REVISTA DE MATEMÁTICA: TEORÍA Y APLICACIONES 2014 21(1) : 55–72
CIMPA – UCR ISSN: 1409-2433
ON THE DESIGN OF MEMBRANES WITH
INCREASING FUNDAMENTAL
FREQUENCY
SOBRE EL DISEÑO DE MEMBRANAS CON
FRECUENCIA FUNDAMENTAL
CRECIENTE
RAÚL B. GONZÁLEZ DE PAZ∗
Received: 30/Jan/2013; Revised: 8/Nov/2013;
Accepted: 15/Nov/2013
∗Departamento de Matematicas, Universidad del Valle de Guatemala, Apdo. Postal 82, 01901
Guatemala, Guatemala. E-Mail: ragopa@ufm.edu
55
56 R.B. GONZÁLEZ DE PAZ
Abstract
By means of a relaxation approach, we study the shape design of a
stiff inclusion with given area in a membrane in order to maximize its fun-
damental frequency. As an eigenvalue control problem, the fundamental
frequency is a concave function of the control, which is not described by
the membrane shape, but by an element in a function space. First order
optimality conditions allow to describe the optimal shape by means of a
free boundary value problem.
Keywords: variational methods for eigenvalues, shape optimization, free
boundary value problems.
Resumen
Mediante un método de relajación, se estudia la forma de una inclusión
rígida de área dada en una membrana de manera que se maximice su fre-
cuencia fundamental. Analizado como un problema de control de valores
propios, la frecuencia fundamental es una función cóncava del control, el
cual no es descrito por la forma de la membrana, sino por un elemento de
un espacio de funciones. Las condiciones de optimalidad de primer or-
den permiten describir la forma óptima mediante un problema de frontera
libre.
Palabras clave: métodos variacionales para valores propios, optimización de
forma, problemas de frontera libre.
Mathematics Subject Classification: 35J20, 35R35, 49R05, 49Q10.
1 Introduction
The subject of the present study was motivated by an article due to Payne and
Weinberger [22] where the following is stated: suppose a two dimensional mem-
brane, defined on a domain Ω, fixed along its outer boundary, perforated by
“holes" with boundaries Γi and along them the membrane is free. For a given
area |Ω| and a given perimeter L of the exterior boundary, the highest funda-
mental frequency is attained when the domain Ω is annular. This classical result,
which was proved by means of isoperimetric inequalities, lies at the origin of the
following problem: let be given a perforated membrane with uniform density,
supposed fixed on the exterior boundary and with the perforation filled by a rigid
inclusion. Assuming that its area |Ω| is a given constant, and the outer boundary
is fixed, we look for the location and shape of the inclusion in order to max-
imize the fundamental frequency of the membrane. This problem falls in two
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(1): 55–72, January 2014
DESIGN OF MEMBRANES WITH INCREASING FUNDAMENTAL FREQUENCY 57
areas: eigenvalue control and optimal shape design, the subject we present is re-
lated to studies published among others, by Buttazo and Dal Maso [2], Cox and
McLaughlin [6], Egnell [8], Eppler [9], Henrot [15], Jouron [17],Rousselet [23],
Tahraoui [26], and Zolesio [30]. We present here a unified perspective by means
of an approach similar as the one used for other shape optimization problems (cf.
Gonzalez de Paz [11] and [12]). We define a regularized problem, where a con-
cave functional is maximized on a convex set. Within this framework, existence
of optimal design, corresponding to the maximal fundamental frequency, so as
differentiability and concavity properties are easily obtained. The functional is
Gateaux differentiable (even Frechet differentiable ) and the analysis of the first
order optimality conditions allows us to describe the boundary Γ of the optimal
set as a free boundary. If the free boundary is regular enough, the results obtained
by our approach in terms of the functional derivative lead to similar properties
published elsewhere concerning shape gradients (cf. Eppler [9], Rousselet [23],
Simon [24] and Zolesio [30]).
