arXiv subjects
Publications and source records attributed to V. I. Burenkov.
We study the eigenvalue problem for the Neumann-Laplace operator in conformal regular planar domains $Ω\subset\mathbb{C}$. Conformal regular domains support the Poincaré inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.
We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $ϕ(Ω)$ parametrized by Lipschitz homeomorphisms $ϕ$ defined on a fixed reference domain $Ω$. Given two open sets $ϕ(Ω)$, $\tilde ϕ(Ω)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $\|\tilde ϕ-ϕ\|_{W^{1,p}(Ω)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.
We prove the equivalence of Hardy- and Sobolev-type inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount. AMS subject classifications: 35P15, 35J25, 47A75, 47B25, 26D10, 46E35. Keywords: Neumann Laplacian, Sobolev inequalities, Hardy inequalities, spectral stability, Hölder continuity.
The norm of an integral operator occurring in the partial wave decomposition of an operator B introduced by Brown and Ravenhall in a model for relativistic one-electron atoms is determined. The result implies that B is non-negative and has no eigenvalue at 0 when the nuclear charge does not exceed a specified critical value.