Heat content asymptotics with inhomogeneous Neumann and Dirichlet boundary conditions (Q5937310)

Let \((M,g)\) be a compact Riemannian manifold with smooth boundary. Additionally is assumed that that the boundary of \(M\) is the union of two disjoint sets, i.e. \(\partial M=C_N \cup C_D\). The heat equation with the initial temperature \(\Phi(x)\), \(x\in M\), Neumann boundary condition on \(C_N\) and Dirichlet boundary condition on \(C_D\) is considered. The authors establish an asymptotic expansion for the heat content asymptotics of \(M\) with inhomegeneous Neumann and Dirichlet boundary conditions. For the physical motivation of this problem see \textit{H. S. Carslaw} and \textit{J. C. Jaeger} [`Conduction of Heat in Solids', 2nd ed., Oxford Science Publications, Oxford University Press (1960; Zbl 0095.30201); see also (1921; JFM 48.0573.09), 2nd ed. (paperback) (1986; Zbl 0584.73001)]. Moreover, it is shown that all coefficients are locally determined and the first several terms in the asymptotic expansion are computed in terms of some integral formulas. This is based on the results of van der Berg, Desjardins and Gilkey. An important technical consideration in the proof is the creation of an additional Neumann boundary component to ensure that the average value of \(\Psi\) vanishes (\(\Psi\) is the function from the Dirichlet boundary condition).