Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 3, Article 91, 2005
THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR
IDRISSA LY
L ABORATOIRE DE M ATHÉMATIQUES ET A PPLICATIONS DE M ETZ
I LE DU S AULCY, 57045 M ETZ C EDEX 01, F RANCE .
idrissa@math.univ-metz.fr
Received 12 February, 2005; accepted 17 June, 2005
Communicated by S.S. Dragomir
A BSTRACT. In this paper, using the Hausdorff topology in the space of open sets under some
capacity constraints on geometrical domains we prove the strong continuity with respect to the
moving domain of the solutions of a p-Laplacian Dirichlet problem. We are also interested in
the minimization of the first eigenvalue of the p-Laplacian with Dirichlet boundary conditions
among open sets and quasi open sets of given measure.
Key words and phrases: p-Laplacian, Nonlinear eigenvalue problems, Shape optimization.
2000 Mathematics Subject Classification. 35J70, 35P30, 35R35.
1. I NTRODUCTION
Let Ω be an open subset of a fixed ball D in RN , N ≥ 2 and 1 < p < +∞. Consider the
Sobolev space W01,p (Ω) which is the closure of C ∞ functions compactly supported in Ω for the
norm
Z
Z
||u||p1,p =
|u(x)|p dx +
Ω
|∇u(x)|p dx.
Ω
The p-Laplacian is the operator defined by
∆p : W01,p (Ω) −→ W −1,q (Ω)
u 7−→ ∆p u = div(|∇u|p−2 ∇u),
where W −1,q (Ω) is the dual space of W01,p (Ω) and we have 1 < p, q < ∞,
We are interested in the nonlinear eigenvalue problem
(
−∆p u − λ|u|p−2 u = 0 in Ω,
(1.1)
u
= 0 on ∂Ω.
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
037-05
1
p
+
1
q
= 1.
2
I DRISSA LY
Let u be a function of W01,p (Ω), not identically 0. The function u is called an eigenfunction if
Z
Z
p−2
|∇u(x)| ∇u∇φdx = λ |u(x)|p−2 uφdx
Ω
Ω
C0∞ (Ω).
for all φ ∈
The corresponding real number λ is called an eigenvalue.
Contrary to the Laplace operator, the p-Laplacian spectrum has not been proved to be discrete.
In [15], the first eigenvalue and the second eigenvalue are described.
Let D be a bounded domain in RN and c > 0. Let us denote λp1 (Ω) as the first eigenvalue for
the p-Laplacian operator. The aim of this paper is to study the isoperimetric inequality
min{λp1 (Ω), Ω ⊆ D and |Ω| = c}
and its continuous dependance with respect to the domain. We extend the Rayleigh-FaberKhran inequality to the p-Laplacian operator and study the minimization of the first eigenvalue
in two dimensions when D is a box. By considering a class of simply connected domains, we
study the stability of the minimizer Ωp of the first eigenvalue with respect to p that is if Ωp
is a minimizer of the first eigenvalue for the p-Laplacian Dirichlet, when p goes to 2, Ω2 is
also a minimizer of the first eigenvalue of the Laplacian Dirichlet. Thus we will give a formal
justification of the following conjecture: "Ω is a minimizer of given volume c, contained in a
fixed box D and if D is too small to contain a ball of the same volume as Ω. Are the free parts
of the boundary of Ω pieces of circle?"
Henrot and Oudet solved this question and proved by using the Hölmgren uniqueness theorem, that the free part of the boundary of Ω cannot be pieces of circle, see [10].
The structure of this paper is as follows: The first section is devoted to the definition of two
eigenvalues. In the second section, we study the properties of geometric variations for the first
eigenvalue. The third section is devoted to the minimization of the first eigenvalue among open
(or, if specified, quasi open) sets of given volume. In the fourth part we discuss the minimization
of the first eigenvalue in a box in two dimensions.
Let D be a bounded open set in RN which contains all the open (or, if specified, quasi open)
subsets used.
2. D EFINITION OF THE F IRST
AND
S ECOND E IGENVALUES
The first eigenvalue is defined by the nonlinear Rayleigh quotient
R
R
|∇φ(x)|p dx
|∇u1 (x)|p dx
ΩR
Ω
R
λ1 (Ω) =
min
,
=
|φ(x)|p
|u1 (x)|p dx
φ∈W01,p (Ω),φ6=0
Ω
Ω
where the minimum is achieved by u1 which is a weak solution of the Euler-Lagrange equation
−∆p u − λ|u|p−2 u = 0 in Ω
(2.1)
u
= 0 on ∂Ω.
