Abstract

We use entropy methods to show that the heat equation with Dirichlet boundary conditions on the complement of a compact set in R^d shows a self-similar behaviour much like the usual heat equation on R^d, once we account for the loss of mass due to the boundary. Giving good bounds for the fundamental solution on these sets is surprisingly a relatively recent result, and we find improvements using some advances in logarithmic Sobolev inequalities. The behaviour is strongly different in dimensions d=1 and 2, with the effect of the "hole" becoming gradually less important in higher dimensions. Entropy methods give useful L^1 estimates, and then by using regularisation properties of the heat equation we are able to improve these to uniform estimates.