Physically, by adding a small enough regularization term, we will handle a
larger membrane defined on the whole domain D = Ω ∪ Ωe with two compo-
nents: the original one defined on Ω and a membrane vibrating on Ωe, which
is affected by a stiffness factor. Mathematically, this is described by means of
elliptic operators of the type −∆ + q where the stiffness factor (regularization
term) q is a function defined on a certain class. We prove that, by increasing the
stiffness factor, in the limit the lowest eigenvalue of the operator is maximized by
q = λχΩ∗e where λ is the first eigenvalue and χΩ∗e is the characteristic function
of the optimal set Ω∗e. A similar result was obtained by Egnell [8] in a context of
quantum mechanics.
Though this problem has been the subject of research during decades, we
think that our approach adds some new perspectives concerning the characteri-
zation of the optimal inclusion. Besides, it is constructive and well adapted for
numerical calculations applying gradient-type algorithms.
2 The regularized problem
Let D = Ω ∪ Ωe be an open, bounded, star shaped, connected set in R2, with a
piecewise smooth boundary ∂D. LetΩ ⊂ D be a subset such that ∂Ω = ∂D∪Γ,
where Γ = ∂Ωe denotes the boundary of the “inclusion”Ωe.
We recall that for the case that the membrane is fixed along its boundary, the
eigenvalue problem for the laplacian onΩwith homogeneous Dirichlet condition
on its boundary describes mathematically the vibration problem. Furthermore,
for the fundamental frequency λ0 (lowest or first eigenvalue) the Ritz-Rayleigh
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(1): 55–72, January 2014
58 R.B. GONZÁLEZ DE PAZ
principle states that:
λ0 = minu∈S(Ω) ‖∇u‖
2
.
Here, S(Ω) =
{
u ∈ H10 (Ω) | ‖u‖
2 = 1
}
, the double bars denote the usual
L2 norm in Ω, andH10 (Ω) the usual Sobolev space (cf. for example Neças [21]).
For the penalized problem we introduce a “stifness factor" which will be
described the following way: let µ be a positive element of the unit ball in
L∞ (D), such that for a given positive constant A < |D| : |µ|L1 = A. Recall
that C = {µ ∈ L∞ (D) | 0 ≤ µ ≤ 1, |µ|L1 = A} is a convex set.
For a given positive constant β we define the functional Jµ : H10 (D) −→ R
u )→ Jµ (u) = ‖*u‖
2 + β
〈
µ, u2
〉
. (1)
The brackets denote the usual
(
L∞, L1
)
duality. It is well known that by
minimizing the functional 1 defined above on the set
S =
{
u ∈ H10 (D) | ‖u‖
2 = 1
}
the existence of optimal solution uµ ∈ S is a classical fact. The optimal value
Jµ (uµ) given by the functional is the lowest eigenvalue of the boundary value
problem P (µ):
−∆u + βµu = λu in D, (2)
u = 0 on ∂D. (3)
For a fixed, positive β we define the functional on C:
µ→ Λβ (µ) = Jµ (uµ) = λβ . (4)
The mapping 4 is well defined if the “stiffness factor” βµ is a “small" per-
turbation for the Laplacian. This is precised in the following sense:
Lemma 1 For the differential operator −∆ + βµ with first eigenvalue λβ , as
defined for the boundary value problem P (µ), in the case β < λβ , the corre-
sponding eigenfunction uβ is superharmonic and strictly positive, so that λβ is
a simple eigenvalue.
Proof. First note that any eigenfunction u is a C1,1loc function (cf. Jensen [16]).
The partial differential equation has a sense a.e. in D. The function v = |u| is
also a continuous eigenfunction and we have −∆v = (λβ − βµ)v > 0 a.e. in
D, i.e. v is superharmonic. Suppose there exists an x0 ∈ D such that v(x0) =
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(1): 55–72, January 2014
DESIGN OF MEMBRANES WITH INCREASING FUNDAMENTAL FREQUENCY 59
0, then there exists a ball centered in x0 where v = 0. The Hopf Maximum
principle states that in this case v = 0 in D (cf. for example Miranda [20]).
