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.
Isoperimetric Inequalities; Eigenvalues of the Laplacian; Riesz Potentials1. 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:
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
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
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
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,
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]
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
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
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
satisfies the boundary condition
Conversely, if a function satisfies
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
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. 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
As above, applying Green’s formula to, we see that
Passing the limit as, we obtain
From (5.6), it implies
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
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).
REFERENCESReferencesJ. W. Rayleigh, “The Theory of Sound,” Dover Publishing, New York, 1945.A. Henrot, “Extremum Problems for Eigenvalues of Elliptic Operators,” Birkhauser, Basel, 2006.D. Daners, “A Faber—Krahn Inequality for Robin Problems in Any Space Dimension,” Mathematische Annalen, Vol. 335, 2006, pp. 767-785.
doi:10.1007/s00208-006-0753-8T. Sh. Kalmenov and D. Suragan, “Boundary Conditions for the Volume Potential for the Polyharmonic Equation,” Differential Equations, Vol. 48, No. 4, 2012, pp. 595-599.A. Burchard, “A Short Course on Rearrangement Inequalities,” 2009.
www.math.toronto.edu/almut/rearrange.pdfF. Riesz, “Sur Une Inregalitre Intregrale,” Journal of the London Mathematical Society, Vol. 5, No. 3, 1930, pp. 162-168. doi:10.1112/jlms/s1-5.3.162B. S. Vladimirov, “Equations of Mathematical Physics,” Nauka, Moscow, 1981.N. S. Landkoff, “Foundations of Modern Potential Theory,” Springer-Verlag, Berlin, 1972.
doi:10.1007/978-3-642-65183-0B. Dittmar, “Sums of Reciprocal Eigenvalues of the Laplacian,” Mathematische Nachrichten, Vol. 237, No. 1, 2002, pp. 45-61.
doi:10.1002/1522-2616(200204)237:1<45::AID-MANA45>3.0.CO;2-MT. Sh. Kalmenov and D. Suragan, “To Spectral Problems for the Volume Potential,” Doklady Mathematics, Vol. 428, No. 1, 2009, pp. 16-19.
, 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.
of 
the following spectral eigenvalue problem of the Riesz potential has discrete spectrum:
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
)
.
and 
Riesz potential coincides with the classical Newtonian potential and for 
satisfies a polyharmonic equation in 
(cf. [
(enumerate their eigenvalues in decreasing order) each time repeated according to multiplicity. For the spectral problem (2.1), we obtain the following results:
open simple-connected bounded domain and 
is a ball of the same measure as the
, i.e.
, then
open simple-connected bounded domain and 
is a ball of the same measure as the
, i.e.
, and the following series has convergence then
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,
can be used the layer-cake decomposition (see, for example, [
in terms of its level sets as
is the characteristic function of the corresponding domain.
.
is lower semicontinuous function, and it is uniquely determined by the distribution function
is equimeasurable with
, i.e. corresponding level sets of the two functions have the same volume,
from
, we have
and 
symmetric and non-increasing rearrangement of positive measurable functions 
and 
respectively.
is positive and simple, and the corresponding eigenfunction 
can be chosen positive in
.
.
is positive, symmetric and decreasing for all
, we have
, we have
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.
is a symmetric and positive decreasing function, we obtain
depending on the dimension of the Euclidean space 
and
.
and
. In this case, the Riesz potential coincides with the classical Newton potential, that is, the kernel of the Riesz potential is
is a natural number.
suppf 
the Newton potential
satisfies
coincides with the Newton potential (5.1), here
denotes the outer normal derivative on the boundary.
. A direct calculation shows that, for any
, we have
and 
are components of the unit normal.
, we get
, we get
. Thus, the Newton potential (5.1) satisfies the boundary condition (5.2).
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
, we see that
, we obtain
for all
, that is, 
, 
coincides with the Newton potential. This completes the proof of Lemma 2.
of the Laplacian boundary value problems (5.8) and (5.9).