The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms

TL;DR

This paper establishes precise estimates for the Dirichlet heat kernel in various inner uniform domains, including non-symmetric forms and complex geometries, advancing understanding of heat distribution in irregular spaces.

Contribution

It provides sharp heat kernel estimates in inner uniform domains for non-symmetric bilinear forms, extending classical results to more general and complex geometries.

Findings

Sharp heat kernel bounds in inner uniform domains

Applicability to non-symmetric bilinear forms

Extension to complex geometries like snowflake domains

Abstract

This paper provides sharp Dirichlet heat kernel estimates in inner uniform domains, including bounded inner uniform domains, in the context of certain (possibly non-symmetric) bilinear forms resembling Dirichlet forms. For instance, the results apply to the Dirichlet heat kernel associated with a uniformly elliptic divergence form operator with symmetric second order part and bounded measurable coefficients in inner uniform domains in $\mathbb R^n$. The results are applicable to any convex domain, to the complement of any convex domain, and to more exotic examples such as the interior and exterior of the snowflake.