Conformal spectral stability estimates for the Neumann Laplacian

Abstract

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.

On growth monotonicity estimates of the principal Dirichlet-Laplacian eigenvalue

“…In works [3,4,8,9,12,13] using approaches are based on the geometric theory of composition operators on Sobolev spaces were obtained the spectral estimates for Dirichlet and Neumann eigenvalues of the Laplace operator for a large class of rough domains satisfying quasihyperbolic boundary conditions. These composition operators are generated by conformal mappings, quasiconformal mappings and their generalizations.…”

Space quasiconformal mappings and Neumann eigenvalues in fractal type domains

“…In the recent works [2,3,15,16,17] the spectral stability problem and lower estimates of Neumann eigenvalues in planar domains were considered. In [18] spectral estimates in space domains using the theory of weak p-quasiconformal mappings were obtained.…”

Sobolev mappings and moduli inequalities on Carnot groups

“…Q-homeomorphisms are closely connected with mappings that generate bounded composition operators on Sobolev spaces (p, q-quasiconformal mappings) [12,36,47,48] which were studied on Carnot groups in [39,40,47,49]. In the recent decade the geometric theory of composition operators on Sobolev spaces was applied to spectral estimates of the Laplace operator in Euclidean non-convex domains (see, for example, [5,6,11,13,15,16]) and so results of this article have applications to the Sobolev mappings theory, to the spectral theory of (sub)elliptic operators and to the non-linear elasticity problems associated with vector fields that satisfy Hörmander's hypoellipticity condition.…”