It follows: u = 0 in D, which contradicts the fact that u ∈ S, so u > 0
on D. Consequently, as every function has fixed sign, the first eigenvalue λβ
is simple (let us remark that this property can also be proved by means of the
Krein-Rutman Theorem).
Remark 2 Note that the mapping µ→ Λβ (µ) = minu∈S ‖*u‖2+β
〈
µ, u2
〉
is
the lower envelope of affine functions related to µ, which implies it is a concave
function respect to µ. This means, the first eigenvalue is a concave function of
the “stiffness” factor.
The next step will be to find the best factor among a certain class in order to
maximize the fundamental frequency of the relaxed problem.
For this goal, we consider now the optimization problem:
sup
µ∈C
Λβ (µ) . (5)
Let us recall that the convex setC is compact for the σ
(
L∞, L1
)
−topology.
In order to obtain existence results for the solution of optimization we prove in
a similar way as in Gonzalez de Paz [11]:
Theorem 3 The mapping 4 µ→ Λβ (µ) is σ
(
L∞, L1
)
−continuous, so that for
β small enough there exists an element µβ ∈ C such that
Λβ
(
µβ
)
= max
µ∈C
Λβ (µ) = λβ. (6)
Proof. For a fixed β > 0, the functional µ → Λβ (µ) is bounded on C. We
remark that there exists a constant Cβ > 0 such that for every µ ∈ C and every
u ∈ H10 (D):
‖*u‖2 + β
〈
µ, u2
〉
≤ Cβ ‖u‖
2
H1
0
. (7)
For a fixed u0 ∈ S, we have for every µ ∈ C:
Λβ (µ) ! Cβ ‖u0‖
2
H10
. (8)
As Λβ is bounded on C, there exists a ball B ⊂ H10 (D) such that for every
µ ∈ C:
min
u∈S
Jµ (u) = min
u∈S∩B
Jµ (u) . (9)
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60 R.B. GONZÁLEZ DE PAZ
Because of the Rellich-Kondrasov injection theorem, the set
W =
{
w | w = u2, u ∈ B
}
is strongly compact in L1(D). We define the set
of affine mappings: F = {Ju : µ→ Jµ(u) | u ∈ B}, it follows that Λβ is the
lower envelope of F .
Let be given δ > 0 and a fixed µ0 ∈ C. Furthermore, let be a µ ∈ C such
that for every u ∈ B:
β
∣∣〈µ− µ0, u2〉∣∣ ! δ. (10)
This means that µ − µ0 ∈
(
β
δW
)0
⊂ L∞ (D), which is the polar set of
β
δW ⊂ L
1 (D) . It follows that µ lies in a neighborhood of µ0 for the topology
of the uniform convergence on the strong compact sets of L1(D), noted also as
the τ -topology. (see for ex. Bourbaki [1]). For ε = δ and for every u ∈ B,∣∣Jµ (u)− Jµ0 (u)∣∣ ! ε. (11)
This means that the mappings collection F is τ−equicontinuous. Being Λβ
the lower envelope of affine τ−equicontinuous functions, it is also τ−continuous.
We remark that the τ−topology and the σ
(
L∞, L1
)
−topology are equivalent on
the unit ball in L∞ ( cf. [1]) which proves our claim.
3 Optimality conditions
As we can see, the eigenvalues depend on a perturbation for the Laplacian; as-
suming a small enough perturbation, the first eigenvalue remains simple and we
may proceed to calculate the functional derivative for Λβ. Other authors have re-
marked that the eigenvalues are differentiable respective to domain deformations
in the case they are simple (cf. for example Rousselet [23], Zolesio [30]).
Similar as in Gonzalez de Paz [11], we have the following theorem.
Theorem 4 For every µ ∈ C :the functional µ → Λβ (µ) has a Gateaux-
derivative, so that for every α = µ− µβ and uβ ∈ S, solution of P (µβ):
Λ′β
(
µβ;α
)
= β
〈
u2β ,α
〉
. (12)
A result due to M. Valadier [27] proves that the mapping µ → Λβ (µ) is
Frechet-differentiable. The gradient is defined as*Λβ(µβ) = βu2β ∈ L1 (D) .
