Journal of Inequalities in Pure and
Applied Mathematics
THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR
IDRISSA LY
volume 6, issue 3, article 91,
2005.
Laboratoire de Mathématiques et Applications de Metz
Ile du Saulcy, 57045 Metz Cedex 01, France.
Received 12 February, 2005;
accepted 17 June, 2005.
EMail: idrissa@math.univ-metz.fr
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
037-05
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Abstract
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.
The First Eigenvalue for the
p-Laplacian Operator
2000 Mathematics Subject Classification: 35J70, 35P30, 35R35.
Key words: p-Laplacian, Nonlinear eigenvalue problems, Shape optimization.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Definition of the First and Second Eigenvalues . . . . . . . . . . . . 6
3
Properties of the Geometric Variations . . . . . . . . . . . . . . . . . . . 7
4
Shape Optimization Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5
Domain in Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
References
Idrissa Ly
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1.
Introduction
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
p
p
|u(x)| dx + |∇u(x)|p dx.
||u||1,p =
Ω
Ω
The p-Laplacian is the operator defined by
The First Eigenvalue for the
p-Laplacian Operator
∆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 < ∞,
1
+ 1q = 1.
p
We are interested in the nonlinear eigenvalue problem
(
−∆p u − λ|u|p−2 u = 0 in Ω,
(1.1)
u
= 0 on ∂Ω.
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
Idrissa Ly
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Ω
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for all φ ∈ C0∞ (Ω). The corresponding real number λ is called an eigenvalue.
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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-Faber-Khran 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)
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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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.
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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2.
Definition of the First and Second Eigenvalues
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 EulerLagrange 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
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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C2 := {C ∈ W01,p (Ω) : C = −C such that genus(C) ≥ 2}.
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In [1], Anane and Tsouli proved that there does not exist any eigenvalue between
the first and the second ones.
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3.
Properties of the Geometric 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
x∈RN
|d(x, Ωc1 )
−
The First Eigenvalue for the
p-Laplacian Operator
d(x, Ωc2 )|.
Idrissa Ly
The H c -topology has some good properties for example the space of the open
Hc
subsets of D is 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
cap(K, B) := inf
|∇φ|p , φ ∈ C0∞ (B), φ ≥ 1 on
B
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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.
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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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).
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.
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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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
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
Ωn
|∇φn (x)|p dx
R
Ωn
|φn (x)|p
R
=
Ωn
|∇un (x)|p dx
R
Ωn
|un (x)|p
,
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where the minimum is attained by un , and
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 .
By the Bucur and Trebeschi theorem, Ωn γp converges to Ω. This implies
1,p
W0 (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).
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Using the weak lower semicontinuity of the Lp −norm, we have the inequality
R
lim inf
n−→+∞
|∇un (x)|p dx
≥
|un (x)|p
D
R
DR
|∇u(x)|p dx
≥
|u(x)|p
Ω
ΩR
R
|∇u1 (x)|p dx
,
|u1 (x)|p
Ω
ΩR
then
lim inf λ1 (Ωn ) ≥ λ1 (Ω).
(3.1)
n→+∞
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
λ1 (Ωn ) ≤ DR
|v (x)|p
D n
this implies that
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p
|∇vn (x)| dx
|vn (x)|p
n−→+∞
R D
R
|∇vn (x)|p dx
|∇u1 (x)|p dx
DR
ΩR
= lim
=
n−→+∞
|v (x)|p
|u1 (x)|p
D n
Ω
lim sup λ1 (Ωn ) ≤ lim sup
DR
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then
(3.2)
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R
n−→+∞
The First Eigenvalue for the
p-Laplacian Operator
lim sup λ1 (Ωn ) ≤ λ1 (Ω).
n→+∞
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By the relations (3.1) and (3.2) we conclude that λ1 (Ωn ) converges to λ1 (Ω).
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4.
Shape Optimization Result
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,
We have the following lemma.
N
Lemma 4.1. Let Ω be an open subset in R .
Let ψ be any continuous function on R∗+ , we have
R
R
1. Ω ψ(u(x))dx = Ω∗ ψ(u∗ (x))dx
u∗ is equi-mesurable with u.
R
R
2. Ω u(x)v(x)dx ≤ Ω∗ u∗ (x)v ∗ (x)dx.
W01,p (Ω), p
W01,p (Ω∗ )
> 1 then u∗ ∈
and
Z
Z
|∇u(x)|p dx ≥
|∇u∗ (x)|p dx Pòlya inequality.
Ω
Idrissa Ly
{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.
3. If u ∈
The First Eigenvalue for the
p-Laplacian Operator
Ω∗
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Proof of Lemma 4.1. See [12].
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 RayleighFaber-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.
Theorem 4.2. Let B be the ball of the same volume as Ω, then
λ1 (B) = min{λ1 (Ω), Ω
open set of
B
The Pòlya inequality implies that
Z
Z
p
|∇u1 (x)| dx ≥
|∇u∗1 (x)|p dx.
Ω
Idrissa Ly
RN , |Ω| = |B|}.
