Heat flow in polygons with reflecting edges

Abstract

We investigate the heat flow in an open, bounded set $D$ in ${\mathbb{R}}^2$ with polygonal boundary $\partial D$. We suppose that $D$ contains an open, bounded set $\tilde{D}$ with polygonal boundary $\partial \tilde{D}$. The initial condition is the indicator function of $\tilde{D}$ and we impose a Neumann boundary condition on the edges of $\partial D$. We obtain an asymptotic formula for the heat content of $\tilde{D}$ in $D$ as time $t\downarrow 0$.

28 pages, 6 figures