Asymptotics of the heat flow on a manifold with smooth boundary
The main results of the paper are small-time asymptotics of solutions of the heat equation on manifolds with boundary with Dirichlet boundary conditions. The paper provides full asymptotics in powers of \(\sqrt{t}\) and an algorithm for computing coefficients of the asymptotics in terms of initial data. In order to describe the results in more detail let us consider the following situation. Let \(\Omega\) be a smooth, compact Riemannian manifold with boundary \(\partial\Omega\), and let \(\phi(t, x)\) be the solution of the heat equation with initial condition \(\phi(x)\in C^\infty(\Omega)\) and zero boundary condition (i.e., \(\phi(t, x)= 0\) for \(x\in\partial\Omega\) and \(t> 0\)); let \(v\) be the inward unit normal vector field on \(\partial\Omega\). For a test function \(\psi(x)\in C^\infty(\Omega)\) let \[ \beta[\phi, \psi](t)= \int_\Omega \phi(t, x)\psi(x)\,dx \] be the heat content. The paper addresses the following problems: First, compute the asymptotics, as \(t\to 0\), of the heat flow through the boundary, i.e. of \({\partial\phi\over\partial v}(t, x)\). Second, compute asymptotics of the heat content \(\beta\) (also in the inhomogeneous case of non-zero boundary condition, i.e. for the manifold being ``heated'' at the boundary in a prescribed way, and zero initial condition). Using Green's formula one reduces the second problem to the first one, and an asymptotic for the heat flow through the boundary is the main result of the paper. The asymptotic itself reads as follows: Assume \(t\to\infty\); then for any \(x\in\partial\Omega\), \[ {\partial\phi\over\partial v}(t, x)\sim{\phi(x)\over\sqrt{\pi}}\cdot t^{-1/2}+ \sum^\infty_{k=0}\widetilde D_k\phi(x)\cdot t^{k/2}, \] where \(\widetilde D_k\) are certain differential operators defined exclusively in terms of the Laplacian and an operator \(N\) defined by \(N\phi= 2\langle\nabla\phi, \nabla\rho\rangle- \phi\Delta\rho\) (where \(\rho\) is the distance from the boundary). The author gives an algorithm for computing the operators \(\widetilde D_k\) and provides explicit formulas for the first 8 operators (the formulas become more and more complicated as \(k\) rises). Next, using the asymptotic for the heat flow, the author gives analogous asymptotics for both homogeneous and inhomogeneous heat content.
scientific article; zbMATH DE number 2109266