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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 $\Omega$., Comment: 17 pages

Let $\Omega$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_{\Omega}$ denote the torsion function for $\Omega$. It is known that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $\Leb^2(\Omega)$, denoted by $\lambda(\Omega)$, is bounded away from $0$. It is shown that the previously obtained bound $\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)\ge 1$ is sharp: for $m\in\{2,3,...\}$, and any $\epsilon>0$ we construct an open, bounded and connected set $\Omega_{\epsilon}\subset \R^m$ such that $\|v_{\Omega_{\epsilon}}\|_{\Leb^{\infty}(\Omega_{\epsilon})} \lambda(\Omega_{\epsilon})<1+\epsilon$. An upper bound for $v_{\Omega}$ is obtained for planar, convex sets in Euclidean space $M=\R^2$, which is sharp in the limit of elongation. For a complete, non-compact, $m$-dimensional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary it is shown that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator acting in $\Leb^2(\Omega)$ is bounded away from $0$., Comment: 13 pages, 1 figure

Let (M,g) be a compact Riemannian manifold without boundary. Let D be a compact subdomain of M with smooth boundary. We examine the heat content asymptotics for the heat flow from D into M where both the initial temperature and the specific heat are permitted to have controlled singularities on the boundary of D. The operator driving the heat process is assumed to be an operator of Laplace type

We study the asymptotic behavior of the heat content on a compact Riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions imposing Robin boundary conditions. Assuming the existence of a complete asymptotic series we determine the first three terms in that series. In addition to the general setting, the interval is studied in detail as are recursion relations among the coefficients and the relationship between the Dirichlet and Robin settings.

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