Clasically, a necessary optimality condition states that for every α = µ −
µβ , µ ∈ C:
Λ′β
(
µβ ;α
)
! 0
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DESIGN OF MEMBRANES WITH INCREASING FUNDAMENTAL FREQUENCY 61
which implies that for the optimal µβ :∫
D
µβu
2
βd* "
∫
D
µu2βd*. (13)
for every µ ∈ C. The optimality condition 13 is a continuous linear programming
problem, its solution is a standard procedure. As it will be shown next, we look
how to “place” the integrand in a domain Ωe,β in order to maximize the integral
value. To describe the optimal domain Ωe,β, we remark first with following
Lemma.
Lemma 5 Let β be a positive constant such that β < λβ and let uβ > 0 be
the corresponding solution for the boundary value problem P (µβ). For every
constant p > 0 such that the level set Sp = {x ∈ D | uβ(x) = p} is not empty,
the Lebesgue measure of Sp is zero.
Proof. We remark that for ∂D regular enough, uβ ∈ C1,1loc (D) ∩ H
2(D). The
partial differential equation is solved in the sense almost every where inD. Thus,
−∆uβ = (λβ − βµ)uβ > 0 a.e. in D. On the other side, on every Sp we have
−∆uβ = 0, in the sense a.e. in D. So it follows: meas(Sp) = 0.
We are now able to describe the optimal set Ωe,β .
Proposition 6 Let β be a positive constant such that β < λβ, uβ as above, then
there exists a scalar p > 0 so that µβ = χΩe,β and
Ωe,β = {x ∈ D | uβ(x) > p} (14)
∂Ωe,β = {x ∈ D | uβ(x) = p} . (15)
Sketch of the proof: We consider the maximization of the linear mapping
µ →
∫
D µu
2
βd* on C. Then there exists a Lagrange multiplier p related to the
measure constraint |µ|L1 = A (cf. Cea-Malanowski [4]) so that
uβ(x) > p implies µβ(x) = 1
uβ(x) = p implies µβ(x) ∈ [0, 1]
uβ(x) < p implies µβ(x) = 0.
The Lebesgue measure of the set Sp is zero, so that µβ = χΩe,β almost ev-
erywhere in D. The structure of the boundary as Γβ = ∂Ωe,β = Sp follows
from the fact that uβ is continuous and superharmonic. The function µβ is a
characteristic function, so is an extremal point of C. (cf. Castaing-Valadier [3]).
Hence, we can constraint our search for maximizing functions among the ex-
tremal points of C, in other words, we look a set Ωe where to place the integrand
in order to maximize the integral
∫
Ωe
u2βd*.
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62 R.B. GONZÁLEZ DE PAZ
Remark 7 The set Ωe,β is unique.
Recall that the gradient βu2β is a positive and non constant in every set in D
with positive measure. So for every α = µ−µβ ,= 0 a.e. it followsΛ′β
(
µβ;α
)
<
0. Therefore the mapping µ → Λβ (µ) is strictly concave and consequently the
maximizing point is unique.
Remark 8 The domain functional 4 to be optimized can be interpreted in our
framework as the first eigenvalue of the membrane defined on Ωβ, i.e. for
µβ = χΩe,β :
λβ(Ωβ) = Λβ
(
χΩe,β
)
. (16)
Remark 9 In another context, Delfour [7] and Zolesio [29] calculate the so-
called shape derivative of functionals based on the continuous deformations of
domains. These authors base their result using the so-called deformation speed
θ, which is nothing but the gradient vector field of the function ϕt : D → D
describing the continuous deformation of the domainϕt(Ω) = Ωt. In our frame-
work, the Gateaux derivative already calculated becomes to the limit a shape-
derivative dλβ(Ω0; θ) in the sense that, formally, for a boundary Γβ regular
enough:
dλβ(Ω0; θ) = lim
t→0+
1
t
Λ′β
(
χΩ0 ;χΩt − χΩ0
)
= lim
t→0+
β
t
〈
u2β,χΩt − χΩ0
〉
= β
∫
Γβ
u2βθndσ. (17)
In this case, the term θn describes the normal component ot the vector field
θ = Dtϕt on the boundary Γβ (see also Eppler [9]).
