A Hardy inequality on Riemannian manifolds and classification of discrete Dirichlet spectra

2014/01/20 by Nils Rautenberg, Rautenberg, Nils
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories #math-ph #math.DG #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.1401.5010

arxiv created 2014/01/20 · openalex publication_date 2014/01/20 · arxiv updated 2014/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains Ω with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B. Davies for the euclidean case, we show that it implies a sufficient geometric criterion under which the Laplace- Beltrami operator with Dirichlet boundary conditions Δ^D has purely discrete spectrum on Ω. We proceed to classify all non-compact Ω with discrete spectrum up to a boundary regularity condition and show that these include for example polygons with ideal vertices in manifolds of negative curvature. This a new result for non-constant curvature.

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