This paper is dedicated to the classical Hadamard formula for asymptotics of eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We prove that the Hadamard formula still holds for C-1-domains with C-1-perturbations. We also derive an optimal estimate for the remainder term in the C-1,C-alpha-case. Furthermore, if the boundary is merely Lipschitz, we show that the Hadamard formula is not valid. (C) 2020 Elsevier Masson SAS. All rights reserved.

The eigenvalue problem for linear differential operators is important since eigenvalues correspond to the possible energy levels of a physical system. It is also important to have good estimates of the error in the computed eigenvalues. In this work, we use spline interpolation to construct approximate eigenfunctions of a linear operator using the corresponding eigenvectors of a discretized approximation of the operator. We show that an error estimate for the approximate eigenvalues can be obtained by evaluating the residual for an approximate eigenpair. The interpolation scheme is selected in such a way that the residual can be evaluated analytically. To demonstrate that the method gives useful error bounds, we apply it to a problem originating from the study of graphene quantum dots where the goal was to investigate the change in the spectrum from incorporating electron–electron interactions in the potential.

It is known that radial symmetry is preserved by the Riesz potential operators and also by the hypersingular Riesz fractional derivatives typically used for inversion. In this paper, we collect properties, asymptotics, and estimates for the radial and spherical parts of Riesz potentials and for solutions to the Riesz potential equation of order one. Sharp estimates for spherical functions are provided in terms of seminorms, and a careful analysis of the radial part of a Riesz potential is carried out in elementary terms. As an application, we provide a two weight estimate for the inverse of the Riesz potential operator of order one acting on spherical functions.

This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the reference domain and the perturbed domain, and the size of eigenfunctions outside the intersection of the two domains. This construction enables the possibility of comparing both nonsmooth domains and domains with different topology. An abstract framework is presented, where the main result is an asymptotic formula where the remainder is expressed in terms of the proximity quantity described above when this is relatively small. As an application, we develop a theory for the Laplacian in Lipschitz domains. In particular, if the domains are assumed to be C-1,C-alpha regular, an asymptotic result for the eigenvalues is given together with estimates for the remainder, and we also provide an example which demonstrates the sharpness of our obtained result.

This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class C-1. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case, however, in the C-1-case we derive an asymptotic formula for the eigenvalues when the domains are of class C-1. Moreover, as an application we consider the case of a C-1-perturbation when the reference domain is of class C-1,C-alpha.

This article considers two weight estimates for the single layer potential - corresponding to the Laplace operator in R (N+1) - on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.

This paper considers to the equation integral(S) U(Q)/vertical bar P - Q vertical bar(N-1) dS(Q) = F(P), P is an element of S, where the surface S is the graph of a Lipschitz function phi on R-N, which has a small Lipschitz constant. The integral on the left-hand side is the single layer potential corresponding to the Laplacian in RN+1. Let Lambda(r) be the Lipschitz constant of phi on the ball centered at the origin with radius 2r. Our analysis is carried out in local L-p-spaces and local Sobolev spaces, where 1 less than p less than infinity, and results are presented in terms of Lambda. Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behavior of the solutions near a point on the surface. These estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Local estimates are provided and some applications are supplied.

For a locally convex space , where the topology is given by a familyof seminorms, we study the existence and uniqueness of fixed points for a mapping defined on some set . We require that there exists a linear and positive operator , acting on functions defined on the index set , such that for every

Under some additional assumptions, one of which is the existence of a fixed point for the operator, we prove that there exists a fixed point of . For a class of elements satisfying as , we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points.We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudo-differential equations with nonlinear terms.

We consider the following equation for the Riesz potential of order one:

Uniqueness of solutions is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove anexistence result and derive an asymptotic formula for solutions near the origin.Our analysis is carried out in local L^p-spaces and Sobolev spaces, which allows us to obtain optimal results concerning the class of right-hand sides and solutions. We also apply our results to weighted L^p-spaces and homogenous Sobolev spaces.

This work is devoted to the equation

where S is the graph of a Lipschitz function φ on R^N with small Lipschitz constant, and dS is the Euclidian surface measure. The integral in the left-hand side is referred to as a simple layer potential and f is a given function. The main objective is to find a solution u to this equation along with estimates for solutions near points on S. Our analysis is carried out in local L^p-spaces and local Sobolev spaces, and the estimates are given in terms of seminorms.

In Paper 1, we consider the case when S is a hyperplane. This gives rise to the classical Riesz potential operator of order one, and we prove uniqueness of solutions in the largest class of functions for which the potential in (1) is defined as an absolutely convergent integral. We also prove an existence result and derive an asymptotic formula for solutions near a point on the surface. Our analysis allows us to obtain optimal results concerning the class of right-hand sides for which a solution to (1) exists. We also apply our results to weighted L^p- and Sobolev spaces, showing that for certain weights, the operator in question is an isomorphism between these spaces.

In Paper 2, we present a fixed point theorem for a locally convex space , where the topology is given by a family of seminorms. We study the existence and uniqueness of fixed points for a mapping defined on a set . It is assumed that there exists a linear and positive operator K, acting on functions defined on the index set Ω, such that for every ,

Under some additional assumptions, one of which is the existence of a fixed point for the operator K + p( ; · ), we prove that there exists a fixed point of . For a class of elements satisfying Kⁿ (p(u ; · ))(α) → 0 as n → ∞, we show that fixed points are unique. This class includes, in particular, the solution we construct in the paper. We give several applications, proving existence and uniqueness of solutions for two types of first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms.

In Paper 3, we treat equation (1) in the case when S is a general Lipschitz surface and 1 < p < ∞. Our results are presented in terms of Λ(r), which is the Lipschitz constant of φ on the ball centered at the origin with radius 2r. Estimates of solutions to (1) are provided, which can be used to obtain knowledge about behaviour near a point on S in terms of seminorms. We also show that solutions to (1) are unique if they are subject to certain growth conditions. Examples are given when specific assumptions are placed on Λ. The main tool used for both existence and uniqueness is the fixed point theorem from Paper 2.

In Paper 4, we collect some properties and estimates of Riesz potential operators, and also for the operator that was used in Paper 1 and Paper 3 to invert the Riesz potential of order one on R^N, for the case when the density function is either radial or has mean value zero on spheres. It turns out that these properties define invariant subspaces of the respective domains of the operators in question.

We consider the following equation for the Riesz potential of order one:

The analysis is done in local L^p and Sobolev spaces, where the topologies are described by a family of semi-norms depending on a positive real parameter. Uniqueness and existence results are proved and asymptotic properties of solutions near the origin are established. Furthermore, we investigate properties of these operators in invariant subspaces.