Gradient estimates and their optimality for heat equation in an exterior domain

TL;DR

This paper investigates gradient estimates for the heat equation in exterior domains, establishing decay rates for derivatives and applying these results to commutator estimates involving the Laplace operator with Dirichlet boundary conditions.

Contribution

It provides new gradient estimates and decay rates for solutions to the heat equation in exterior domains, extending understanding of boundary effects and derivative behavior.

Findings

Derived decay rates for derivatives of heat equation solutions

Established gradient estimates for Dirichlet problems in exterior domains

Applied estimates to commutator estimates for Laplace operators

Abstract

This paper is devoted to the study of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain of a compact set. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem. Applications of these estimates to bilinear type commutator estimates for Laplace operator with Dirichlet boundary condition in exterior domain are discussed too.