The first eigenvalue has many special properties, it is strictly positive, simple in any bounded
connected domain see [15]. And u1 is the only positive eigenfunction for the p-Laplacian
Dirichlet see also [15].
In [15], the second eigenvalue is defined by
R
|∇φ(x)|p dx
ΩR
λ2 (Ω) = inf max
,
C∈C2 C
|φ(x)|p
Ω
where
C2 := {C ∈ W01,p (Ω) : C = −C such that genus(C) ≥ 2}.
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
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T HE F IRST E IGENVALUE FOR THE p-L APLACIAN O PERATOR
3
In [1], Anane and Tsouli proved that there does not exist any eigenvalue between the first and
the second ones.
3. P ROPERTIES OF THE G EOMETRIC VARIATIONS
In this section we are interested in the continuity of the map
Ω 7−→ λ1 (Ω).
Then, we have to fix topology on the space of the open subsets of D. On the family of the open
subsets of D, we define the Hausdorff complementary topology, denoted H c given by the metric
dH c (Ωc1 , Ωc2 ) = sup |d(x, Ωc1 ) − d(x, Ωc2 )|.
x∈RN
The H c -topology has some good properties for example the space of the open subsets of D is
Hc
compact. Moreover if Ωn → Ω, then for any compact K ⊂⊂ Ω we have K ⊂⊂ Ωn for n large
enough.
However, perturbations in this topology may be very irregular and in general situations the
continuity of the mapping Ω 7−→ λ1 (Ω) fails, see [4].
In order to obtain a compactness result we impose some additional constraints on the space
of the open subsets of D which are expressed in terms of the Sobolev capacity. There are many
ways to define the Sobolev capacity, we use the local capacity defined in the following way.
Definition 3.1. For a compact set K contained in a ball B,
Z
|∇φ|p , φ ∈ C0∞ (B), φ ≥ 1 on
cap(K, B) := inf
K .
B
Definition 3.2.
(1) It is said that a property holds p-quasi everywhere (abbreviated as p − q.e) if it holds
outside a set of p-capacity zero.
(2) A set Ω ⊂ RN is said to be quasi open if for every > 0 there exists an open set Ω
such that Ω ⊆ Ω , and cap(Ω \Ω) < .
(3) A function u : RN −→ R is said p-quasi continuous if for every > 0 there exists an
open set Ω such that cap(Ω ) < and u|R\Ω is continuous in R\Ω .
It is well known that every Sobolev function u ∈ W 1,p (RN ) has a p-quasi continuous representative which we still denote u. Therefore, level sets of Sobolev functions are p-quasi open
sets; in particular Ωv = {x ∈ D; |v(x)| > 0} is quasi open subsets of D.
Definition 3.3. We say that an open set Ω has the p − (r, c) capacity density condition if
∀x ∈ ∂Ω,
∀0 < δ < r,
cap(Ωc ∩ B̄(x, δ), B(x, 2δ))
≥c
cap(B̄(x, δ), B(x, 2δ))
where B(x, δ) denotes the ball of raduis δ, centred at x.
Definition 3.4. We say that the sequence of the spaces W01,p (Ωn ) converges in the sense of
Mosco to the space W01,p (Ω) if the following conditions hold
(1) The first Mosco condition: For all φ ∈ W01,p (Ω),there exists a sequence φn ∈ W01,p (Ωn )
such that φn converges strongly in W01,p (D) to φ.
(2) The second Mosco condition: For every sequence φnk ∈ W01,p (Ωnk ) weakly convergent
in W01,p (D) to a function φ, we have φ ∈ W01,p (Ω).
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
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4
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Definition 3.5. We say a sequence (Ωn ) of open subsets of a fixed ball D γp -converges to Ω if
for any f ∈ W −1,q (Ω) the solutions of the Dirichlet problem
−∆p un = f
in
Ωn , un ∈ W01,p (Ωn )
converge strongly in W01,p (D), as n −→ +∞, to the solution of the corresponding problem in
Ω, see [7], [8].
