=\begin{cases}+1&\text{ with probability }p_{ij}(t),\\\ -1&\text{
with probability }[1-p_{ij}(t)],\end{cases}$ (4a) where
$p_{ij}(t)=\frac{\exp[\beta\xi_{ij}(t)]}{\exp[\beta\xi_{ij}(t)]+\exp[-\beta\xi_{ij}(t)]},$
(4b) $\beta=1/T$ (4c) and $\xi_{ij}(t)=\sum_{k}a_{ik}x_{ik}(t)\cdot
a_{kj}x_{kj}(t).$ (4d)

###  II.3 Calculations of $U(T)$

An ergodic Markov chain based on repeating local steps done with the heat-bath
transition probability

$p(A\rightarrow B)=\frac{e^{-\beta E_{B}}}{\sum_{C}e^{-\beta E_{C}}}$ (5)

leads to a stationary state with the partition function

$Z=\sum_{C}e^{-\beta E_{C}}.$ (6)

For one triad, we have $2^{3}$ microscopic states $C$, four of them with
energy $U=-1$ and four with energy $U=+1$; hence the partition function is

$Z_{1}=\sum_{C}\exp(-\beta E_{C})=8\cosh(\beta),$ (7)

and the mean value of energy is

$U_{1}=-\frac{d}{d\beta}\ln Z_{1}=-\tanh(\beta).$ (8)

For two triads with a common link, we have $2^{5}$ microscopic states. The
energy spectrum contains eight states of energy equal $-2$, eight states of
energy equal $+2$, and sixteen states of energy equal zero. The direct
summation over these states gives the partition function $Z_{2}=Z_{1}^{2}$.
Accordingly,

$U_{2}=2U_{1}.$ (9)

Figure 3: Chain of triangles appearing in $(3^{3},4^{2})$ lattice

For a chain of triangles shown in Figure 3, we can distinguish two kinds of
links: internal (dashed) and closing (solid) ones. For each configuration of
internal links, two states of each triad are possible, with opposing energies
of this triad: $-1$ for a balanced triad, and $+1$ for an imbalanced triad.
The energy of the triad is determined by the state of the third, closing link
in this triad. Summarizing, the partition function of the chain includes:

  * •

all configurations of internal links;

  * •

two states of each closing link, with opposite energies, for each triad.

In this sense, the energies of the configurations depend only on the states of
closing links,

$Z_{N}=\sum_{i}\sum_{c}\exp(-\beta E(i,c))=2^{N}[2\cosh(\beta)]^{N},$ (10)

where $i$ and $c$ are related to the states of internal and closing links,
respectively. As a consequence, the energy per one triangle is equal to
$-\tanh(\beta)$ again.

##  III Results

For all the considered lattices, the initial state of each of the links
$x_{ij}$ is chosen randomly to be $+1$ or $-1$ with equal probabilities, so
that $U/\Delta\approx 0$ at time $t=0$. Then, one of the two update
procedures, synchronous or asynchronous, is applied for $t_{\text{max}}$ time
steps, as given by Equation 4. In the case of the synchronous updating scheme,
in one time step all links undergo this procedure simultaneously so that a new
matrix $\mathbf{X}$ is created, which then replaces the old matrix before the
next time step is performed. In the asynchronous approach, a single link is
chosen randomly for an attempted update and its new value is determined
instantly from Equation 4; in one time step this procedure is repeated as many
times as the total number of links, which means that statistically each link
is taken into account once in a time step. In both cases, the first 10% of the
performed time steps are not taken into account when calculating the results
presented below, so that the system can relax from the initial state. For most
calculations $t_{\text{max}}=10^{4}$, but in some cases it is larger for
$T<1$, as described below. In addition, all simulations are repeated 100 times
and the mean value over those simulations is denoted by
$\langle\cdots\rangle$.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 4:  $\langle U_{\text{av}}\rangle/\Delta$ for synchronous (4(a), 4(c),
4(e), 4(g)), and asynchronous (4(b), 4(d), 4(f), 4(h)) update scheme. Nodes
degree (on average, for the diluted triangular lattice and classical random
graphs) $k=4$ (4(c), 4(d)), $k=5$ (4(e), 4(f)), $k=6$ (4(g), 4(h))

The energy per triangle defined in Equation 3 is therefore calculated in the
final 90% of the time steps and then averaged to find $U_{\text{av}}$, and
then averaged again over all the simulations performed. The results, denoted
as $\langle U_{\text{av}}\rangle/\Delta$ are presented in LABEL:{fig:U_vs_T}
as a function of temperature $T$, together with the curve given by

$U/\Delta=-\tanh(1/T),$ (11)

corresponding to the theoretical results of Section II.3.

