Laplace Dirichlet heat kernels in convex domains

TL;DR

This paper derives precise exponential bounds for Laplace Dirichlet heat kernels in convex domains, identifying conditions under which these bounds are sharp, thus advancing understanding of heat kernel behavior in geometric settings.

Contribution

It provides general bounds for heat kernels in convex $ ext{C}^{1,1}$ domains and characterizes sets where these bounds are sharp, extending known special cases.

Findings

Established exponential bounds for heat kernels in convex domains.

Identified classes of sets with sharp heat kernel estimates.

Extended known results to more general convex sets.

Abstract

We provide general lower and upper bounds for Laplace Dirichlet heat kernel of convex $\mathcal C^{1,1}$ domains. The obtained estimates precisely describe the exponential behaviour of the kernels, which has been known only in a few special cases so far. Furthermore, we characterize a class of sets for which the estimates are sharp, i.e. the upper and lower bounds coincide up to a multiplicative constant. In particular, this includes sets of the form $\{x\in \R^n: x_n>a|(x_1,...,x_{n-1})|^p\}$ where $p\geq2$, $n\geq2 $ and $a>0$.