By Op−(r,c) (D), we denote the family of all open subsets of D which satisfy the p − (r, c)
capacity density condition. This family is compact in the H c topology see [4]. In [2], D. Bucur
and P. Trebeschi, using capacity constraints analogous to those introduce in [3] and [4] for the
linear case, prove the γp -compactness result for the p-Laplacian. In the same way, they extend
the continuity result of Šveràk [19] to the p-Laplacian for p ∈ (N − 1, N ], N ≥ 2. The reason
of the choice of p is that in RN the curves have p positive capacity if p > N − 1. The case
p > N is trivial since all functions in W 1,p (RN ) are continuous.
Let us denote by
Ol (D) = {Ω ⊆ D, ]Ωc ≤ l}
where ] denotes the number of the connected components. We have the following theorem.
Theorem 3.1 (Bucur-Trebeschi). Let N ≥ p > N − 1. Consider the sequence (Ωn ) ⊆ Ol (D)
and assume that Ωn converges in Hausdorff complementary topology to Ω. Then Ω ⊆ Ol (D)
and Ωn γp −converges to Ω.
Proof of Theorem 3.1. See [2].
For N = 2 and p = 2, Theorem 3.1 becomes the continuity result of Šveràk [19].
Back to the continuity result, we use the above results to prove the following theorem.
Theorem 3.2. Consider the sequence (Ωn ) ⊆ Ol (D). Assume that Ωn converges in Hausdorff
complementary topology to Ω. Then λ1 (Ωn ) converges to λ1 (Ω).
Proof of Theorem 3.2. Let us take
R
λ1 (Ωn ) =
min
1,p
φn ∈W0 (Ωn ),φn 6=0
|∇φn (x)|p dx
Ωn
R
=
|φn (x)|p
Ωn
where the minimum is attained by un , and
R
λ1 (Ω) =
min
φ∈W01,p (Ω),φ6=0
R
Ωn
|∇un (x)|p dx
R
Ωn
|un (x)|p
,
R
|∇φ(x)|p dx
|∇u1 (x)|p dx
RΩ
=
,
|φ(x)|p
|u1 (x)|p dx
Ω
Ω
ΩR
where the minimum is achieved by u1 .
By the Bucur and Trebeschi theorem, Ωn γp converges to Ω. This implies W01,p (Ωn ) converges in the sense of Mosco to W01,p (Ω).
If the sequence (un ) is bounded in W01,p (D), then there exists a subsequence still denoted
un such that un converges weakly in W01,p (D) to a function u. The second condition of Mosco
implies that u ∈ W01,p (Ω).
Using the weak lower semicontinuity of the Lp −norm, we have the inequality
R
R
R
|∇u(x)|p dx
|∇u1 (x)|p dx
|∇un (x)|p dx
ΩR
ΩR
DR
≥
≥
,
lim inf
p
p
p
n−→+∞
|u
(x)|
|u(x)|
|u
(x)|
n
1
D
Ω
Ω
then
(3.1)
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
lim inf λ1 (Ωn ) ≥ λ1 (Ω).
n→+∞
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T HE F IRST E IGENVALUE FOR THE p-L APLACIAN O PERATOR
5
Using the first condition of Mosco, there exists a sequence (vn ) ∈ W01,p (Ωn ) such that vn
converges strongly in W01,p (D) to u1 .
We have
R
|∇vn (x)|p dx
DR
λ1 (Ωn ) ≤
|v (x)|p
D n
this implies that
R
|∇vn (x)|p dx
DR
lim sup λ1 (Ωn ) ≤ lim sup
|vn (x)|p
n−→+∞
n−→+∞
R D
|∇vn (x)|p dx
= lim DR
n−→+∞
|v (x)|p
D n
R
|∇u1 (x)|p dx
= ΩR
|u1 (x)|p
Ω
then
lim sup λ1 (Ωn ) ≤ λ1 (Ω).
(3.2)
n→+∞
By the relations (3.1) and (3.2) we conclude that λ1 (Ωn ) converges to λ1 (Ω).
4. S HAPE O PTIMIZATION R ESULT
We extend the classical inequality of Faber-Krahn for the first eigenvalue of the Dirichlet
Laplacian to the Dirichlet p-Laplacian. We study this inequality when Ω is a quasi open subset
of D.