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
The First Eigenvalue for the
p-Laplacian Operator
B
By the two conditions, it becomes
R
R
|∇u∗1 (x)|p dx
|∇u1 (x)|p dx
B
Ω
R
R
≤
= λ1 (Ω).
|u∗1 (x)|p dx
|u1 (x)|p dx
B
Ω
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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 1. 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
The First Eigenvalue for the
p-Laplacian Operator
J(Id + tΩ) − J(Ω)
.
t−→0
t
dJ(Ω; V ) = lim
Idrissa Ly
We have the variation calculation
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p−2
p−2
−div(|∇u|
∇u) = λ|u| u
Z
p−2
− div(|∇u| ∇u)φdx =
λ|u|p−2 uφdx,
ΩZ
ZΩ
|∇u|p−2 ∇u∇φdx =
λ|u|p−2 uφdx,
Contents
Z
Ω
for all
φ ∈ D(Ω)
for all φ ∈ D(Ω)
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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
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J
Ω
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We have
Z
Z
p−2
0
(|∇u| ∇u∇φ) dx = (|∇u|p−2 )0 ∇u∇φdx
Ω
Z Ω
Z
p−2
0
+ |∇u| ∇u(∇φ) dx + |∇u|p−2 (∇u)0 ∇φdx.
Ω
Ω
We have the expression
p−2 0
(|∇u|p−2 )0 = (|∇u|2 ) 2
The First Eigenvalue for the
p-Laplacian Operator
p−4
p−2
(|∇u|2 )0 (|∇u|2 ) 2
2
p−2 0
(|∇u| ) = (p − 2)∇u∇u0 |∇u|p−4 .
Idrissa Ly
=
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Then
Z
dJ(Ω; V ) = (p − 2)
ZΩ
−
p−2
div(|∇u|
0
dJ(Ω, V ) = (p − 2)
−
Ω
div(|∇u|p−2 (∇u)0 )φdx
Ω
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p−2
|∇u|
ZΩ
Z
∇u)φ dx −
Ω
Z
JJ
J
|∇u|p−4 |∇u|2 ∇u0 ∇φdx
0
∇u ∇φdx
div(|∇u|p−2 ∇u)φ0 dx −
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Z
Ω
div(|∇u|p−2 (∇u)0 )φdx
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because
Z
Z
p−2
div(|∇u| ∇u∇φ · V (0))dx =
Ω
|∇u|p−2 ∇u∇φ · V (0) · νds = 0.
∂Ω
We obtain
Z
dJ(Ω; V ) = −(p − 1)
p−2
div(|∇u|
Ω
0
Z
∇u )φdx −
div(|∇u|p−2 ∇u)φ0 dx
Ω
We have also
Z
0
Z
uφdx + λ|u|p−2 u0 φdx
Ω
Z Ω
Z
p−2
0
+ λ|u| uφ dx + (p − 2) λ|u|p−2 u0 φdx,
Ω
Ω
Z
Z
Z
0
p−2
p−2
0
dJ1 (Ω; V ) =
λ |u| uφdx + λ|u| uφ dx + (p − 1) λ|u|p−2 u0 φdx
p−2
λ |u|
dJ1 (Ω; V ) =
Ω
Ω
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
ZΩ
Z
Z
0
p−2
p−2
0
=
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
Z Ω
Z Ω
=
λ0 |u|p−2 uφdx + (p − 1) λ|u|p−2 u0 φdx, for all φ ∈ D(Ω).
Ω
Ω
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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This implies that
(4.1) −(p − 1)div(|∇u|p−2 ∇u0 ) = λ0 |u|p−2 u + (p − 1)λ|u|p−2 u0 in D0 (Ω)
We multiply the equation (4.1) by u and by Green ’s formula we get
Z
− (p − 1)
p−2
div(|∇u|
Z
0
∇u)u dx +
Ω
0
p−2
|∇u|
∇u · νu ds
Z
0
= λ + (p − 1) λ|u|p−2 uu0 dx.
∂Ω
Ω
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
Finally we obtain the expression of
Z
0
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|∇u|p ∇u · νu0 ds
λ (Ω; V ) = −(p − 1)
∂Ω
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V (0) · ν on ∂Ω. Then
where u0 satisfies u0 = − ∂u
∂ν
Z
0
λ (Ω, V ) = −(p − 1)
|∇u|p V · νds.
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∂Ω
R
We have a similar formula
R 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
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Then we obtain
|∇u| =
−a
p−1
p1
on
∂Ω.
Since Ω is C 2 −regular and u = 0 on ∂Ω, then we get
∂u
−
=
∂ν
−a
p−1
p1
on
∂Ω.
We are also interested the existence of a minimizer for the following problem
The First Eigenvalue for the
p-Laplacian Operator
min{λ1 (Ω), Ω ∈ A, |Ω| ≤ c},
Idrissa Ly
where A is a family of admissible domain defined by
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A = {Ω ⊆ D, Ω
is quasi open}
and λ1 (Ω) is defined by
R
λ1 (Ω) =
min
φ∈W01,p (Ω),φ6=0
R
|∇φ(x)|
dx
|∇u1 (x)|p dx
ΩR
Ω
R
=
.
|φ(x)|p
|u1 (x)|p dx
Ω
Ω
p
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.
