Abstract
Let \(\Omega \) be an open set in a complete, smooth, non-compact, m-dimensional Riemannian manifold M without boundary, where M satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if \(\Omega \) has infinite measure, and if \(\Omega \) has finite heat content \(H_{\Omega }(T)\) for some \(T>0\), then \(H_{\Omega }(t)<\infty \) for all \(t>0\). Comparable two-sided bounds for \(H_{\Omega }(t)\) are obtained for such \(\Omega \).
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van den Berg, M. Heat Content in Non-compact Riemannian Manifolds. Integr. Equ. Oper. Theory 90, 8 (2018). https://doi.org/10.1007/s00020-018-2440-z
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DOI: https://doi.org/10.1007/s00020-018-2440-z