Bounds for the Heat Diffusion through Windows of Given Area

We consider the lowest eigenvalue for the Laplacian in a given Lipschitz domain under mixed boundary conditions: Dirichlet in a subset of the boundary nonin- sulated window., Neumann otherwise. This eigenvalue can be interpreted as heat leakage rate due to diffusion. We give an explicit calculation for a model problem, a rigorous lower bound that depends only on the area, but not on the geometry of the window. This bound confirms the observations from the model problem. Finally, we show that no nontrivial upper bound is possible; i.e., any small area for the window being prescribed, its geometry can be made bad enough to cause heat leak rates arbitrarily close to the ones for no insulation anywhere. The most important techniques are the Aronszajn]Weinstein method of intermediate variational problems and the Gaussian upper bounds for the heat kernel by E. B. Davies.