The two approaches used in our calculations, synchronous and asynchronous,
produce results that differ significantly at lower $T$. Figure 4(a) shows that
the synchronous updating leads to $\langle U_{\text{av}}(T)\rangle/\Delta$
dependencies which are unique to each of the discussed Archimedean lattices.
In the low-temperature limit, we observe perfectly balanced states, $\langle
U_{\text{av}}\rangle/\Delta=-1$, in the $(3,6,3,6)$, $(3^{3},4^{2})$ and
$(3^{4},6)$ lattices, but the decrease of energy with decreasing temperature
occurs along a different curve for each of those lattices, and in all cases
the values of the work function per triangle remain significantly larger than
$-\tanh(1/T)$. On the other hand, only a slight decrease in the work function
is found for the triangular lattice $(3^{6})$ and somewhat larger, to $\langle
U_{\text{av}}(T\to 0)\rangle/\Delta\approx-0.5$, for the $(3^{2},4,3,4)$
lattice. For the asynchronous updating, the results in Figure 4(b) prove that
for all types of the considered Archimedean lattices the temperature
dependence of the work function per triad is exactly the same and identical to
the theoretically predicted $U/\Delta=-\tanh(1/T)$.

For comparison, we match the Archimedean lattices with the corresponding
diluted triangular lattices (DTL) [12] and classical random graphs (CRG) [14]
having the same average node degree $k=4$, $5$, or $6$ as the connectivity of
the considered Archimedean lattices. The only Archimedean lattice discussed
here with $k=4$ is $(3,6,3,6)$ and for synchronous updating DTL and CRG of the
same $k$ reveal similar but not identical characteristics, with non-perfect
balance $\langle U_{\text{av}}(T)\rangle/\Delta\approx-0.9$ at $T\to 0$ as
visible in Figure 4(c). It should be noted, though, that the random nature of
the DTL and CRG graphs means that the simulations are performed on lattices
with the same average node degree and number of nodes, but with different
structure of connections (graph egdes) which results, among other things, in
the number of triads in those lattices that is not constant in all simulations
but found to be $\Delta=37\pm 3$ for DTL or $\Delta=11\pm 3$ for CRG, with
$N=64$ in both cases. For $k=5$ (Figure 4(e)) and $k=6$ (Figure 4(g)) with
synchronous updating the results for DTL and CRG are similar to those for
$k=4$, which also means that they are not exactly in line with those for
Archimedean lattices, in particular in the case of $(3^{2},4,3,4)$ lattice
where the balanced state is not observed even in the low-temperature limit. At
higher $T$, the results for all Archimedean, DTL and CRG lattices agree. For
asynchronous updating, the results obtained for DTL and CRG at $k=4,5,6$ and
presented in Figures 4(d), 4(f) and 4(h) do not differ from those of
Archimedean lattices.

(a)

(b)

Figure 5:  Values of $U_{\text{av}}/\Delta$ obtained in 100 simulations for
each temperature $T$, for various simulation times $t_{\text{max}}=10^{4}$,
$10^{6}$ and $10^{8}$, for (5(a)) $(3^{4},6)$ and (5(b)) $(3^{2},4,3,4)$
lattice and synchronous scheme of link updates

