Revista de Matema´tica: Teor´ıa y Aplicaciones 2009 16(1) : 127–136
cimpa – ucr issn: 1409-2433
minimization of the first eigenvalue in
problems involving the bi-laplacian
Claudia Anedda∗ Fabrizio Cuccu† Giovanni Porru‡
Recibido/Received: 20 Feb 2008 — Aceptado/Accepted: 25 Jul 2008
Abstract
This paper concerns the minimization of the first eigenvalue in problems involving
the bi-Laplacian under either homogeneous Navier boundary conditions or homoge-
neous Dirichlet boundary conditions. Physically, in case ofN = 2, our equation models
the vibration of a non homogeneous plate Ω which is either hinged or clamped along
the boundary. Given several materials (with different densities) of total extension |Ω|,
we investigate the location of these materials inside Ω so to minimize the first mode
in the vibration of the corresponding plate.
Keywords: bi-Laplacian, first eigenvalue, minimization.
Resumen
Este art´ıculo trata de la minimizacio´n del primer autovalor en problemas relativos
al bi-Laplaciano bajo condiciones de frontera homoge´neas de tipo Navier o Dirichlet.
F´ısicamente, en el problema bi-dimensional, nuestra ecuacin modela la vibracio´n de
una placa inhomoge´nea Ω fija con goznes a lo largo de su borde. Dados varios ma-
teriales (de diferentes densidades) y extensio´n total |Ω|, investigamos cua´l debe ser
la localizacio´n de tales materiales en la placa para minimizar el primer modo de su
vibracio´n.
Palabras clave: bi-Laplaciano, primer autovalor, minimizacio´n.
Mathematics Subject Classification: 35P15, 47A75, 49K20.
∗Dipartimento di Matematica e Informatica, Universita´ di Cagliari Via Ospedale 72, 09124 Cagliari,
Italy. E-Mail: canedda@unica.it
†Misma direccio´n que C. Anedda. E-Mail: fcuccu@unica.it
‡Misma direccio´n que C. Anedda. E-Mail: porru@unica.it
127
128 C. Anedda – F. Cuccu – G. Porru Rev.Mate.Teor.Aplic. (2009) 16(1)
1 Introduction
Let Ω be a bounded smooth domain in RN and let g0 be a measurable function satisfying
0 ≤ g0 ≤ M in Ω, where M is a positive constant. To avoid trivial situations, we always
assume g0 6≡ 0 and g0 6≡M . Define G as the family of all measurable functions defined in
Ω which are rearrangements of g0. Consider the following eigenvalue problems
∆2u = λgu, in Ω, u = ∆u = 0 on ∂Ω, (1)
and
∆2v = Λgv, in Ω, v =
∂v
∂ν
= 0 on ∂Ω, (2)
where g ∈ G, λ = λg, Λ = Λg are the first eigenvalues and u, v are the corresponding
eigenfunctions. The operator ∆2 stands for the usual bi-Laplacian, that is ∆2u = ∆(∆u).
The first eigenvalue λ of problem (1) is obtained by minimizing the associate Rayleigh
quotient
λ = inf
{∫
Ω(∆z)
2dx∫
Ω gz
2dx
: z, ∆z ∈ H10 (Ω), z 6≡ 0
}
. (3)
The first eigenvalue Λ of problem (2) is obtained by minimizing the quotient
Λ = inf
{∫
Ω(∆z)
2dx∫
Ω gz
2dx
: z ∈ H20 (Ω), z 6≡ 0
}
. (4)
It is well known that the inferior is attained in both cases [14]. The minimum of (3)
satisfies problem (1) in the weak sense, that is∫
Ω
∆u∆z dx = λ
∫
Ω
guz dx, ∀z : z,∆z ∈ H10 (Ω).
The minimum of (4) satisfies problem (2) in the sense∫
Ω
∆v∆z dx = Λ
∫
Ω
gvz dx, ∀z ∈ H20 (Ω).
By regularity results (see [1]) the solutions to problems (1) and (2) belong to H4loc(Ω).
In this paper we investigate the problems
min
g∈G
λg, and min
g∈G
Λg. (5)
Let us give a motivation for the study of these problems in case of N = 2. Physically,
our equations model the vibration of a non homogeneous plate Ω which is either hinged
or clamped along the boundary ∂Ω. Given several materials (with different densities) of
total extension |Ω|, we investigate the location of these materials inside Ω so to minimize
the first mode in the vibration of the plate. The corresponding problem for second order
equations has been discussed in several papers, see for example [6], [7], [9].