Definition 4.1. Let Ω be an open subset and bounded in RN . We denote by B the ball centred
at the origin with the same volume as Ω. Let u be a non negative function in Ω, which vanishes
on ∂Ω. For all c > 0, the set {x ∈ Ω, u(x) > c} is called the level set of u.
The function u∗ which has the following level set
∀c > 0,
{x ∈ B, u∗ (x) > c} = {x ∈ Ω, u(x) > c}∗
is called the Schwarz rearrangement of u. The level sets of u∗ are the balls that we obtain by
rearranging the sets of the same volume of u.
We have the following lemma.
Lemma 4.1. Let Ω be an open subset in RN .
Let ψ be any continuous function on R∗+ , we have
R
R
(1) RΩ ψ(u(x))dx = RΩ∗ ψ(u∗ (x))dx
u∗ is equi-mesurable with u.
∗
∗
(2) Ω u(x)v(x)dx ≤ Ω∗ u (x)v (x)dx.
(3) If u ∈ W01,p (Ω), p > 1 then u∗ ∈ W01,p (Ω∗ ) and
Z
Z
p
|∇u(x)| dx ≥
|∇u∗ (x)|p dx Pòlya inequality.
Ω
Proof of Lemma 4.1. See [12].
Ω∗
The basic result for the minimization of eigenvalues is the conjecture of Lord Rayleigh: “The
disk should minimize the first eigenvalue of the Laplacian Dirichlet among every open set of
given measure”. We extend the Rayleigh-Faber-Krahn inequality to the p-Laplacian operator.
Let Ω be any open set in RN with finite measure. We denote by λ1 (Ω) the first eigenvalue for
the p-Laplacian operator with Dirichlet boundary conditions. We have the following theorem.
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
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6
I DRISSA LY
Theorem 4.2. Let B be the ball of the same volume as Ω, then
open set of RN , |Ω| = |B|}.
λ1 (B) = min{λ1 (Ω), Ω
Proof of Theorem 4.2. Let u1 be the first eigenfunction of λ1 (Ω), it is strictly positive see [15].
By Lemma 4.1, equi-mesurability of the function u1 and its Schwarz rearrangement u∗1 gives
Z
Z
p
|u1 (x)| dx =
|u∗1 (x)|p dx.
Ω
B
The Pòlya inequality implies that
Z
Z
p
|∇u1 (x)| dx ≥
|∇u∗1 (x)|p dx.
Ω
B
By the two conditions, it becomes
R
R
∗
p
|∇u1 (x)|p dx
|∇u
(x)|
dx
RB ∗ 1
RΩ
≤
= λ1 (Ω).
|u1 (x)|p dx
|u1 (x)|p dx
B
Ω
This implies that
R
R
p
|∇v(x)|
dx
|∇u∗ (x)|p dx
RB
RB ∗ 1
λ1 (B) =
min
≤
≤ λ1 (Ω).
|v(x)|p dx
|u1 (x)|p dx
v∈W01,p (B),v6=0
B
B
Remark 4.3. The solution Ω must satisfy an optimality condition. We suppose that ΩC 2 − regular
to compute the shape derivative. We deform the domain Ω with respect to an admissible vector
field V to compute the shape derivative
J(Id + tΩ) − J(Ω)
.
t−→0
t
dJ(Ω; V ) = lim
We have the variation calculation
−div(|∇u|p−2 ∇u) = λ|u|p−2 u
Z
p−2
− div(|∇u| ∇u)φdx =
λ|u|p−2 uφdx,
Ω
Z
ZΩ
|∇u|p−2 ∇u∇φdx =
λ|u|p−2 uφdx,
Z
Ω
for all
φ ∈ D(Ω)
for all
φ ∈ D(Ω)
Ω
R
R
Let us take J(Ω) = Ω |∇u|p−2 ∇u∇φdx and J1 (Ω) = Ω λ|u|p−2 uφdx. We have dJ(Ω; V ) =
dJ1 (Ω; V ).
We use the classical Hadamard formula to compute the Eulerian derivative of the functional
J at the point Ω in the direction V.
Z
Z
p−2
0
dJ(Ω; V ) = (|∇u| ∇u∇φ) dx + div(|∇u|p−2 ∇u∇φ.V (0))dx.
