An isoperimetric inequality for a nonlinear eigenvalue problem
Abstract.
We prove an isoperimetric inequality of the RayleighFaberKrahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue. More precisely, we show that the minimizer among sets of given volume is the union of two equal balls.
Key words and phrases:
Shape optimization, eigenvalues, symmetrization, Euler equation, shape derivative1991 Mathematics Subject Classification:
35J60, 35P30, 47A75, 49R50, 52A401. Introduction
In this note we study a generalized version of the so called twisted Dirichlet eigenvalue problem. More precisely, for an open bounded subset of we set
(1.1) 
Among the sets with fixed volume, we are interested in characterizing those which minimize . In other words we look for an isoperimetric inequality of RayleighFaberKrahn type. This kind of inequality is related to the optimization of the first eigenvalue for the Dirichlet problem associated to nonlinear operators in divergence form and have been widely studied for functionals that do not involve mean constraints. In such cases a rearrangement technique proves that the minimizing set is a ball and several results concerning its stability are also available (see for instance [23],[21],[15]). When mean type constraints are considered together with the Dirichlet boundary condition in an eigenvalue problem, the optimization problem becomes more difficult, since one is lead to deal with non local problems. Due to the fact that an eigenfunction for is forced to change sign inside , and hence has at least two nodal domains, one cannot expect in general to have a radial optimizer.
The adjective twisted was introduced by Barbosa and Bérard in [1], in the study of spectral properties of the second variation of a constant mean curvature immersion of a Riemannian manifold. In that framework a Dirichlet eigenvalue problem arose naturally with a vanishing mean constraint. The condition on the mean value comes from the fact that the variations under consideration preserve some balance of volume.
Further results in this direction can be found in the paper of Freitas and Henrot [14], where, dealing with the linear case, the authors solved the shape optimization problem for the first twisted Dirichlet eigenvalue. In particular they considered , and they proved that the only optimal shape is given by a pair of balls of equal measure. The onedimensional case has also attracted much interest. In [6], Dacorogna, Gangbo and Subía studied the following generalization of the Wirtinger inequality
(1.2) 
for proving that the optimizer is an odd function. Moreover they explained the connection between the value of , where , and an isoperimetric inequality. Indeed, let whose boundary is a simple closed curve with . Let
and
Then . The case of equality holds if and only if , up to a translation and a dilation.
Several other results are available in the onedimensional case, see for instance [5], [4], [2], [19], [11] and the references therein for further details.
Our aim here, as in [14], is to prove that the optimal shape for is a pair of equal balls. The main result can be stated as follows.
Theorem 1.1.
Let be an open bounded subset of . Then, for
(1.3) 
we have
where and are disjoint balls of measure .
The rest of the paper is devoted to the proof of Theorem 1.1 and it is divided into two steps. In the first one, using the symmetrization technique, we show that it is enough to minimize the functional on sets given by the union of two disjoint balls and (not necessarily equal) and to identify the minimizing pairs. Moreover we write the Euler equation for a minimizer of , proving that the Lagrange multiplier associated to the constraint
is zero (cf. Theorem 2.5).
The second step, which consists in showing that the two optimal balls have to be equal, is more subtle. In the case solved in [14], the proof is based on the explicit formula for the (radial) solutions to the Euler equation of the functional and on fine properties of the zeroes of Bessel functions. Here we use a more geometric argument obtaining as a byproduct a simpler proof of the results of Freitas and Henrot. More precisely is attained at a function , with and radial positive functions. If we look at as a function of sets we obtain the following optimality condition from the domain derivative (cf. Theorem 3.2):
From the other hand, the divergence theorem applied to the Euler equation gives that
This, combined with the previous condition, implies that and have the same measure.
2. The first generalized twisted eigenvalue
We start our study proving that the the value is attained for any choice of a bounded open set .
Lemma 2.1.
Assume (1.3). Then and there exists a bounded function such that
Proof.
Let
with , and
By definition of infimum, for every there exists such that
Without loss of generality we can assume that By Poincaré inequality, is uniformly bounded. Since , up to a subsequence, converges weakly to some . By hypotheses (1.3) on and , in and then
This implies that
By definition of , necessarily we have
To prove that and , it is sufficient to pass to the limit in to get
We are now going to prove that is bounded. For , let
Let be such that
if such does not exist one would have which is a contradiction. Then, set
The hypotheses on imply that . By the implicit function theorem applied to , there exists a function such that =0 and . Since is a minimizer for , we deduce that
(2.1) 
Let now be such that
and set
Our aim is to prove that we can reduce to the case of two balls. We will use a technique used in [14] based on the Schwarz rearrangement. Here we recall just the definition and the properties that we will need in the proof. For more details on rearrangement techniques we refer to [16] and [18].
Definition 2.2.
For a measurable set , we denote by the ball of same measure as . If is a nonnegative measurable function defined on a measurable set and on , let
The Schwarz rearrangement of is the function defined on by
The next theorem summarizes some of the main properties of the Schwarz symmetrization.
Theorem 2.3.
Let be a nonnegative measurable function defined on a measurable set with on . Then

