**Applied Mathematics** Vol.4 No.7A(2013), Article ID:33946,4 pages DOI:10.4236/am.2013.47A001

On Isoperimetric Inequalities of Riesz Potentials and Applications

Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

Email: bolus91.ark@mail.ru

Copyright © 2013 Tynysbek Sh. Kalmenov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received March 17, 2013; revised April 18, 2013; accepted April 25, 2013

**Keywords:** Isoperimetric Inequalities; Eigenvalues of the Laplacian; Riesz Potentials

ABSTRACT

In this article, we prove certain isoperimetric inequalities for eigenvalues of Riesz potentials and show some applications of the results to a non-local boundary value problem of the Laplace operator.

1. Introduction

Historically, the minimization of the first eigenvalue of the Dirichlet Laplacian is probably the first such problem which appeared in the scientific literature. In Rayleigh’s famous book “Theory of Sound” [1] (first published in 1877), by using some explicit computation and physical interpretations, he stated that a circle minimizes (among all domains of the same area) the first eigenvalue of the Laplacian with the Dirichlet boundary condition. The proof of this conjecture was obtained only after 30 years later, simultaneously (and independently) by G. Faber and E. Krahn. Nowadays, the Rayleigh-Faber-Krahn inequality has been expanded many other boundary value spectral problems and operators; see [2,3] for further references.

In the present paper, we give simple proofs of some isoperimetric inequalities for the eigenvalues of Riesz potential by using methods of symmetrical decreasing rearrangements of positive measurable functions and variational principles. Riesz potentials, that is convolution operators with fractional powers of the distance to a point in, have important roles in fractional calculus theory. We also apply these results for the Laplacian with a nonlocal boundary conditions, in particular, we prove Rayleigh-Faber-Krahn inequality for the obtained non-local boundary value spectral problem of the Laplacian.

2. Main Results

Let in an open bounded domain of the following spectral eigenvalue problem of the Riesz potential has discrete spectrum:

(2.1)

where

is the distance between and

in the -dimensional Euclidean space

, and Г is the gamma function. The potential satisfies in the distributional sense (is the characteristic function of the set)

.

Note that when and Riesz potential coincides with the classical Newtonian potential and for satisfies a polyharmonic equation in (cf. [4]). We denote eigenvalues of the Reisz potential by (enumerate their eigenvalues in decreasing order) each time repeated according to multiplicity. For the spectral problem (2.1), we obtain the following results:

Theorem 1. Let open simple-connected bounded domain and is a ball of the same measure as the, i.e., then

(2.2)

Theorem 2. Let open simple-connected bounded domain and is a ball of the same measure as the, i.e., and the following series has convergence then

(2.3)

3. Preliminary

Let bounded measurable set in. Its symmetric rearrangement is an open ball originated at 0 with a volume equal to the volume of, i.e. and

where is the surface area of the unit sphere in. Let be a nonnegative measurable function vanishing at infinity, in the sense that all its positive level sets have finite measure,

In the definition of the symmetric decreasing rearrangement of can be used the layer-cake decomposition (see, for example, [5]), which expresses a nonnegative function in terms of its level sets as

(3.1)

where is the characteristic function of the corresponding domain.

Definition. A function

is called a symmetric decreasing rearrangement of.

As its level sets are open domains is lower semicontinuous function, and it is uniquely determined by the distribution function

By construction, is equimeasurable with, i.e. corresponding level sets of the two functions have the same volume,

(3.2)

Lemma 1. For each non-negative function of from, we have

Proof. Using the layer-cake decomposition (3.1), Fubini’s theorem and (3.2), we obtain

Lemma 1 is proved.

In proofs of the theorems we use the following F. Riesz’s inequality [5,6]

(3.3)

where and symmetric and non-increasing rearrangement of positive measurable functions and respectively.

4. Proofs of Theorems 1 and 2

Since Riesz kernel of the potential (2.1) is symmetric and positive, by Ench theorem [7], its largest eigenvalue is positive and simple, and the corresponding eigenfunction can be chosen positive in.

Let us use the denotation.

Using the Riesz inequality (3.3) and the fact that is positive, symmetric and decreasing for all, we have

Hence by Lemma 1 and the variational principle for, we have

Theorem 1 is completely proved.

Note 1. One may wonder whether the ball is only maximizer of among all domains of the same volume. But the answer is no. For example, if we remove a set of zero capacity from the ball, a new domain also maximizes the value of since the Hilbert space does not change if we remove from a set of zero capicity.

Now we prove Theorem 2. By bilinear decomposition of repeated kernel, we have

(4.1)

From (3.3) and the fact that is a symmetric and positive decreasing function, we obtain

According to (4.1),

Theorem 2 is proved.

Note 2. We can generalize Theorem 2 writing in the following form

(4.2)

but obviously, in this case we need some restrictions on depending on the dimension of the Euclidean space and.

5. On Applications of Results for Boundary Value Problems of the Laplacian

Let and. In this case, the Riesz potential coincides with the classical Newton potential, that is, the kernel of the Riesz potential is

where is a natural number.

Lemma 2. For any function suppf the Newton potential

(5.1)

satisfies the boundary condition

(5.2)

Conversely, if a function satisfies

(5.3)

and the boundary condition (5.2), then the function coincides with the Newton potential (5.1), here

denotes the outer normal derivative on the boundary.

Proof. Suppose that. A direct calculation shows that, for any, we have

where and are components of the unit normal.

This implies, for, we get

(5.4)

Applying properties of single-layer and double-layer potentials [8] to Formula (5.4) with, we get

(5.5)

i.e. (5.5) is a boundary condition for the Newton potential (5.1). Passing to the limit we can easily show that (5.5) remains valid for all. Thus, the Newton potential (5.1) satisfies the boundary condition (5.2).

Conversely, if the function satisfies and the boundary condition (5.2), then it coincides with the Newtonian potential (5.1). Indeed, if this is not so, then the function, where is the Newton potential (5.1) satisfies the homogeneous equation with the boundary condition of

(5.6)

As above, applying Green’s formula to, we see that

Passing the limit as, we obtain

From (5.6), it implies

(5.7)

By virtue of the uniqueness of a solution of the Dirichlet problem for the Laplace equation, we have for all, that is, , coincides with the Newton potential. This completes the proof of Lemma 2.

Lemma 2 shows that the eigenvalue problem of the Newton potential is equivalent to spectral boundary problem for the Laplace equation

(5.8)

(5.9)

Finally, by Theorem 1, we obtain Rayleigh-FaberKrahn inequality for the first eigenvalue of the Laplacian boundary value problems (5.8) and (5.9).

Proposition 1. A ball minimizes the first eigenvalue of the Laplacian (5.8) with a special type of non-local boundary condition (5.9) among all domains of the same measure.

And from Theorem 2 we obtain the following analogue of Dittmar’s result [9].

Proposition 2. Series made up of squares of reciprocal eigenvalues of the Laplacian with the boundary condition (5.9),

is minimized in a ball among all domains of the same measure.

Note that in [10] in the case of two-dimensional ball and three-dimensional ball we calculated all eigenvalues of the the Laplacian with the boundary condition (5.9).

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