Ω
Ω
We have
Z
(|∇u|p−2 ∇u∇φ)0 dx
Ω
Z
Z
Z
p−2 0
p−2
0
= (|∇u| ) ∇u∇φdx + |∇u| ∇u(∇φ) dx + |∇u|p−2 (∇u)0 ∇φdx.
Ω
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
Ω
Ω
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T HE F IRST E IGENVALUE FOR THE p-L APLACIAN O PERATOR
7
We have the expression
p−2 0
(|∇u|p−2 )0 = (|∇u|2 ) 2
p−4
p−2
(|∇u|2 )0 (|∇u|2 ) 2
2
p−2 0
(|∇u| ) = (p − 2)∇u∇u0 |∇u|p−4 .
=
Then
Z
|∇u|p−4 |∇u|2 ∇u0 ∇φdx
Ω
Z
Z
p−2
0
− div(|∇u| ∇u)φ dx − div(|∇u|p−2 (∇u)0 )φdx
dJ(Ω; V ) = (p − 2)
Ω
Z
dJ(Ω, V ) = (p − 2)
Ω
Ω
|∇u|p−2 ∇u0 ∇φdx
Z
Z
p−2
0
− div(|∇u| ∇u)φ dx − div(|∇u|p−2 (∇u)0 )φdx
Ω
Ω
because
Z
p−2
div(|∇u|
Z
|∇u|p−2 ∇u∇φ · V (0) · νds = 0.
∇u∇φ · V (0))dx =
Ω
∂Ω
We obtain
Z
dJ(Ω; V ) = −(p − 1)
p−2
div(|∇u|
Z
0
∇u )φdx −
Ω
div(|∇u|p−2 ∇u)φ0 dx
Ω
We have also
Z
0
p−2
λ |u|
dJ1 (Ω; V ) =
Z
uφdx +
Ω
λ|u|p−2 u0 φdx
Ω
Z
Z
p−2
0
+ λ|u| uφ dx + (p − 2) λ|u|p−2 u0 φdx,
Ω
Z
dJ1 (Ω; V ) =
λ0 |u|p−2 uφdx +
Z
Ω
λ|u|p−2 uφ0 dx + (p − 1)
Z
Ω
Ω
λ|u|p−2 u0 φdx
Ω
dJ(Ω; V ) = dJ1 (Ω, V ) implies
Z
Z
p−2
0
− (p − 1) div(|∇u| ∇u )φdx − div(|∇u|p−2 ∇u)φ0 dx
Ω
Ω
Z
Z
Z
p−2
0
0
p−2
λ |u| uφdx + λ|u| uφ dx + (p − 1) λ|u|p−2 u0 φdx.
=
Ω
Ω
Ω
By simplification we get
Z
Z
p−2
0
− (p − 1) div(|∇u| ∇u )φdx − div(|∇u|p−2 ∇u)φ0 dx
Ω
Ω
Z
Z
0
p−2
=
λ |u| uφdx + (p − 1) λ|u|p−2 u0 φdx,
Ω
for all
This implies that
−(p − 1)div(|∇u|p−2 ∇u0 ) = λ0 |u|p−2 u + (p − 1)λ|u|p−2 u0
(4.1)
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
φ ∈ D(Ω).
Ω
in
D0 (Ω)
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8
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We multiply the equation (4.1) by u and by Green ’s formula we get
Z
Z
Z
p−2
0
p−2
0
0
−(p−1)
div(|∇u| ∇u)u dx +
|∇u| ∇u · νu ds = λ +(p−1) λ|u|p−2 uu0 dx.
Ω
∂Ω
Ω
Finally we obtain the expression of
Z
0
|∇u|p ∇u · νu0 ds
λ (Ω; V ) = −(p − 1)
∂Ω
where u0 satisfies u0 =
− ∂u
V
∂ν
(0) · ν on ∂Ω. Then
Z
0
λ (Ω, V ) = −(p − 1)
|∇u|p V · νds.
∂Ω
R
We have Ra similar formula for the variation of the volume dJ2 (Ω, V ) = ∂Ω V · νds, where
J2 (Ω) = Ω dx − c.
If Ω is an optimal domain then there exists a Lagrange multiplier a < 0 such that
Z
Z
p
−(p − 1)
|∇u| V · νds = a
V · νds.
∂Ω
∂Ω
Then we obtain
1
−a p
|∇u| =
on ∂Ω.
p−1
Since Ω is C 2 −regular and u = 0 on ∂Ω, then we get
1
∂u
−a p
−
=
on ∂Ω.