The dispersion of the values obtained in 100 simulations is usually relatively
small, especially when asynchronous updating is applied. In most cases, it is
also true for synchronous updating, except for the $(3^{2},4,3,4)$ and
$(3^{4},6)$ lattices at $T<1$ where it was required to perform
$t_{\text{max}}=10^{8}$ time steps, and for DTL and CRG lattices where
$t_{\text{max}}=10^{6}$ was used. In particular in the case of $(3^{4},6)$
lattice shorter simulations do not allow us to see that the balanced state is
achieved at $T\to 0$, as visible in Figure 5(a) showing the values
$U_{\text{av}}/\Delta$ obtained in all performed simulations. Figure 5(b)
confirms that on the $(3^{2},4,3,4)$ lattice the balanced state is not
observed even in the low-temperature limit, and additionally the values of the
work function obtained in the simulations performed are quantized and
distributed over a relatively long interval. The values of the work function
observed at the lowest $T$ are spaced in the intervals of $2/\Delta$, which
corresponds to the difference related to the energetically nearest states. The
same quantization of the work function applies for any other two-dimensional
lattice that contains triangles as it is associated with changing a single
term ($x_{ij}x_{jk}x_{ik}$—where $i$, $j$, $k$ are triangle vertices) in sum
in nominator of Equation 3 from $+1$ to $-1$ (or vice versa).

(a)

(b)

(c)

(d)

Figure 6: Thermal evolution (in the low temperature limit $T<1$) of fractions
of triads presented in Figure 1 in the case of asynchronous updating and
different Archimedean lattices. The panels (6(a)), (6(b)), (6(c)), and (6(d))
correspond to the triangles presented in Figures 1(a), 1(b), 1(c), and 1(d),
respectively. The error bars show the standard deviation of the results in 100
simulations. Dashed line for asynchronous updating (common result for all
lattices)

If any of the systems discussed is in the balanced state, it means that all
triads are balanced, that is, they are in one of two possible configurations:
with all positive links ($[+++]$, Figure 1(a)) or with one positive and two
negative links ($[+--]$, Figure 1(c)). The fraction of triads of a given kind,
$\langle f\rangle$, as a function of $T$ is shown in Figure 6. It reveals that
in the cases where the balanced state is reached at $T\to 0$, about 25% of the
triads are in the $[+++]$ configuration, and the rest are in the $[+--]$
state. However, in that limit (only for $T<0.2$) those fractions vary
substantially among the simulations, as shown by the error bars indicating the
standard deviation, which is also the case for the $(3^{2},4,3,4)$ lattice
which does not reach the balanced state and for all triads in that case
including $[++-]$ (Figure 1(b)) and $[---]$ (Figure 1(d)). On the other hand,
the $(3^{6})$ triangular lattice produces almost identical results in all
simulations. In the balanced state observed in the low-temperature limit, the
fractions of triads obtained using the asynchronous updating scheme (dashed
lines in Figure 6) are the same as for the balanced state reached with
synchronous updating. However, in the case of asynchronous updating the
increasing temperature causes much slower decrease of the number of balanced
triads than for synchronous approach, which is also visible in slower increase
of the work function with increasing $T$.

##  IV Discussion

Here we show that the Heider model (1) on a two-dimensional lattice with the
asynchronous updates is equivalent to a one-dimensional Ising model with the
nearest-neighbor interactions.

Let us apply the following transformation, where $s_{i}=-1$,

$x^{\prime}_{ij}=s_{i}x_{ij},$ (12)

to all links incident with the node $i$. The total energy of the system (1)
remains the same $E_{A^{\prime}}=E_{A}$. The gauge transformation (12) leaves
the spin products $x_{ij}x_{jk}\ldots x_{si}$ unchanged in any closed loop.
The transformation (12) can be repeated for any node of the lattice. The
configurations obtained in this way form an equivalence class with the same
energy. Let $N$ be the number of nodes. It is easy to see that each
equivalence class contains $2^{N-1}$ configurations since one can apply the
transformation (12) to $(N-1)$ nodes. Applying it to the $N$-th node would
undo all changes as each spin $x_{ij}$ would be multiplied twice
$s_{i}x_{ij}s_{j}$. Degeneracy can be removed by fixing values $x_{ij}$ on any
spanning tree of the lattice, for example by assigning all edges of the tree
the value one: $x_{ij}=1$. The spanning tree has $N-1$ edges, so fixing the
gauge completely eliminates degeneracy. Once the gauge is fixed, the partition
function (6) can be reduced to

$Z=2^{N-1}\sum_{a}e^{-\beta E_{a}},$ (13)

where $a$ are configurations of $x_{ij}$ for all edges except those which were
fixed to one on the spanning tree. We will illustrate this equivalence in
detail on a triangular lattice with twisted boundary conditions, as shown in
Figure 7, but it can be easily modified for any two-dimensional lattice.

