Laplace Dirichlet heat kernels in convex domains

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 $\{x\in {\mathbb{R}}^n:x_n>a|(x_1,...,x_{n-1}){|}^p\}$ where $p\ge 2$, $n\ge 2$ and $a>0$.

30 pages