∂ν
p−1
We are also interested the existence of a minimizer for the following problem
min{λ1 (Ω), Ω ∈ A, |Ω| ≤ c},
where A is a family of admissible domain defined by
A = {Ω ⊆ D, Ω
is quasi open}
and λ1 (Ω) is defined by
R
λ1 (Ω) =
min
φ∈W01,p (Ω),φ6=0
R
|∇u1 (x)|p dx
|∇φ(x)|p dx
RΩ
=
.
|φ(x)|p
|u1 (x)|p dx
Ω
Ω
ΩR
The Sobolev space W01,p (Ω) is seen as a closed subspace of W01,p (D) defined by
W01,p (Ω) = {u ∈ W01,p (D) : u = 0p − q.e
on
D\Ω}.
The problem is to look for weak topology constraints which would make the class A sequentially compact. This convergence is called weak γp -convergence for quasi open sets.
Definition 4.2. We say that a sequence (Ωn ) of A weakly γp -converges to Ω ∈ A if the sequence
un converges weakly in W01,p (D) to a function u ∈ W01,p (D) (that we may take as quasicontinuous) such that Ω = {u > 0}.
We have the following theorem.
Theorem 4.4. The problem
(4.2)
min{λ1 (Ω), Ω ∈ A, |Ω| ≤ c}
admits at least one solution.
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
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T HE F IRST E IGENVALUE FOR THE p-L APLACIAN O PERATOR
9
Proof of Theorem 4.4. Let us take
R
λ1 (Ωn ) =
min
1,p
Ωn
φn ∈W0 (Ωn ),φn 6=0
R
|∇φn (x)|p dx
|φn (x)|p dx
Ωn
R
= RΩn
|∇un (x)|p dx
|un (x)|p dx
Ωn
.
Suppose that (Ωn )(n∈N) is a minimizing sequence
of domain for the problem (4.2). We denote
R
by un a first eigenfunction on Ωn , such that Ωn |un (x)|p dx = 1.
Since un is the first eigenfunction of λ1 (Ωn ), un is strictly positive, cf [15], then the sequence
(Ωn ) is defined by Ωn = {un > 0}.
If the sequence (un ) is bounded in W01,p (D), then there exists a subsequence still denoted by
1,p
un such
R that up n converges weakly in W0 (D) to a function u. By compact injection, we have
that Ω |u(x)| dx = 1.
Let Ω be quasi open and defined by Ω = {u > 0}, this implies that u ∈ W01,p (Ω). As the
sequence (un ) is bounded in W01,p (D), then
R
R
R
p
p
|∇u
(x)|
dx
|∇u(x)|
dx
|∇u1 (x)|p dx
n
RΩ
RΩ
lim inf RΩn
≥
≥
= λ1 (Ω).
n→+∞
|un (x)|p dx
|u(x)|p dx
|u1 (x)|p dx
Ωn
Ω
Ω
Now we show that |Ω| ≤ c.
We know that if the sequence Ωn weakly γp - converges to Ω and the Lebesgue measure
is weakly γp -lower semicontinuous on the class A (see [5]), then we obtain |{u > 0}| ≤
lim inf |{un > 0}| ≤ c this implies that |Ω| ≤ c.
n→+∞
5. D OMAIN IN B OX
Now let us take N = 2. We consider the class of admissible domains defined by
C = {Ω, Ω
open subsets of D and simply connected, |Ω| = c}.
• For p > 2, the p-capacity of a point is stricly positive and every W01,p function has a
continuous representative. For this reason, a property which holds p − q.e with p > 2
holds in fact everywhere. For p > 2, the domain Ωp is a minimizer of the problem
min{λp1 (Ωp ), Ωp ∈ C}.
Consider the sequence (Ωpn ) ⊆ C and assume that Ωpn converges in Hausdorff complementary topology to Ω2 , when pn goes to 2 and pn > 2. Then Ω2 ⊆ C and Ωpn
γ2 -converges to Ω2 .
By the Sobolev embedding theorem, we have W01,pn (Ωpn ) ,→ H01 (Ωpn ). The γ2 convergence implies that H01 (Ωpn ) converges in the sense of Mosco to H01 (Ω2 ). For
pn > 2, by the Hölder inequality we have
21
Z
p1
Z
n
1
1
−
≤ |Ωpn | 2 pn
|∇upn |pn dx
|∇upn |2 dx
Z
2
21
|∇upn |
1
1
dx ≤ c 2 − pn λp1n (Ωpn ).