The paper is organized as follows. In Section 2 we collect some definitions and known
results. In Section 3 we investigate the problems (5) proving results of existence and
results of representation of minimizers. In case Ω is a ball we prove uniqueness for both
problems.
minimization of the first eigenvalue in problems involving the ... 129
2 Preliminaries
Denote with |E| the Lebesgue measure of the (measurable) set E ⊂ RN . Given a mea-
surable function g0(x) defined in Ω we say that g(x), defined in Ω, belongs to the class of
rearrangements G = G(}′) if |{x ∈ Ω : g(x) ≥ β}| = |{x ∈ Ω : g0(x) ≥ β}| ∀β ∈ R.
We make use of the following results.
Lemma 2.1 Let g ∈ L1(Ω) and let u ∈ L1(Ω). Suppose that every level set of u (that is,
sets of the form u−1({α})), has measure zero. Then there exists an increasing function φ
such that φ(u) is a rearrangement of g.
Proof. The assertion follows by Lemma 2.9 of [4]. 
Lemma 2.2 Let G be the set of rearrangements of a fixed function g0 ∈ Lr(Ω), r > 1,
g0 6≡ 0, and let G denote the weak closure of G in Lr(Ω). If u ∈ Ls(Ω), s = r/(r − 1),
u 6≡ 0, and if there is an increasing function φ such that φ(u) ∈ G then∫
Ω
g u dx ≤
∫
Ω
φ(u)u dx ∀g ∈ G,
and the function φ(u) is the unique maximizer relative to G.
Proof. The assertion follows by Lemma 2.4 of [4]. 
Lemma 2.3 Let G be the set of rearrangements of a fixed function g0 ∈ Lr(Ω), r > 1,
g0 6≡ 0, and let u ∈ Ls(Ω), s = r/(r − 1), u 6≡ 0. There exists g ∈ G such that∫
Ω
g u dx ≤
∫
Ω
g u dx ∀g ∈ G.
Proof. It follows by Lemma 2.4 of [4]. See also [5]. 
Next we recall a well known rearrangement inequality. For u non negative in Ω, u]
denotes the decreasing Schwarz rearrangement of u; that is, u] is defined in Ω], the ball
centered in the origin with measure equal to |Ω|, is radially symmetric, decreases as |x|
increases, and satisfies
|{x ∈ Ω : u(x) ≥ β}| = |{x ∈ Ω] : u](x) ≥ β}| ∀β ≥ 0.
If u ∈ H10 (Ω) is non-negative and if u] is the decreasing Schwarz rearrangement of u then
u] ∈ H10 (Ω]) and the inequality∫
Ω]
|∇u]|2 dx ≤
∫
Ω
|∇u|2 dx (6)
holds. The case of equality in (6) has been considered in [3]. We have
Lemma 2.4 Let u ∈ H10 (Ω) be non-negative, and suppose equality holds in (6). If
|{x ∈ Ω] : ∇u∗(x) = 0, 0 < u∗(x) < sup
Ω
u(x)}| = 0
then u is a translate of u].
Proof. See Theorem 1.1 of [3] or the monograph [13]. 
130 C. Anedda – F. Cuccu – G. Porru Rev.Mate.Teor.Aplic. (2009) 16(1)
3 Main results
Let Ω ⊂ RN be a bounded smooth domain and let M > 0 be a real number. Let G be the
family of all functions defined in Ω which are rearrangements of a given function g0 with
0 ≤ g0(x) ≤M , g0(x) 6≡ 0, g0(x) 6≡M . For g ∈ G, let λg be the first eigenvalue of problem
(1), and let Λg be the first eigenvalue of problem (2). We investigate the problems
min
g∈G
λg, and min
g∈G
Λg.
Recalling (3) and (4) we can formulate the previous problems as
min
g∈G
λg = min
{∫
Ω(∆z)
2dx∫
Ω g z
2dx
: g ∈ G, z ∈ H10 (Ω), ∆z ∈ H10 (Ω)
}
, (7)
and
min
g∈G
Λg = min
{∫
Ω(∆z)
2dx∫
Ω g z
2dx
: g ∈ G, z ∈ H20 (Ω)
}
. (8)
Theorem 3.1 Let 0 ≤ g0(x) ≤ M , g0(x) 6≡ 0, g0(x) 6≡ M , and let G be the class of all
rearrangements of g0. Then
a) there exists g ∈ G such that
λg = min
g∈G
λg;
b) there exists g˜ ∈ G such that
Λg˜ = min
g∈G
Λg.
