Spectral stability of the Neumann Laplacian

By V. Burenkov, E. Davies. 2001

64 citations recorded by the source corpus.

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 Holder category, then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/67680c49efedb1beb74c3ed2c1b8b6c9c2facd28

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In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Spectral convergence and nonlinear dynamics of reaction–diffusion equations under perturbations of the domain (2004, background): “Also in [10] the case of Hölder perturbations of a domain is studied.”
Domain Perturbation for Linear and Semi-Linear Boundary Value Problems (2008, unknown): “Other references include [32,33,47,49,81].”
Geometrical Structure of Laplacian Eigenfunctions (2012, background): “The spectral stability of elliptic operators under domain perturbations has been thoroughly investigated [105, 106, 107, 108, 109, 226, 240].”
Weyl Asymptotic Formula for the Laplacian on Domains with Rough Boundaries (2003, background): “Remark 1.11. In [ BD ] the authors proved that”
Spectral stability of Dirichlet second order uniformly elliptic operators (2008, methodology): “In [1] the following transformation”
Hardy inequalities on Riemannian manifolds and applications (2012, background): “The preeminent role of Hardy inequalities and the knowledge of the best constants involved is a well known fact, as the reader can recognize from the wide literature that uses such a tool in Euclidean or in subelliptic setting as well as on manifolds ([4,6,9,10,12,22,37,46] just to cite a few).”