Hardy Inequality and Heat Semigroup Estimates for Riemannian Manifolds with Singular Data |
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| Authors: | M van den Berg P Gilkey A Grigor'yan K Kirsten |
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| Institution: | 1. School of Mathematics , University of Bristol , Bristol , UK M.vandenBerg@bris.ac.uk;3. Mathematics Department , University of Oregon , Eugene , Oregon , USA;4. Fakult?t für Mathematik , Universit?t Bielefeld , Bielefeld , Germany;5. Department of Mathematics , Baylor University Waco , Texas , USA |
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| 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 ?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 δ2, (ii) the initial temperature distribution, and the specific heat of D are given by δ?α and δ?β respectively, where δ is the distance to ?D, and 1 < α <2, 1 < β <2. |
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| Keywords: | Hardy inequality Heat content Singular data |
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