4 The free boundary value problem
As a consequence of the optimality conditions, the function uβ ∈ H10 (D) ∩
C1,1loc (D) solves the free boundary problem:
−∆uβ = λβuβ in Ωβ = {x ∈ D | 0 < uβ(x) < p} , (18)
−∆uβ + βuβ = λβuβ in Ωe,β = {x ∈ D | p < uβ(x)} , (19)
uβ(x) = p on Γβ , (20)
uβ(x) = 0 on ∂D. (21)
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DESIGN OF MEMBRANES WITH INCREASING FUNDAMENTAL FREQUENCY 63
Physically, we can interpret this system as a membrane with two compo-
nents defined on D = Ωβ ∪ Ωe,β . In the subdomain Ωβ it has the fundamental
frequency λβ , it is fixed on ∂D and interacts along the free boundary Γβ with the
membrane defined on Ωe,β , which vibrates with fundamental frequency λβ − β.
Remark 10 The constraint β < λβ allows the corresponding uβ to be a su-
perharmonic function. If the domain D is star-shaped, the level sets are simply-
connected (cf. Kawohl [18]) and this implies, the setΩe,β is a connected domain.
Remark 11 Because of the C1,1loc regularity of the solution, the free boundary
is locally Lipschitz continuous (cf. Kinderlehrer-Stampacchia [19]). Recalling
that the free boundaryΓβ is a level set, we have for every neighborhood of points
in Γβ where |*uβ | > 0 :
∂u+
β
∂n
=
∂u_
β
∂n
onΓβ . (22)
Here the restrictions of uβ to Ωβ and Ωe,β are denoted by u+β and u_β re-
spectively. This “transmission” condition describes the interaction along the
boundary between the vibrating membranes occupying each subdomain.
5 A global existence result
We have seen that for each set Ωe ⊂ D with a given measure A describing an
inclusion in the membrane, we are able to calculate a “relaxed” fundamental
frequency in the sense that for the domain Ω = D\Ωe: λβ(Ω) = Λβ
(
χΩe
)
.
For a fixed β, the maximization of the mapping µ → Λβ (µ) on the convex set
C provides a set Ωe,β and the corresponding first eigenvalue λβ(Ωβ). We will
show in this section that for a family of bounded stiffness factors (βn)n and a
fixed domain Ωe ⊂ D, there exists a global optimum.
First we prove a monotonicity property.
Lemma 12 Let Ωe ⊂ D be a fixed domain, let β′ > β two stiffness factors
then we have for the first eigenvalues corresponding to the relaxed problems:
λβ′ > λβ .
Proof. Recall that for a fixed β and a fixed χΩ ∈ C :
Λβ
(
χΩe
)
= ‖*uβ‖
2 + β
〈
χΩe , u
2
β
〉
with uβ ∈ S solution of P (χΩ). Then for β′ > β :
Λβ′
(
χΩe
)
>
∥∥*uβ′∥∥2+β 〈χΩe , u2β′〉 " ‖*uβ‖2+β 〈χΩe , u2β〉 = Λβ (χΩe) .
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(1): 55–72, January 2014
64 R.B. GONZÁLEZ DE PAZ
That is, λβ′ > λβ .
This means, as the stiffness factor grows, so it does the eigenvalue. Now,
what we aim to show is that the fundamental frequency corresponding to the
vibrating membrane with a stiff inclusion on Ωe bounds the relaxed frequencies
we studied before.
Lemma 13 Let Ωe ⊂ D be a given, simply connected set with a piecewise
smooth boundary Γ. Let be given a sequence of increasing stiffness factors β
such that β < λβ = Λβ(χΩe). Then there exists a λ˜ > 0, such that for some
u˜ ∈ H10 (D),
−∆u˜ = λ˜u˜ in Ω = D\Ωe. (23)
∆u˜ = 0 in Ωe (24)
in the weak sense, and for every λβ:
λ˜ " λβ . (25)
Proof. Let λ0 be the fundamental frequency of a membrane defined on Ω and
fixed on its boundary ∂Ω = ∂D ∪ Γ ( i.e. a homogenous boundary condition of
the Dirichlet type is prescribed for the corresponding eigenfuntion). We remark
that, for every λβ as defined above: λβ ! λ0. This implies that the relaxed
eigenfrequencies remain bounded and this allows to define: λ˜ = supβ λβ <∞.