Figure 7: An example of a triangular lattice with a toroidal topology with
twisted boundary conditions. The lattice has $N=25$ nodes, indexed by $(i,j)$,
with $i=0,\ldots,4$ and $j=0,\ldots,4$. It has $\Delta=50$ triangles and
$L=75$ links. The twisted boundary condition is encoded in a shift by one in
the coordinate $j$ between the first and the last columns

Let us introduce a gauge fixing as shown in Figure 8: $x_{ij}=1$ for all edges
$ij$ on the line drawn in red.

Figure 8: Gauge fixing: for all red edges $x_{ij}=1$. Any white triangle has
two free edges, for which $x_{ij}=\pm 1$. There are only two triangles with
three free edges. They are drawn in yellow and turquoise

Let us label the values of spins for the free links (which do not belong to
the gauge tree) as $\sigma_{1},\ldots,\sigma_{50}$ as shown in Fig. 8, and
denote the value of $\sigma$ on the link $(4,4)-(0,0)$ by $\sigma_{*}$. We can
write the partition function (6) for this gauge fixing as

$Z=2^{N-1}(Z_{+}+Z_{-}),$ (14)

where $Z_{+}$ and $Z_{-}$ are sums over configurations with $\sigma_{*}=+1$
and $\sigma_{*}=-1$, respectively. It is easy to see that

$Z_{+}=\sum_{\sigma}e^{\beta\sum_{i=1}^{50}\sigma_{i}\sigma_{i+1}},$ (15)

where $\sigma_{51}=\sigma_{1}$. This is a partition function of the one-
dimensional nearest-neighbor Ising model with periodic boundary conditions.
For $\sigma_{*}=-1$ we have

$Z_{-}=\sum_{\sigma}e^{\beta(\sum_{i=1}^{50}\sigma_{i}\sigma_{i+1}-E)},$ (16)

where $E=2\sigma_{39}\sigma_{40}+2\sigma_{50}\sigma_{1}$. Up to the last term
$E$, $Z_{-}$ is also equal to the partition function of the one-dimensional
Ising model. This term introduces some finite size corrections which can be
neglected when the size $n\times n$ of the lattice increases $n\to\infty$, so
we see that for large lattices the original model is mapped into a one-
dimensional Ising model on a chain of length $2n$. Energy per node for the
Ising chain is known to be $-\tanh\beta$, so therefore in the Heider model
with asynchronous update scheme energy per triangle should reproduce this
result $U/\Delta\rightarrow-\tanh\beta$ in the limit $n\to\infty$. This is not
the case for synchronous update scheme which violates the detailed balance
condition.

###### Acknowledgements.

The authors thank Zdzisław Burda for a fruitful discussion of the gauge
invariance theory applied to show that the perfectly balanced state is
unreachable. We gratefully acknowledge the Polish high-performance computing
infrastructure PLGrid (HPC Center: ACK Cyfronet AGH) for providing computer
facilities and support within computational grant no. PLG/2023/016259.