Then the sequence (upn ) is uniformly bounded in H01 (Ωpn ). There exists a subsequence
still denoted upn such that upn converges weakly in H01 (D) to a function u. The second
condition of Mosco implies that u ∈ H01 (Ω2 ).
For p > 2, we have the Sobolev embedding theorem W01,p (D) ,→ C 0,α (D̄).
Ascoli’s theorem implies that upn −→ u and ∇upn −→ ∇u locally uniformly in Ω2 ,
when pn goes to 2 and pn > 2.
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10
I DRISSA LY
Now show that
Z
lim
pn −→2
Z
2
|upn | dx = 1 i.e.
|u|2 dx = 1.
For > 0 small, we have pn > 2 − . Noting that
1
Z
Z
p1
2−
n
1
1
1
1
−
p
2−
n
|∇upn | dx
|∇upn | dx
= c 2− − pn λp1n (Ωpn ),
≤ |Ωpn | 2− pn
this implies that the sequence upn is uniformly bounded in W01,2− (Ωpn ). Then there
exists a subsequence still denoted upn such that upn is weakly convergent in H01 (D) to
u. By the second condition of Mosco we get u ∈ W01,2− (Ω2 ). It follows that
Z
2−
Z
Z
pn
1− 2−
p
2−
2−
n
p
|upn | dx
|u| dx = lim
|upn | dx ≤ lim |Ωpn | n
= c2 .
pn −→2
pn −→2
R
Letting −→ 0, we obtain |u|2 dx ≤ 1.
On the other hand, Lemma 4.2 of [14] implies that
Z
Z
Z
pn
pn
|u| dx ≥ |upn | dx + pn |u|pn −2 dxupn (u − upn ).
(5.1)
The
integral on the right-hand
R second
R side approaches 0 as pn −→ 2. Thus we get
|u|2 dx ≥ 1, and we conclude that |u|2 dx = 1.
In [11, Theorem 2.1 p. 3350], λpk is continuous in p for k = 1, 2, where λpk is the
k − th eigenvalue for the p-Laplacian operator.
We have
Z
Z
pn −2
|∇upn |
∇upn ∇φdx = λp1n |upn |pn −2 upn φdx, for all φ ∈ D(Ω2 ).
Letting pn go to 2, pn > 2 in (5.1), and noting that upn converges uniformly to u on the
support of φ, we obtain
Z
Z
∇u∇φdx = λ21 uφdx, for all φ ∈ D(Ω2 ),
whence we have
−∆u = λ21 u
u
= 0
in
on
D0 (Ω2 )
∂Ω2 .
We conclude that when p −→ 2 and p > 2 the free parts of the boundary of Ωp cannot
be pieces of circle.
• For p ≤ 2, we consider the sequence (Ωpn ) ⊆ C and assume that Ωpn converges in
Hausdorff complementary topology to Ω2 , when pn goes to 2 and pn ≤ 2. Then by
Theorem 3.1, we get Ω2 ⊆ C and Ωpn γ2− -converges to Ω2 .
In [16], the sequence (upn ) is bounded in W 1,2− (D), 0 < < 1 that is ∇upn con2−
2−
verges weakly
R in L2 (D) to ∇u Rand 2upn converges strongly in L (D) to u. In [16],
we get also |∇u| dx ≤ β and |u| dx < ∞.
By Lemma 4.2 of [14], we have
Z
Z
Z
pn
pn
|u| dx ≥ |upn | dx + pn |u|pn −2 dxupn (u − upn ).
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
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T HE F IRST E IGENVALUE FOR THE p-L APLACIAN O PERATOR
(5.2)
11
The
integral on the right-hand side approaches 0 as pn −→ 2. Thus we get
R second
2
|u| dx ≥ 1. This implies that
Z
Z
2−
lim
|upn | dx = |u|2− dx = 1.
pn −→2
R
Letting −→ 0, we obtain |u|2 dx = 1.