Let be given an increasing sequence of stiffness factors (βn)n, let λn be the
corresponding relaxed eigenfrequency. The eigenvalue sequence is monotone
increasing and it follows: λn → λ˜. Recalling the fact that βn < λn, let the
sequence (βn)n be such that βn → λ˜, if n→∞.
For the corresponding eigenfunctions un ∈ H10 (D) it follows that the se-
quence (‖*un‖)n is bounded. This implies the existence of a weakly convergent
subsequence in H10 (D), noted also (un)n, i.e. un ⇀ u˜ ∈ H10 (D) .
We have therefore in the weak sense for every test function ϕ ∈ D and
every n:
(*un,*ϕ) + βn
(
χΩeun,ϕ
)
= λn (un,ϕ) .
To the limit it becomes,
(*u˜,*ϕ) + λ˜
(
χΩe u˜,ϕ
)
= λ˜ (u˜,ϕ) ,
which is nothing but the weak formulation of the partial differential equation
system 23 and 24 stated above.
Physically, to the limit we have a vibrating membrane onΩ fixed on the outer
boundary ∂D and free along Γ.
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DESIGN OF MEMBRANES WITH INCREASING FUNDAMENTAL FREQUENCY 65
Theorem 14 Let D be a star-shaped domain with a piecewise smooth boundary,
then there exists a simply connected set Ω∗e ⊂ D with meas(Ω∗e) = A such that
the fundamental frequency λ(Ω∗) for the membrane defined on Ω∗ = D\Ω∗e,
fixed on ∂D and an inclusion defined in Ω∗e has the property:
λ(Ω∗) " λ(Ω) (26)
for every fundamental frequency λ(Ω) corresponding to a domain Ω = D\Ωe,
with a stiff inclusion defined on Ωe, with measure A.
Proof. Let us consider an increasing sequence of β′s. such that β < λβ. Remark
that for β′ > β:
Λβ′
(
µβ′
)
" Λβ′
(
µβ
)
> Λβ
(
µβ
)
.
The corresponding
{
Λβ
(
µβ
)}
β
build therefore an increasing sequence.
The sequence of optimal control functions (µβ)β is L2−bounded, so there
exists a subsequence weakly convergent to µ∗ ∈ C.
Let W =
{
u ∈ H10 (D) |
〈
µ∗, u2
〉
= 0
}
which is closed in H10 (D). Then
there exists an element u0 ∈ S ∩W such that:
‖*u0‖
2 = min
u∈S∩W
‖*u‖2 .
Thus, for every β: Λβ
(
µβ
)
! ‖*u0‖
2 , i.e. the sequence of optimalΛβ
(
µβ
)
=
λβ(Ωβ) is bounded.
We note: λ∗ = supβ Λβ
(
µβ
)
, remark that λβ(Ωβ)→ λ∗ for any sequence
of increasing β, provided that β < λβ
Let us choose a subsequence (βk)k such that βk → λ∗, for k →∞. For the
corresponding eigenfunctions uk we have:
‖*uk‖
2 < Λβ
(
µβk
)
< λ∗.
Choosing a suitable subsequence noted also (uk)k , it converges weakly inH10 (D).
Besides, for every test function ϕ ∈ D:
(*uk,*ϕ) + βk
(
µβkuk,ϕ
)
= λk (uk,ϕ) .
Because of the Rellich-Kondrasov injection theorem, the sequence (uk)k
converges also in the strong topology in L2(D) so that to the limit we obtain
the variational equation:
(*u∗,*ϕ) + λ∗ (µ∗u∗,ϕ) = λ∗ (u∗,ϕ) .
