Dirichlet heat kernel for the Laplacian in a ball

TL;DR

This paper derives precise two-sided estimates for the Dirichlet heat kernel of the Laplacian in a ball, capturing its exponential behavior for small times and improving upon previous results.

Contribution

It provides the first full description of the Dirichlet heat kernel in a ball with sharp global two-sided estimates, enhancing understanding of its behavior.

Findings

Sharp two-sided estimates for the heat kernel in a ball

Accurate description of exponential behavior for small times

Significant improvement over previous qualitative results

Abstract

We provide sharp two-sided estimates on the Dirichlet heat kernel $k_1(t,x,y)$ for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel $k_1(t,x,y)$ in terms of its global two-sided sharp estimates.