Stability results for the first eigenvalue of the Laplacian on domains in space forms (original) (raw)

Spectral stability of the -Laplacian

Nonlinear Analysis: Theory, Methods & Applications, 2009

We study the dependence of the eigenvalues of the p-Laplacian upon domain perturbation. We prove Lipschitz-type estimates for the deviation of the eigenvalues following a domain perturbation. Such estimates are expressed in terms of suitable measures of vicinity between open sets, such as the 'atlas distance' and the 'lower Hausdorff-Pompeiu deviation'. In the case of open sets with Hölder continuous boundaries, our results improve a result known for the first eigenvalue.

Isoperimetric inequalities for eigenvalues of the Laplacian

Contemporary Mathematics, 2011

This paper deals with eigenvalue optimization problems for a family of natural Schrödinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eigenvalues. We show that under suitable assumptions on the potential, the first and the second eigenvalues are maximized by (round) spheres.

Conformal spectral stability estimates for the Dirichlet Laplacian

Mathematische Nachrichten

We study the eigenvalue problem for the Dirichlet Laplacian in bounded simply connected plane domains Ω ⊂ C using conformal transformations of the original problem to the weighted eigenvalue problem for the Dirichlet Laplacian in the unit disc D. This allows us to estimate the variation of the eigenvalues of the Dirichlet Laplacian upon domain perturbation via energy type integrals for a large class of "conformal regular" domains which includes all quasidiscs, i.e. images of the unit disc under quasiconformal homeomorphisms of the plane onto itself. Boundaries of such domains can have any Hausdorff dimension between one and two.

Conformal spectral stability estimates for the Neumann Laplacian

Mathematische Nachrichten

We study the eigenvalue problem for the Neumann-Laplace operator in conformal regular planar domains Ω ⊂ 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.

Spectral stability estimates for the eigenfunctions of second order elliptic operators

Mathematische Nachrichten, 2012

Dirichlet boundary conditions, stability estimates for the eigenfunctions, perturbation of an open set, gap between linear operators MSC (2010) 47F05, 35J40, 35B30, 35P15 Stability of the eigenfunctions of nonnegative selfadjoint second-order linear elliptic operators subject to homogeneous Dirichlet boundary data under domain perturbation is investigated. Let Ω, Ω ⊂ R n be bounded open sets. The main result gives estimates for the variation of the eigenfunctions under perturbations Ω of Ω such that Ωε = {x ∈ Ω : dist(x, R n \Ω) > ε} ⊂ Ω ⊂ Ω ⊂ Ω in terms of powers of ε, where the parameter ε > 0 is sufficiently small. The estimates obtained here hold under some regularity assumptions on Ω, Ω. They are obtained by using the notion of a gap between linear operators, which has been recently extended by the authors to differential operators defined on different open sets.

SPECTRAL STABILITY OF THE NEUMANN LAPLACIAN

Journal of Differential Equations. , 2002

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 H¨older category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.