The γ2− -convergence implies that upn converges strongly in W01,2− (Ω2 ) to u. According to P. Lindqvist see [16], we have u ∈ H 1 (D), and we can deduce that u ∈
H01 (Ω2 ). As the first eigenvalue for the p-Laplacian operator is continuous in p cf [11],
we have
Z
Z
pn −2
|∇upn |
∇upn ∇φdx = λp1n |upn |pn −2 upn φdx, for all φ ∈ D(Ω2 ).
Letting pn go to 2, pn ≤ 2 in (5.2), and noting that upn converges uniformly to u on the
support of φ, we obtain
Z
Z
∇u∇φdx = λ21 uφdx, for all φ ∈ D(Ω2 ),
whence we have
−∆u = λ21 u in D0 (Ω2 )
u
= 0
on ∂Ω2 .
We conclude that when p −→ 2 and p ≤ 2 the free parts of the boundary of Ωp cannot
be pieces of circle.
R EFERENCES
[1] A. ANANE AND N. TSOULI, The second eigenvalue of the p-Laplacian, Nonlinear Partial Differential Equations (Fès, 1994), Pitman Research Notes Math. Ser., 343, Longman, Harlow, (1996),
1–9.
[2] D. BUCUR AND P. TREBESCHI, Shape optimization problem governed by nonlinear sate equation, Proc. Roy. Edinburgh, 128 A(1998), 945–963.
[3] D. BUCUR AND J.P. ZOLÉSIO, Wiener’s criterion and shape continuity for the Dirichlet problem,
Boll. Un. Mat. Ital., B 11 (1997), 757–771.
[4] D. BUCUR AND J.P. ZOLÉSIO, N-Dimensional shape Optimization under Capacity Constraints,
J. Differential Equations, 123(2) (1995), 504–522.
[5] D. BUCUR AND G. BUTTAZZO, Variational Methods in Shape Optimization Problems, SNS Pisa
2002.
[6] G. BUTTAZZO AND G. DAL MASO, An existence result for a class of shape optimization problems, Arch. Rational. Mech. Anal., 122 (1993), 183–195.
[7] G. DAL MASO, Γ-convergence and µ capacities, Ann. Scuola Norm. Sup. Pisa,14 (1988), 423–
464.
[8] G. DAL MASO AND A. DEFRANCESCHI, Limits of nonlinear Dirichlet problems in varying
domains, Manuscripta Math., 61 (1988), 251–278.
[9] G. FABER, Bewiw, dass unter allen homogenen Membranen von gleicher Fläche und gleicher
Spannung die kreisförmige den tiefsten Grundton gibt, Sitz. Ber. Bayer. Akad. Wiss, (1923), 169–
172.
[10] A. HENROT AND E. OUDET, Minimizing the second eigenvalue of the Laplace operator with
Dirichlet boundary conditions, Arch. Ration. Mech. Anal., 169(1) (2003), 73–87.
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
http://jipam.vu.edu.au/
12
I DRISSA LY
[11] Y.X. HUANG, On the eigenvalue of the p-Laplacian with varying p, Proc. Amer. Math. Soc.,
125(11) (1997), 3347–3354.
[12] B. KAWOHL, Rearrangements and Convexity of Level Sets in PDE, (Lecture Notes In Mathematics
1150), Springer-Verlag, Heidelberg.
[13] E. KRAHN, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 94
(1924), 97–100.
[14] P. LINDQVIST, On the equation div(|∇u|p−2 ∇u)+λ|u|p−2 u = 0, Proc. Amer. Math. Soc., 109(1)
(1990), 157–164. Addendum, ibiden, 116(2) (1992), 583–584.
[15] P. LINDQVIST, On a nonlinear eigenvalue problem, Berichte Univ. Jyvaskyla Math. Inst., 68
(1995), 33–54.
[16] P. LINDQVIST, Stability for the solutions of div(|∇u|p−2 ∇u) = f with varying p, J. Math. Anal.
Appl., 127(1) (1987), 93–102.
[17] J. SIMON, Differential with respect to the domain in boundary value problems, Numer. Funct.
Anal. and Optimiz., 2(7-8) (1980), 649–687.
[18] J. SOKOLOWSKI AND J.P. ZOLESIO, Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer series in Computational Mathematics, 16. Springer-Verlag, Berlin (1992).
[19] V. ŠVERÀK, On optimal shape design, J. Math. Pures Appl., 72 (1993), 537–551.
J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005
http://jipam.vu.edu.au/