## References

  * Heider [1946] F. Heider, Attitudes and cognitive organization, The Journal of Psychology 21, 107 (1946).
  * Harary [1953] F. Harary, On the notion of balance of a signed graph, Michigan Mathematical Journal 2, 143 (1953).
  * Cartwright and Harary [1956] D. Cartwright and F. Harary, Structural balance: A generalization of Heider’s theory, Psychological Review 63, 277–293 (1956).
  * Harary [1959] F. Harary, On the measurement of structural balance, Behavioral Science 4, 316 (1959).
  * Davis [1967] J. A. Davis, Clustering and structural balance in graphs, Human Relations 20, 181 (1967).
  * Harary  _et al._ [1965] F. Harary, R. Z. Norman, and D. Cartwright,  _Structural Models: An Introduction to the Theory of Directed Graphs_ , 3rd ed. (John Wiley and Sons, New York, 1965).
  * Bonacich and Lu [2012] P. Bonacich and P. Lu,  _Introduction to Mathematical Sociology_ (Princeton University Press, Princeton, 2012).
  * Kułakowski [2007] K. Kułakowski, Some recent attempts to simulate the Heider balance problem, Computing in Science & Engineering 9, 80 (2007).
  * Belaza  _et al._ [2017] A. M. Belaza, K. Hoefman, J. Ryckebusch, A. Bramson, M. van den Heuvel, and K. Schoors, Statistical physics of balance theory, Plos One 12, e0183696 (2017).
  * Malarz and Wołoszyn [2020] K. Malarz and M. Wołoszyn, Expulsion from structurally balanced paradise, Chaos 30, 121103 (2020).
  * Malarz  _et al._ [2020] K. Malarz, M. Wołoszyn, and K. Kułakowski, Towards the Heider balance with a cellular automaton, Physica D 411, 132556 (2020).
  * Wołoszyn and Malarz [2022] M. Wołoszyn and K. Malarz, Thermal properties of structurally balanced systems on diluted and densified triangulations, Physical Review E 105, 024301 (2022).
  * Malarz and Kułakowski [2021a] K. Malarz and K. Kułakowski, Heider balance of a chain of actors as dependent on the interaction range and a thermal noise, Physica A 567, 125640 (2021a).
  * Malarz and Wołoszyn [2023] K. Malarz and M. Wołoszyn, Thermal properties of structurally balanced systems on classical random graphs, Chaos 33, 073115 (2023).
  * Malarz and Hołyst [2022] K. Malarz and J. A. Hołyst, Mean-field approximation for structural balance dynamics in heat bath, Physical Review E 106, 064139 (2022).
  * Malarz and Kułakowski [2021b] K. Malarz and K. Kułakowski, Comment on ‘Phase transition in a network model of social balance with Glauber dynamics’, Physical Review E 103, 066301 (2021b).
  * Rabbani  _et al._ [2019] F. Rabbani, A. H. Shirazi, and G. R. Jafari, Mean-field solution of structural balance dynamics in nonzero temperature, Physical Review E 99, 062302 (2019).
  * Shojaei  _et al._ [2019] R. Shojaei, P. Manshour, and A. Montakhab, Phase transition in a network model of social balance with Glauber dynamics, Physical Review E 100, 022303 (2019).
  * Manshour and Montakhab [2021] P. Manshour and A. Montakhab, Reply to “Comment on ‘Phase transition in a network model of social balance with Glauber dynamics’”, Physical Review E 103, 066302 (2021).
  * Bae  _et al._ [2024] M. Bae, T. Shimada, and S. K. Baek, Exact cluster dynamics of indirect reciprocity in complete graphs (2024), arXiv:2404.15664 [physics.soc-ph] .
  * Kepler [1619] J. Kepler,  _Harmonices Mundi_ (Linz, 1619).
  * Brace  _et al._ [2002] P. Brace, K. Sims-Butler, K. Arceneaux, and M. Johnson, Public opinion in the American states: New perspectives using national survey data, American Journal of Political Science 46, 173 (2002).
  * Antal  _et al._ [2005] T. Antal, P. L. Krapivsky, and S. Redner, Dynamics of social balance on networks, Physical Review E 72, 036121 (2005).
  * Krawczyk  _et al._ [2017] M. J. Krawczyk, S. Kaluzny, and K. Kułakowski, A small chance of paradise—Equivalence of balanced states, EPL (Europhysics Letters) 118, 58005 (2017).

##  Appendix A Archimedean lattices mapped into square lattice and the shapes
of their adjacency matrices

In Figures 13(a), 10(a), 11(a), 12(a) and 9(a) shapes of Archimedean lattices
embedded in a distorted triangular lattice are presented for kagomé,
$(3^{2},4,3,4)$, $(3^{3},4^{2})$, maple-leaf, and triangular lattices,
respectively.

In Figures 13(b), 10(b), 11(b), 12(b) and 9(b) examples of adjacency matrices
associated with lattices presented in Figures 13(a), 10(a), 11(a), 12(a) and
9(a) are displayed.