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(1): 55–72, January 2014
66 R.B. GONZÁLEZ DE PAZ
Setting ϕ = u∗ ∈ S:
‖*u∗‖2 + λ∗
〈
µ∗, u∗2
〉
= λ∗.
In a weak sense:
−∆u∗ + λ∗µ∗u∗ = λ∗u∗ in D.
Note that u∗ ∈ C0,1loc (D) ∩ H2 (D), so the equation above has a sense also
almost everywhere in D.
It follows:
µβuβ → µ
∗u∗ a.e. in D.
As for every βk : uk > 0 a.e. in D, this implies µβ → µ∗ a.e. in D and
consequently:
µ∗ = χΩ∗e a.e. in D for a set Ω
∗
e ⊂ D.
Up to a null measure set Ω∗e = lim infβ Ωe,β . As all sets Ωe,β are in a metric
space, the limit set Ω∗e is closed and connected.
In order to obtain more information on the set Ω∗e, we apply the first order
optimality condition for µβ . We know that for each βk as defined before we have
the condition 13: ∫
D
µβku
2
kd* "
∫
D
µu2kd* for every µ ∈ C.
To the limit k →∞ we obtain:∫
D
µ∗u∗2d* "
∫
D
µu∗2d* for every µ ∈ C.
Similar as before, there exists a Lagrange multiplier p∗ related to the measure
constraint |µ∗ |L1 = A such that
u∗ " p a.e. on Ω∗e
u∗ < p a.e. on Ω∗ = D\Ω
∗
e.
As u∗ is continuous on D, the boundary ∂Ω∗e = Γ∗ is included in the level set
Sp∗ = {x ∈ D | u∗(x) = p∗}. Recall that for star-shaped D, the subdomain Ω∗e
is simply connected. The function u∗ is harmonic in the interior of Ω∗e, applying
the maximum principle for harmonic functions we obtain:
u∗ = p ∈ Ω∗e (27)
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(1): 55–72, January 2014
DESIGN OF MEMBRANES WITH INCREASING FUNDAMENTAL FREQUENCY 67
and
−∆u∗ = λ∗u∗ ∈ Ω∗ = D\Ω∗e = {x ∈ D | 0 < u
∗(x) < p∗} . (28)
Physically in the domain Ω∗ we have a vibrating membrane with fundamen-
tal frequency λ∗. For the subdomain Ω∗e the function u∗ describes the corre-
sponding vibration mode for the fundamental frequency of a free membrane de-
fined on Ω∗e . It is a classical result that in this case, the fundamental frequency
is equal to zero with associated eigenfunctions u = const. (cf. Courant-Hilbert
[5]). In other words, λ∗ = λ(Ω∗) is the fundamental frequency of a membrane
fixed along the outer boundary ∂D with a stiff inclusion fixed on the free bound-
ary Γ∗.
Finally, this means that for every simply connected set Ωe ⊂ D with
meas(Ωe) = A and for every β < λβ :
λ(Ω∗) " Λβ
(
µβ
)
" Λβ
(
χΩe
)
.
Letting β → λ˜ and applying previous Lemma, we conclude for every
Ω = D\Ωe:
λ(Ω∗) " λ˜(Ω) " λ(Ω). (29)
Remark 15 In another context, assuming that Γ is a regular enough, Rousse-
let [23], Simon [24], Sokolowski and Zolesio[25] calculate the so-called shape
derivative of the fundamental frequency of a membrane with free boundary Γ
(in other words, the classical Hadamard formula), so that for the corresponding
eigenfunction u and the gradient vector field θ of the function ϕt : D → D
describing the continuous deformation of the domain Ω→ ϕt(Ω) = Ωt:
dλ(Ω; θ) = −
1
2
∫
Γ
∣∣∣∣∂u∂n
∣∣∣∣2 θndσ. (30)
Here, θn describes the normal component of the vector field θ = Dt ϕt on Γ.
