459 research outputs found
Spectral Stability of the Neumann Laplacian
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.
AMS subject classifications: 35P15, 35J25, 47A75, 47B25, 26D10, 46E35.
Keywords: Neumann Laplacian, Sobolev inequalities, Hardy inequalities,
spectral stability, H\"older continuity.Comment: 23 page
Mollifying operators with variable step and their application to approximation by infinitely differentiable functions
Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains
We consider general second order uniformly elliptic operators subject to
homogeneous boundary conditions on open sets parametrized by
Lipschitz homeomorphisms defined on a fixed reference domain .
Given two open sets , we estimate the
variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm
for finite values of , 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.Comment: 34 pages. Minor changes in the introduction and the refercenes.
Published in: Around the research of Vladimir Maz'ya II, pp23--60, Int. Math.
Ser. (N.Y.), vol. 12, Springer, New York 201
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