On Isoperimetric Inequalities of Riesz Potentials and Applications

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.


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.

Main Results
Let in an open bounded domain of the following spectral eigenvalue problem of the Riesz potential has discrete spectrum: tion.The potential satisfies in the distributional sense ( u   is the characteristic function of the set Note that when .

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.
is the surface area of the unit sphere in .Let u 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 where  is the characteristic function of the corresponding domain.
is called a symmetric decreasing rearrangement of .u As its level sets are open domains is lower semicontinuous function, and it is uniquely determined by the distribution function By construction, u is equimeasurable with u , i.e. corresponding level sets of the two functions have the same volume, Using the layer-cake decomposition (3.1), Fubini's theorem and (3.2), we obtain In proofs of the theorems we use the following where , f g   and h  symmetric and non-increasing rearrangement of positive measurable functions , f g and respectively.h

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 1

   
Hence by Lemma 1 and the variational principle for x Theorem 1 is completely proved.
Copyright © 2013 SciRes.AM Note 1.One may wonder whether the ball is only maximizer of 1  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 1  since the Hilbert space does not change if we remove from a set of zero capicity.

  H  
Now we prove Theorem 2. By bilinear decomposition of repeated kernel, we have From (3.3) and the fact that is a symmetric and positive decreasing function, we obtain Theorem 2 is proved.Note 2. We can generalize Theorem 2 writing in the following form , but obviously, in this case we need some restrictions on depending on the dimension of the Euclidean space and n d  .

On Applications of Results for Boundary Value Problems of the Laplacian
Let and 3 d  2   .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.d Lemma 2. For any function suppf the Newton potential satisfies the boundary condition and the boundary condition (5.2), then the function   u x coincides with the Newton potential (5.1), here y n  denotes the outer normal derivative on the  boundary.
Proof.Suppose that A direct calculation shows that, for any , we have This implies, for x   , we get Applying properties of single-layer and double-layer potentials [8] to Formula (5.4) with , we get 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   2 u H   .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 (5.2) As above, applying Green's formula to 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).


is positive and simple, and the corresponding eigenfunction symmetric and decreasing for all   , x y    , we have 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 (