A pointwise inequality for derivatives of solutions of the heat equation in bounded domains

Abstract

Let $u(t,x)$ be a solution of the heat equation in ${\mathbb{R}}^n$. Then its kth derivative also solves the heat equation and satisfies a maximum principle: the largest kth derivative of $u(t,x)$ cannot be larger than the largest kth derivative of $u(0,x)$. We prove an analogous statement for the solution of the heat equation on bounded domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with Dirichlet boundary conditions. As an application, we give a new and fairly elementary proof of the sharp growth of the second derivatives of Laplacian eigenfunction $-\mathrm{\Delta }{\mathit{\varphi }}_k={\mathit{\lambda }}_k{\mathit{\varphi }}_k$ with Dirichlet conditions on smooth domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$.