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.

The authors 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, they 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.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in analysis.

Table of Contents

• Introduction
• Harnack-type Dirichlet spaces
• The Neumann heat kernel in inner uniform domains
• The harmonic profile of an unbounded inner uniform domain
• The Dirichlet heat kernel in inner uniform domains
• Examples
• Bibliography