Several authors have obtained as optimality condition (provided that the
boundary Γ of the optimal domain is regular enough) an additional boundary
condition of the Neumann type on the free boundary:
∣∣∣∣∂u∂n
∣∣∣∣ = const. on Γ. (31)
Thus, the optimal domain Ω∗ and the corresponding eigenfunction u∗ solve
an overdetermined boundary value problem of the Cauchy type for the eigen-
value equation.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(1): 55–72, January 2014
68 R.B. GONZÁLEZ DE PAZ
6 Some remarks on the numerical analysis of the
optimization problem
6.1 A sketch of the algorithm:
The approach we have proposed seems well suited for numerical implementa-
tion, as the functional µ → Λβ (µ) is differentiable and strictly concave, a gra-
dient method of the Frank-Wolfe type is proposed to solve a finite dimensional
approximation. (cf. Valadier [28], Gonzalez de Paz-Tiihonen [13]).
1. For a sequence (βk, µk)k, generate the corrresponding (uk, λk) solving
the eigenvalue problem:
−∆uk + βkµkuk = λkuk in D
uk = 0 on ∂D.
2. Define the set Sk+1 = {x ∈ D | uk(x) " pk}. The multiplier pk being
such that meas(Sk+1) = A.
3. Define µk+1 by µk+1(x) =
{
1
0
if x ∈ Sk+1
elsewhere.
4. If |λk+1 − λk| > .tol, set βk+1 = λk and go to step 1. Else, if |λk+1 − λk| ≤
.tol set Ω∗e = Sk+1.
6.2 A test problem
As a matter of illustration, we consider a membrane defined on a rectangular
domain D = [0, 2] × [0, 2]. Numerical calculations were carried out using a
finite difference approximation (discretization parameter h = 0.1 ) for the lapla-
cian operator. As initial domain Ωe the best choice seems to be given by the set
bounded by the equipotential curve of the eigenfunction for β = 0 satisfying the
measure constraint. In fact, in this case there is no need to change the domain
again. Convergence was achieved quite fast, after three iterations the vibration
model remains constant on the domainΩ∗e . We stopped short before the tolerance
threshold was achieved, because the numerical stability of the limit case when
β = λβ was affected. The numerical results are presented in the next table:
#Iter.step β λβ
1 0 4.92
2 4.92 5.78
3 5.78 5.92
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(1): 55–72, January 2014
DESIGN OF MEMBRANES WITH INCREASING FUNDAMENTAL FREQUENCY 69
We present in Figures 1, 2 and 3 some ilustrations with the equipotential lines
and the profile of the first vibration mode corresponding to the results calculated
by the three iterations. The optimal inclusion in Ω∗e is well identified in the third
iteration step as the set with positive measure where the function u∗ is constant.
Figure 1: First iteration, β = 0, λ = 4.92.
Figure 2: Second iteration, β = 4.92, λ = 5.78.
Figure 3: Third iteration, β = 5.8, λ = 5.92.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(1): 55–72, January 2014
70 R.B. GONZÁLEZ DE PAZ
7 Conclusions
We have performed an analysis on the shape and location of stiff inclusions for
membranes with maximal fundamental frequency. This is done by means of
a regularization approach in a Convex Analysis framework. The functional to
be maximized, which is equivalent to the lowest eigenvalue of the regularized
problem, is concave and differentiable, and our results concerning the existence
and description of the optimizing elements are related to other research already
known (cf. Buttazzo and Dal Maso [2] and Egnell [8]). The regularization ap-
proach allows these results to be presented in a unified way. As we develop
further the analysis of first order optimality conditions, we have shown that the
functional derivative calculated in this context has a relation with the shape-
derivative given by other authors, which under suitable regularity assumptions,
can be interpreted as a limit case. The characterization obtained for the descrip-
tion and location of the optimal inclusions seems related to work due to Har-
rel,Kröger,Kurata [14]. Further applications to the analysis of conjectures raised
by these authors and Henrot (cf. chapter 3 of [15]) could be worthwhile.
From the numerical analysis perspective, due to the structure of the func-
tional derivative, the approach presented is well suited for numerical calculations
and a gradient method is proposed. A main advantage seems to be the stability
of the algorithm, as calculations are performed with a fixed grid.
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