Proof. We prove first part a). Let
I = inf
g∈G
λg = lim
i→∞
λgi = lim
i→∞
∫
Ω(∆ui)
2dx∫
Ω giu
2
i dx
, (9)
where ui = ugi is the eigenfunction corresponding to gi normalized so that∫
Ω
u2i dx = 1.
We may assume that the sequence {λgi} is decreasing. By (9) and the latter equation we
get ∫
Ω
(∆ui)2dx ≤ λg1M. (10)
On the other side, since ui vanishes on ∂Ω, by Lemma 9.17 of [11] we have
‖ui‖H2(Ω) ≤ C‖∆ui‖L2(Ω)
with C independent of i. It follows that the norms ‖ui‖H2(Ω) and ‖∆ui‖L2(Ω) are equiva-
lent. This fact and (10) imply that the sequence {ui} is bounded in the H2(Ω) norm and
some subsequence (still denoted {ui}) converges weakly in H2(Ω) to a function u. We
minimization of the first eigenvalue in problems involving the ... 131
can also assume that {ui} converges strongly to u in L2+(Ω) for some  > 0. Further-
more, since {gi} is bounded in L∞(Ω), it must contain a subsequence (still denoted {gi})
converging weakly to η ∈ Lr(Ω) for any r > 1. We have∫
Ω
giu
2
i dx−
∫
Ω
ηu2dx =
∫
Ω
(gi − η)u2dx+
∫
Ω
gi(u2i − u2)dx.
We find
lim
i→∞
∫
Ω
(gi − η)u2dx = 0,
because u2 ∈ Ls(Ω) for some s > 1 and gi → η weakly in Lr(Ω) for r = s/(s − 1).
Moreover,
lim
i→∞
∫
Ω
gi(u2i − u2)dx = 0.
The latter result can be proved by using Lebesgue’s theorem as follows. Since ui → u in
L2(Ω), we have (up to a subsequence)
lim
i→∞
gi(u2i − u2) = 0 a.e. in Ω,
and
gi|u2i − u2| ≤M(ψ2 + u2),
for some integrable function ψ2. Indeed, since ui converges in L2(Ω) one can find ψ ∈ L2(Ω)
such that ui(x) ≤ ψ(x) a.e. for some subsequence of ui [10]. Hence,
lim
i→∞
∫
Ω
giu
2
i dx =
∫
Ω
η u2dx. (11)
By Lemma 2.3 we can find g ∈ G such that∫
Ω
η u2dx ≤
∫
Ω
g u2dx. (12)
On the other side, from the inequality
0 ≤
∫
Ω
(
∆(ui − u)
)2
dx =
∫
Ω
(∆ui)2dx− 2
∫
Ω
∆ui∆u dx+
∫
Ω
(∆u)2dx
and the weak convergence of {ui} to u in H2(Ω) we find
lim inf
i→∞
∫
Ω
(∆ui)2dx ≥
∫
Ω
(∆u)2dx.
By using the latter result together with (11) and (12) we have
I = lim
i→∞
∫
Ω(∆ui)
2dx∫
Ω giu
2
i dx
≥
∫
Ω(∆u)
2dx∫
Ω η u
2dx
≥
∫
Ω(∆u)
2dx∫
Ω g u
2dx
. (13)
132 C. Anedda – F. Cuccu – G. Porru Rev.Mate.Teor.Aplic. (2009) 16(1)
Our minimizing sequence ui satisfies (in a weak sense)
∆(∆ui) = λgigiui, ∆ui ∈ H10 (Ω).
If we multiply by −∆ui and integrate over Ω, after simplification we find
‖∇(∆ui)‖L2(Ω) ≤ λgi‖giui‖L2(Ω).
Since λgi is decreasing, 0 ≤ gi ≤M and ‖ui‖L2(Ω) = 1 we find that ‖∇(∆ui)‖L2(Ω) ≤ λg1M.
As a consequence, since ∆ui ∈ H10 (Ω) we also have ∆u ∈ H10 (Ω). Therefore, if λg is
the (first) eigenvalue corresponding to g in problem (1), and if ug is a corresponding
eigenfunction then by (3) we have∫
Ω(∆u)
2dx∫
Ω g u
2dx
≥
∫
Ω(∆ug)
2dx∫
Ω g u
2
gdx
= λg ≥ I.
By the latter result and (13) we must have I = λg. Part a) of the theorem is proved.
The proof of part b) is similar. Define
I˜ = inf
g∈G
Λg = lim
i→∞
∫
Ω(∆vi)
2dx∫
Ω giv
2
i dx
,
where vi = vgi is the eigenfunction corresponding to gi normalized so that∫
Ω
v2i dx = 1.
