Heat flow from polygons

Abstract

We study the heat flow from an open, bounded set $D$ in ${\mathbb{R}}^2$ with a polygonal boundary $\partial D$. The initial condition is the indicator function of $D$. A Dirichlet $0$ boundary condition has been imposed on some but not all of the edges of $\partial D$. We calculate the heat content of $D$ in ${\mathbb{R}}^2$ at $t$ up to an exponentially small remainder as $t\downarrow 0$.

17 pages, including 4 figures; to appear in Potential Analysis