Hardy inequality and heat semigroup estimates for Riemannian manifolds with singular data

2010/11/08 by M. van den Berg, Berg, M. van den, Peter Gilkey +5
Mathematics · #35K20 #58J32 #58J35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1011.1726

openalex publication_date 2010/11/08 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on bd(D), and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in L²(D) satisfies a strong Hardy inequality with weight r², (ii) the initial temperature distribution, and the specific heat of D are given by r^-a and r^-b respectively, where r is the distance to the boundary, and 1

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• Heat content, torsional rigidity and generalized Hardy inequalities for complete Riemannian manifolds
• Heat content in non-compact Riemannian manifolds
• Heat content with singular initial temperature and singular specific heat
• Heat flow in Riemannian manifolds with non-negative Ricci curvature
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