Of course, {gi} is not, in general, the same as for part a). Arguing as in the previous
case we find that vi is bounded in the norm of H2(Ω). Therefore, a subsequence (still
denoted {vi}) converges weakly in H2(Ω) to a function v˜ ∈ H20 (Ω). We can also assume
that {vi} converges strongly to v˜ in L2+(Ω) for some  > 0. Furthermore, {gi} must
contain a subsequence (still denoted {gi}) converging weakly to some ζ ∈ Lr(Ω) for any
r > 1. Hence,
lim
i→∞
∫
Ω
giv
2
i dx =
∫
Ω
ζ v˜2dx.
By Lemma 2.3 we can find g˜ ∈ G such that∫
Ω
ζ v˜2dx ≤
∫
Ω
g˜ v˜2dx.
Moreover we have
lim inf
i→∞
∫
Ω
(∆vi)2dx ≥
∫
Ω
(∆v˜)2dx.
Using the last three results we find
I˜ = lim
i→∞
∫
Ω(∆vi)
2dx∫
Ω giv
2
i dx
≥
∫
Ω(∆v˜)
2dx∫
Ω ζ v˜
2dx
≥
∫
Ω(∆v˜)
2dx∫
Ω g˜ v˜
2dx
. (14)
minimization of the first eigenvalue in problems involving the ... 133
Recall that v˜ ∈ H20 (Ω) and g˜ ∈ G. If Λg˜ is the (first) eigenvalue corresponding to g˜ in
problem (2), and if vg˜ is a corresponding eigenfunction then, using (4) we have∫
Ω(∆v˜)
2dx∫
Ω g˜ v˜
2dx
≥
∫
Ω(∆vg˜)
2dx∫
Ω g˜ v
2
g˜dx
= Λg˜ ≥ I˜ . (15)
By (14) and (15) we must have I˜ = Λg˜. The theorem is proved. 
We prove the so called Euler-Lagrange equation for solutions of our minimization
problems. Actually, there is a difference between the two cases. Concerning problem (1),
we know that the first eigenfunction does not change sign, and we can assume that it is
positive in Ω. Concerning problem (2), there are domains Ω such that the corresponding
first eigenfunction is sign changing, and there are domains such that the corresponding
first eigenfunction is positive: see [12] and references therein.
In what follows we write {g(x) > 0} instead of {x ∈ Ω : g(x) > 0}.
Theorem 3.2 a) Suppose g is a solution to problem (7). There exists an increasing
function φ such that
g = φ(ug).
b) Suppose g is a solution to problem (8) and that Ω is such that the corresponding first
eigenfunction of problem (2) is positive. There exists an increasing function ϕ such that
g = ϕ(ug).
Proof. If ug is the positive normalized eigenfunction corresponding to the minimizer g of
problem (7), for any g ∈ G we have∫
Ω(∆ug)
2dx∫
Ω gu
2
gdx
≤
∫
Ω(∆ug)
2dx∫
Ω gu
2
gdx
.
Hence, ∫
Ω
gu2g dx ≤
∫
Ω
gu2g dx (16)
for all g ∈ G.
On the other side, we know that the function ug satisfies the eigenvalue equation
∆2ug = λ gug.
If −∆ug = v, by the above equation we have −∆v ≥ 0 in Ω and v = 0 on ∂Ω. It follows
that v(x) > 0 in Ω. Since −∆ug > 0, the function ug cannot have level sets of positive
measure. Hence, by Lemma 2.1, inequality (16) and Lemma 2.2 we infer the existence of
an increasing function φ1 such that g = φ1(u2g). Thus, part a) of the theorem follows with
φ(t) = φ1(t2).
If ug is the positive normalized eigenfunction corresponding to the minimizer g of
problem (8), inequality (16) holds for all g ∈ G. Moreover, ∆2ug = Λ gug. By this
equation, the function ug cannot have level sets of positive measure on {g(x) > 0}. If the
134 C. Anedda – F. Cuccu – G. Porru Rev.Mate.Teor.Aplic. (2009) 16(1)
set {g(x) = 0} has zero measure, by Lemma 2.1, inequality (16) and Lemma 2.2 we infer
the existence of an increasing function ϕ1 such that g = ϕ1(u2g). Thus, in this case part b)
of the theorem follows with ϕ(t) = ϕ1(t2). Otherwise, setting E = {g(x) = 0}, we define:
S = sup
x∈E
(ug(x))2.