is a radially symmetric nonincreasing function of ;

for any measurable function

if , then and
Using the Schwarz symmetrization and suitable constrained variations we are now able to reduce our problem to the “radial” one. Indeed we have the following theorem.
Theorem 2.4.
Let be a ball of same measure as . Then .
Proof.
Let and . By symmetrizing and respectivelly, by the properties of Schwarz rearrangement (cfr. Theorem 2.3) we can write
Moreover, by equimeasurability ensured by Theorem 2.3.(2), using the volume constraint, we deduce that
If we set
where is defined by
we clearly have
(2.2) 
It is easily seen that is attained in , with . Without loss of generality we can moreover assume that
(2.3) 
For , define
Let be such that
Such choice of is possible, since, if not, one would have , that contradicts (2.3). If we define the functional
the hypotheses on imply that . By the implicit function theorem applied to , there exists a function such that =0 and . Since is a minimizer for ,
that is,
with
It follows that satisfies on the equation
(2.4) 
Now we observe that multiplying (2.4) by , one has
The above inequality and (2.2) imply that . ∎
We are now going to write the Euler equation for in the case where is the union of two disjoint balls. We will make use of a technique introduced in [6] to carefully choose the variations.
Theorem 2.5.
Let where and are two disjoint balls. Let be a bounded function such that and Then
(2.5) 
Proof.
We set
Let and . Let
Then is continuous, and . By continuity there exists such that .
Let fixed. We set . We are going to prove the existence of a sequence such that has a finite limit as (up to a subsequence). If there exists a sequence and such that , then we have the result, since is bounded. If there exists such that, for every , for every , let us show that must change sign in . Otherwise, by the strict convexity of (and then by the strict monotonicity of ) we should have
or
that is,
or
which is a contradiction.
Then, for , on a subset of one has and on its complement . This implies that . Therefore there exists such that and as .
We have
On the other hand,
The previous inequality implies that
for every
∎
3. The shape optimization problem
In this section we are going to find a geometrical necessary condition for a set to be a minimizer of , where is the union of two disjoint balls. We will exploit the derivative with respect to the domain of the set functional and investigate an optimality condition, i.e. we will identify the domains with vanishing domain derivative. Here we briefly recall, for the reader’s convenience, the ideas underlying the concept of domain derivative and we refer for instance to [17] and [25] for a detailed description of the theory and for further details on its applicability.
Roughly speaking the domain derivative can be understood in the following way. Let be a bounded smooth domain in , be a sufficiently smooth vector field, and denote by the image of under the map , where stands for the identity. Let us consider the boundary value problem
(3.1) 
and an integral functional given by
with and are differential operators acting on a space of functions defined in . Under suitable regularity hypotheses, the function , that associates to the solution of problem (3.1), is differentiable and its derivative in zero, denoted by , satisfies the following conditions
(3.2) 
where is the outward unit normal to . Moreover, we can calculate the domain derivative for of the functional in the direction as
(3.3) 
The results of the previous section ensure us that we can restrict our study to the sets where and are two disjoint balls of radius and respectively such that , where is the measure of the unit ball in . Let be the minimizer function realizing the value . Using the Schwarz rearrangement as in Theorem 2.4 we can assume that is attained at a function , with and non negative radial functions on and respectively.
For this kind of domains, by Theorem 2.5, satisfies (2.5). By scaling invariance, it is not restrictive to deal with solutions that satisfy the condition
(3.4) 
Thus we are lead to consider satisfying (3.4), the constraint
and the Dirichlet eigenvalue problem
Observe that in dimension 1, the minimizer of is an antisymmetric function with respect to , as proved in [6]. Therefore in the sequel we will assume that .
Clearly an optimal set, i.e. a set that minimizes , will be a critical set with respect to the domain variations. If we prove that the eigenvalue has a domain derivative , which will be true if is a simple eigenvalue, then (see for example [17] for further details and proof of the differentiability of a simple eigenvalue). This motivates the next theorem.
Theorem 3.1.
Let , where and are two disjoint balls. Then is a simple eigenvalue, i.e. there exists a unique function , modulo a multiplicative constant, that realizes
Proof.
Let and be two functions at which is attained. We can assume that and , for , are radial by Lemma 4.1 in the appendix. Moreover are nonnegative and
(3.5) 
Without loss of generality we can assume that . Therefore
(3.6) 
We remark that, by Theorem 2.5, letting , we have that and satisfy on , the Cauchy problem
with possibly different constants for and , and a similar result holds for and on .