Johan Thim

Linköping University | LiU · Department of Mathematics (MAI)

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 C1-domains with C1-perturbations. We also derive an optimal estimate for the remainder term in the C1,α-case. Furthermore, if the boundary is mere...

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 $C^{1,\alpha}$-case. Furthermore, if the...

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 by using th...

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 d...

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...

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...

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 a...

This paper considers to the equation [\int_{S} \frac{U(Q)}{|P-Q|^{N-1}} dS(Q) = F(P), P \in S,] where the surface S is the graph of a Lipschitz function \phi on R^N, which has a small Lipschitz constant. The integral in the left-hand side is the single layer potential corresponding to the Laplacian in R^{N+1}. Let \Lambda(r) be a Lipschitz constant...

For a locally convex space with the topology given by a family {p(┬; α)} α ∈ ω of 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 operatorK, acting on functions defined on the index set Ω, such that for everyu, Under some additional assumptions,...

We consider the following equation for the Riesz potential of order one: Uniqueness is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove an existence result and derive an asymptotic formula for solutions near the origin. Our analysis is carried out in local L p -spaces and Sobolev spa...

Abstract In the early nineteenth century, most mathematicians believed that a contin- uous function has derivative at a significant set of points. A. M. Amp` ere even tried to give a theoretical justification for this (within the limitations of the definitions of his time) in his paper from 1806. In a presentation before the Berlin Academy on July...