Heat flow and Brownian motion for a region in \({\mathbb{R}}^ 2\) with a polygonal boundary (Q1824285)

Consider the following heat conduction problem. Let D be an open, bounded and connected set in Euclidean space \({\mathbb{R}}^ 2\) with a polygonal boundary. Suppose that D has temperature 1 at time \(t=0\), while the boundary is kept at temperature 0 for all time \(t>0\). We obtain the asymptotic behaviour for the amount of heat in D at time t up to \(O(e^{-q/t})\) as \(t\to 0\).