Description

This monograph focuses on the heat equation with either the Neumann or the Dirichlet boundary condition in unbounded domains in Euclidean space, Riemannian manifolds, and in the more general context of certain regular local Dirichlet spaces. In works by A. Grigor'yan, L. Saloff-Coste and K-T. Sturm, the equivalence between.

- the parabolic Harnack inequality,

- the two-sided Gaussian heat kernel estimate,

- the Poincaré inequality and the volume doubling property,

is established in a very general context. We use this result to provide precise two-sided heat kernel estimates in a large class of domains described in terms of their inner intrinsic metric and called inner (or intrinsically) uniform domains. Perhaps surprisingly, we treat both the Neumann boundary condition and the Dirichlet boundary condition using essentially the same approach albeit with the additional help of a Doob's h-transform in the case of Dirichlet boundary condition.

The main results are new even when applied to Euclidean domains with smooth boundary where they capture the global effect of the condition of inner uniformity as, for instance, in the case of domains that are the complement of a convex set in Euclidean space.