By using (16) one proves that (ug(x))2 ≥ S on {g(x) > 0} a.e. For the proof of this result
we refer to [8], Theorem 3.2. Since ug cannot have level sets of positive measure on Ω \E,
by Lemma 2.1 we infer the existence of an increasing function ϕ1 : (S,∞) → [0,M ] such
that ϕ1(u2g) is a rearrangement of g on Ω \E. Now we define an increasing function ϕ2 as
ϕ2(t) =
{
0 t ≤ S
ϕ1(t) t > S.
Since ϕ2(u2g) is a rearrangement of g on Ω, by inequality (16) and Lemma 2.2 we infer that
g = φ2(u2g). Part b) of the theorem follows taking ϕ(t) = ϕ2(t
2). The theorem is proved.

Remarks. Theorem 3.2 gives some information on the location of the materials
in order to minimize the first eigenvalue of problem (7). Indeed, since the associate
eigenfunction ug vanishes on the boundary ∂Ω, and g = φ(ug) with φ increasing, the
material with higher density must be located where ug is large, that is, far from ∂Ω. The
same remark holds for problem (8) in appropriate domains.
Theorem 3.3 Let B be a ball in RN , and let g be a minimizer of either problem (7) or
problem (8) with Ω = B. Then g = g].
Proof. If g is a minimizer of problem (7) and if u = ug is a corresponding positive
eigenfunction we have
λg =
∫
B(∆u)
2dx∫
B gu
2dx
. (17)
Put
−∆u = z. (18)
Then
−∆z = λggu.
Since u > 0 in B and z = 0 on ∂B we have z > 0 in B. If z] is the Schwarz decreasing
rearrangement of z then z] ∈ H10 (B) and∫
B
(∆u)2dx =
∫
B
(z])2dx. (19)
Furthermore, if u is the solution to the problem
−∆u = z] in B, u = 0 on ∂B (20)
minimization of the first eigenvalue in problems involving the ... 135
then, by a result of G. Talenti ([15], Theorem 1, (iv)) we have
u] ≤ u in B. (21)
By a well known inequality on rearrangements and (21) we find∫
B
gu2dx ≤
∫
B
g](u])2dx ≤
∫
B
g](u)2dx. (22)
Since (z])2 = (∆u)2, by (19), (17) and (22) we find
λg ≥
∫
B(∆u)
2dx∫
B g
](u)2dx
≥
∫
B(∆ug])
2dx∫
B g
](ug])2dx
= λg] .
In the last step we have used the fact that u is admissible (because u = ∆u = 0 on ∂B)
and the variational characterization of λg] . Since λg is a minimizer, we must have λg] = λg
and equality must hold in (22). In particular,∫
B
g](u])2dx =
∫
B
g](u)2dx.
Recalling that g(x) is positive in a set of positive measure we have g](x) > 0 in a ball
B(r0) of radius r0 for some r0 > 0. Therefore the previous equation and (21) imply
that u](0) = u(0). An inspection of the proof of Talenti’s result [15] (see also [2]) yields
u](x) = u(x) in all of B. Moreover by (18), (20) with u = u], and (6) we find∫
B
|∇u|2dx =
∫
B
uzdx ≤
∫
B
u]z]dx =
∫
B
|∇u]|2dx ≤
∫
B
|∇u|2dx.
It follows that ∫
B
|∇u|2dx =
∫
B
|∇u]|2dx.
By Lemma 2.4 we get u(x) = u](x) in B. Furthermore, by Theorem 3.2 a) we have
g = φ(u) for some increasing function φ. This implies that g is radially symmetric and
decreasing, hence g = g]. The theorem is proved in this case.
Let us come to problem (8). Putting −∆v = w and recalling that ∂v∂ν = 0 on ∂B we
find ∫
B
w dx = −
∫
B
∆v dx =
∫
∂B
∂v
∂ν
dσ = 0.
This means that w(x) is sign changing in B. Let w](x) be the signed Schwarz decreasing
rearrangement of w(x) and let
−∆v = w] in B, v = 0 on ∂B.
Since
∫
B w
]dx = 0 the result of Talenti [15] continues to hold as observed also in [2].
Hence,
v] ≤ v in B.
Moreover, since
0 = −
∫
B
w dx = −
∫
B
w]dx =
∫
B
∆v dx =
∫
∂B
∂v
∂ν
dσ =
∂v
∂ν
|∂B|,
we have v ∈ H20 (B). The proof continues as in the previous case. 
136 C. Anedda – F. Cuccu – G. Porru Rev.Mate.Teor.Aplic. (2009) 16(1)
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