Heat Flow in Polygons with Reflecting Edges

Abstract

We investigate the heat flow in an open, bounded set D in R2 with polygonal boundary ∂D . We suppose that D contains an open, bounded set D~ with polygonal boundary ∂D~ . The initial condition is the indicator function of D~ and we impose a Neumann boundary condition on the edges of ∂D . We obtain an asymptotic formula for the heat content of D~ in D as time t↓ 0 .