(a)

(b)

Figure 9: Triangular $(3^{6})$ lattice, $k=6$, (9(a)) $N=W^{2}$ nodes at and
inside green square are labeled as $i=0,\cdots,W^{2}-1$, $i=x+Wy$,
$x,y\in\\{0,1,2,\cdots,W-1\\}$, $W=6$ (9(b)) and its adjacency matrix

(a)

(b)

Figure 10: Archimedean $(3^{2},4,3,4)$ lattice, $k=5$, (10(a)) $N=W^{2}$ nodes
at and inside green square are labeled as $i=0,\cdots,W^{2}-1$, $i=x+Wy$,
$x,y\in\\{0,1,2,\cdots,W-1\\}$, $W=4$ (10(b)) and its adjacency matrix

(a)

(b)

Figure 11: Archimedean $(3^{3},4^{2})$ lattice, $k=5$, (11(a)) $N=W^{2}$ nodes
at and inside green square are labeled as $i=0,\cdots,W^{2}-1$, $i=x+Wy$,
$x,y\in\\{0,1,2,\cdots,W-1\\}$, $W=6$ (11(b)) and its adjacency matrix

(a)

(b)

Figure 12: Maple leaf $(3^{4},6)$ lattice, $k=5$, (12(a)) $N=W^{2}$ nodes at
and inside green square are labeled as $i=0,\cdots,W^{2}-1$, $i=x+Wy$,
$x,y\in\\{0,1,2,\cdots,W-1\\}$, $W=7$ (12(b)) and its adjacency matrix

(a)

(b)

Figure 13: Archimedean $(3,6,3,6)$ lattice (kagomé lattice), $k=4$, (13(a))
$N=W^{2}$ nodes at and inside green square are labeled as
$i=0,\cdots,W^{2}-1$, $i=x+Wy$, $x,y\in\\{0,1,2,\cdots,W-1\\}$, $W=6$ (13(b))
and its adjacency matrix

##  Appendix B Listings

LABEL:lst:module-par presents Fortran module defining constant parameters for
the subroutines supplying the adjacency matrix $\mathbf{A}$. The integer
variable `Z` corresponds to the network connectivity $k$. The integer variable
`W` represents the linear size of the underlying square lattice in which the
distorted triangles are embedded. Then, the integer variable `N` is equal to
the square of `W`. To ensure the possibility of wrapping the lattice according
to the assumed periodic boundary condition, its value must be a multiple of 4
for $(3^{2},4,3,4)$ and 7 for $(3^{4},6)$. The other lattices are presented
for `W`=6, however for $(3^{3},4^{2})$ and $(3,6,3,6)$ any even number is
suitable, while there are no restrictions in the case of the triangular
lattice $(3^{6})$.

LABEL:lst:code-triangular, LABEL:lst:code-al33344, LABEL:lst:code-tri2kagome,
LABEL:lst:code-tri2mapleleaf and LABEL:lst:code-tri2al33434 contain Fortran
subroutines computing the adjacency matrix $\mathbf{A}$ for triangular,
$(3^{2},4,3,4)$, $(3^{3},4^{2})$, $(3^{4},6)$, and $(3,6,3,6)$, respectively.
Subroutines `tri_2_al33434` (LABEL:lst:code-tri2al33434), `tri_2_maple_leaf`
(LABEL:lst:code-tri2mapleleaf), `tri_2_kagome` (LABEL:lst:code-tri2kagome) for
the $(3^{2},4,3,4)$, $(3^{3},6)$, $(3,6,3,6)$ lattices, respectively, require
earlier preparation of adjacency matrix $\mathbf{A}$ as for triangular
lattices using the subroutine given in LABEL:lst:code-triangular. The
adjacency matrix $\mathbf{A}$ is returned in the matrix variable `A`. The
local matrix variable `nn` (if used) stores labels of $k$ neighbors of the
sites in the lattice.

Listing 1: Module including parameters

⬇

1MODULE par

2INTEGER, PARAMETER :: Z=5, W=7, N=W*W

3END MODULE

Listing 2: Subroutine for triangular $(3^{6})$ lattice

⬇

1SUBROUTINE triangular(A)

2!! \-------------------------------------------

3USE par

4IMPLICIT NONE

5INTEGER, DIMENSION(0:N-1,0:Z-1) :: nn

6INTEGER, DIMENSION(0:N-1,0:N-1) :: A

7INTEGER :: i, j

8!! \-------------------------------------------

9DO i=0, N-1

10 nn(i,0) = MOD(N+i -W ,N)