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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 quasi-continuous) such that Ω = {u > 0}.
We have the following theorem.
Theorem 4.3. The problem
min{λ1 (Ω), Ω ∈ A, |Ω| ≤ c}
(4.2)
admits at least one solution.
Proof of Theorem 4.3. Let us take
R
λ1 (Ωn ) =
min
φn ∈W01,p (Ωn ),φn 6=0
R
p
|∇φ
(x)|
dx
|∇un (x)|p dx
n
RΩn
RΩn
=
.
|φn (x)|p dx
|un (x)|p dx
Ωn
Ωn
Suppose that (Ωn )(n∈N) is a minimizing sequence of domainR for the problem
(4.2). We denote 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 un such that un converges
weakly in W01,p (D) to a function u.
R
By compact injection, we have that Ω |u(x)|p dx = 1.
Let Ω be quasi open and defined by Ω = {u > 0}, this implies that u ∈
1,p
W0 (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
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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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→+∞
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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5.
Domain in Box
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
Z
2
|∇upn | dx
Z
2
|∇upn |
12
≤ |Ωpn |
1
− p1
2
n
Z
pn
|∇upn | dx
p1
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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21
dx ≤ c
1
− p1
2
n
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 ).
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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.
Now show that
Z
Z
2
|upn | dx = 1 i.e.
lim
pn −→2
|u|2 dx = 1.
For > 0 small, we have pn > 2 − . Noting that
Z
2−
|∇upn |
1
2−
≤ |Ωpn |
dx
1
− p1
2−
n
1
The First Eigenvalue for the
p-Laplacian Operator
Z
pn
p1
n
|∇upn | dx
1
= c 2− − pn λp1n (Ωpn ),
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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
Z
2−
|u| dx = lim
|upn |2− dx
pn −→2
1− 2−
p
≤ lim |Ωpn |
Z
n
pn −→2
Letting −→ 0, we obtain
R
Idrissa Ly
|u|2 dx ≤ 1.
pn
|upn | dx
2−
p
n
= c2 .
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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 ).
The second
on the right-hand side approaches
0 as pn −→ 2. Thus
R integral
R 2
2
we get |u| dx ≥ 1, and we conclude that |u| 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
(5.1)
|∇upn |pn −2 ∇upn ∇φdx
Z
= λp1n |upn |pn −2 upn φdx,
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
for all
φ ∈ D(Ω2 ).
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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 ),
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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.
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• 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 converges weakly in L2− (D)
R to ∇u2 and upn converges
R 2 strongly in
2−
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 ).
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
The second
on the right-hand side approaches 0 as pn −→ 2. Thus
R integral
2
we get |u| dx ≥ 1. This implies that
Z
Z
2−
lim
|upn | dx = |u|2− dx = 1.
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pn −→2
Letting −→ 0, we obtain
R
|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
(5.2)
|∇upn |pn −2 ∇upn ∇φdx
Z
= λp1n |upn |pn −2 upn φdx, for all φ ∈ D(Ω2 ).
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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 .
The First Eigenvalue for the
p-Laplacian Operator
We conclude that when p −→ 2 and p ≤ 2 the free parts of the boundary
of Ωp cannot be pieces of circle.
Idrissa Ly
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References
[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–
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[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.
The First Eigenvalue for the
p-Laplacian Operator
Idrissa Ly
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[7] G. DAL MASO, Γ-convergence and µ capacities, Ann. Scuola Norm. Sup.
Pisa,14 (1988), 423–464.
Close
[8] G. DAL MASO AND A. DEFRANCESCHI, Limits of nonlinear Dirichlet
problems in varying domains, Manuscripta Math., 61 (1988), 251–278.
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[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
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[11] Y.X. HUANG, On the eigenvalue of the p-Laplacian with varying p, Proc.
Amer. Math. Soc., 125(11) (1997), 3347–3354.
The First Eigenvalue for the
p-Laplacian Operator
[12] B. KAWOHL, Rearrangements and Convexity of Level Sets in PDE, (Lecture Notes In Mathematics 1150), Springer-Verlag, Heidelberg.
Idrissa Ly
[13] E. KRAHN, Über eine von Rayleigh formulierte Minimaleigenschaft des
Kreises, Math. Ann., 94 (1924), 97–100.
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[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)
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[15] P. LINDQVIST, On a nonlinear eigenvalue problem, Berichte Univ. Jyvaskyla Math. Inst., 68 (1995), 33–54.
p−2
[16] P. LINDQVIST, Stability for the solutions of div(|∇u|
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∇u) = f with
[17] J. SIMON, Differential with respect to the domain in boundary value problems, Numer. Funct. Anal. and Optimiz., 2(7-8) (1980), 649–687.
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[18] J. SOKOLOWSKI AND J.P. ZOLESIO, Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer series in Computational Mathematics, 16. Springer-Verlag, Berlin (1992).
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p-Laplacian Operator
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