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25 April 2024

» arxiv » 1011.1726

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Hardy inequality and heat semigroup estimates for Riemannian manifolds with singular data
M. van den Berg ; P. Gilkey ; K. Kirsten ; A. Grigor'yan ;
Date:	8 Nov 2010
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 $L^2(D)$ satisfies a strong Hardy inequality with weight $r^2$, (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<a<2, 1<b<2.
Source:	arXiv, 1011.1726
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