11 nn(i,1) = MOD(N+i -W+1,N)

12 nn(i,2) = MOD(N+i -1 ,N)

13 nn(i,3) = MOD(N+i +1 ,N)

14 nn(i,4) = MOD(N+i +W-1,N)

15 nn(i,5) = MOD(N+i +W ,N)

16 IF(MOD(i,W)==0) THEN

17 nn(i,2) = MOD(N+W+i -1 ,N)

18 nn(i,4) = MOD(N+W+i +W-1,N)

19 ENDIF

20 IF(MOD(i+1,W)==0) THEN

21 nn(i,1) = MOD(N-W+i -W+1,N)

22 nn(i,3) = MOD(N-W+i +1 ,N)

23 ENDIF

24ENDDO

25!! \-------------------------------------------

26A=0

27DO i=0,N-1

28DO j=0,Z-1

29 A(i,nn(i,j))=1

30ENDDO

31ENDDO

32!! \-------------------------------------------

33END SUBROUTINE triangular

Listing 3: Subroutine for Archimedean $(3^{2},4,3,4)$ lattice

⬇

1SUBROUTINE tri_2_al33434(A)

2USE par

3IMPLICIT NONE

4INTEGER, DIMENSION(0:N-1,0:Z-1) :: nn

5INTEGER, DIMENSION(0:N-1,0:N-1) :: A

6INTEGER :: i, x ,y

7!! \-------------------------------------------

8!! neighbors on triangular lattice

9DO i=0, N-1

10 nn(i,0) = MOD(N+i -W ,N)

11 nn(i,1) = MOD(N+i -W+1,N)

12 nn(i,2) = MOD(N+i -1 ,N)

13 nn(i,3) = MOD(N+i +1 ,N)

14 nn(i,4) = MOD(N+i +W-1,N)

15 nn(i,5) = MOD(N+i +W ,N)

16 IF(MOD(i,W)==0) THEN

17 nn(i,2) = MOD(N+W+i -1 ,N)

18 nn(i,4) = MOD(N+W+i +W-1,N)

19 ENDIF

20 IF(MOD(i+1,W)==0) THEN

21 nn(i,1) = MOD(N-W+i -W+1,N)

22 nn(i,3) = MOD(N-W+i +1 ,N)

23 ENDIF

24ENDDO

25!! \-------------------------------------------

26!! adjacency matrix for (3^2,4,3,4) lattice:

27DO x=0,W-1

28DO y=0,W-1

29 i=x+y*W

30 IF(MOD(x,W).EQ.0 .AND. MOD(y,W).EQ.0)

31 A(i,nn(i,1))=0

32 IF(MOD(x,W).EQ.1 .AND. MOD(y,W).EQ.0)

33 A(i,nn(i,5))=0

34 IF(MOD(x,W).EQ.2 .AND. MOD(y,W).EQ.0)

35 A(i,nn(i,1))=0

36 IF(MOD(x,W).EQ.3 .AND. MOD(y,W).EQ.0)

37 A(i,nn(i,5))=0

38

39 IF(MOD(x,W).EQ.0 .AND. MOD(y,W).EQ.1)

40 A(i,nn(i,4))=0

41 IF(MOD(x,W).EQ.1 .AND. MOD(y,W).EQ.1)

42 A(i,nn(i,0))=0

43 IF(MOD(x,W).EQ.2 .AND. MOD(y,W).EQ.1)

44 A(i,nn(i,4))=0

45 IF(MOD(x,W).EQ.3 .AND. MOD(y,W).EQ.1)

46 A(i,nn(i,0))=0

47

48 IF(MOD(x,W).EQ.0 .AND. MOD(y,W).EQ.2)

49 A(i,nn(i,5))=0

50 IF(MOD(x,W).EQ.1 .AND. MOD(y,W).EQ.2)

51 A(i,nn(i,1))=0

52 IF(MOD(x,W).EQ.2 .AND. MOD(y,W).EQ.2)

53 A(i,nn(i,5))=0

54 IF(MOD(x,W).EQ.3 .AND. MOD(y,W).EQ.2)

55 A(i,nn(i,1))=0

56

57 IF(MOD(x,W).EQ.0 .AND. MOD(y,W).EQ.3)

58 A(i,nn(i,0))=0

59 IF(MOD(x,W).EQ.1 .AND. MOD(y,W).EQ.3)

60 A(i,nn(i,4))=0

61 IF(MOD(x,W).EQ.2 .AND. MOD(y,W).EQ.3)

62 A(i,nn(i,0))=0

63 IF(MOD(x,W).EQ.3 .AND. MOD(y,W).EQ.3)

64 A(i,nn(i,4))=0

65ENDDO

66ENDDO

67END SUBROUTINE tri_2_al33434

Listing 4: Subroutine for Archimedean $(3^{3},4^{2})$ lattice

⬇

1SUBROUTINE al_33344(A)

2!! \-------------------------------------------

3USE par

4IMPLICIT NONE

5INTEGER, DIMENSION(0:N-1,0:Z-1) :: nn

6INTEGER, DIMENSION(0:N-1,0:N-1) :: A

7INTEGER :: i, j, x, y

8!! \-------------------------------------------

9DO y=0,W-1,2

10DO x=0,W-1

11 i=x+y*W

12 nn(i,0) = MOD(N+i -1 ,N)

13 nn(i,1) = MOD(N+i +1 ,N)

14 nn(i,2) = MOD(N+i -W ,N)

15 nn(i,3) = MOD(N+i +W ,N)

16 nn(i,4) = MOD(N+i +W-1,N)

17 IF(MOD(i,W)==0) THEN

18 nn(i,0) = MOD(N+W+i -1 ,N)

19 nn(i,4) = MOD(N+W+i +W-1,N)

20 ENDIF

21 IF(MOD(i+1,W)==0) THEN

22 nn(i,1) = MOD(N-W+i +1,N)

23 ENDIF

24ENDDO

25ENDDO

26!! \-------------------------------------------

27DO y=1,W-1,2

28DO x=0,W-1

29 i=x+y*W

30 nn(i,0) = MOD(N+i -1 ,N)

31 nn(i,1) = MOD(N+i +1 ,N)

32 nn(i,2) = MOD(N+i -W ,N)

33 nn(i,3) = MOD(N+i +W ,N)

34 nn(i,4) = MOD(N+i -W+1,N)

35 IF(MOD(i,W)==0) THEN

36 nn(i,0) = MOD(N+W+i -1,N)

37 ENDIF

38 IF(MOD(i+1,W)==0) THEN

39 nn(i,1) = MOD(N-W+i +1,N)

40 nn(i,4) = MOD(N-W+i -W+1,N)

41 ENDIF

42ENDDO

43ENDDO

44!! \-------------------------------------------

45A=0

46DO i=0,N-1

47DO j=0,Z-1

48 A(i,nn(i,j))=1

49ENDDO

50ENDDO

51!! \-------------------------------------------

52END SUBROUTINE al_33344

Listing 5: Subroutine for maple leaf $(3^{3},6)$ lattice

⬇

1SUBROUTINE tri_2_maple_leaf(A)

2!! \-------------------------------------------

3USE par

4IMPLICIT NONE

5INTEGER, DIMENSION(0:N-1,0:N-1) :: A

6INTEGER :: i, j, k, x, y

7!! \-------------------------------------------

8DO y=0,W-1

9DO k=1,S

10 x = MOD(2*y+7*k,W)

11 i = x + W*y

12 DO j=0,N-1

13 A(i,j)=0

14 A(j,i)=0

15 ENDDO

16ENDDO

17ENDDO

18!! \-------------------------------------------

19END SUBROUTINE tri_2_maple_leaf

Listing 6: Subroutine for kagomé $(3,6,3,6)$ lattice

⬇

1SUBROUTINE tri_2_kagome(A)

2!! \-------------------------------------------

3USE par

4IMPLICIT NONE

5INTEGER, DIMENSION(0:N-1,0:N-1) :: A

6INTEGER :: i, j, x, y

7!! \-------------------------------------------

8DO y=1,W,2

9DO x=1,W,2

10 i=x+W*y

11 DO j=0,N-1

12 A(i,j)=0

13 A(j,i)=0

14 ENDDO

15ENDDO

16ENDDO

17!! \-------------------------------------------

18END SUBROUTINE tri_2_kagome