A Pointwise Inequality for Derivatives of Solutions of the Heat Equation in Bounded Domains

TL;DR

This paper extends the maximum principle for derivatives of heat equation solutions from unbounded to bounded domains with Dirichlet boundary conditions, and applies it to analyze eigenfunction growth.

Contribution

It establishes a pointwise inequality for derivatives of heat solutions in bounded domains, generalizing known results from unbounded spaces, and provides a new proof for eigenfunction derivative growth.

Findings

Maximum derivative of heat solutions in bounded domains is bounded by initial data.

New elementary proof for sharp growth of Laplacian eigenfunction derivatives.

Extension of maximum principle to bounded domain heat equations.

Abstract

Let $u(t,x)$ be a solution of the heat equation in $\mathbb{R}^n$. Then, each $k-$th derivative also solves the heat equation and satisfies a maximum principle, the largest $k-$th derivative of $u(t,x)$ cannot be larger than the largest $k-$th derivative of $u(0,x)$. We prove an analogous statement for the solution of the heat equation on bounded domains $\Omega \subset \mathbb{R}^n$ 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 $-\Delta \phi_k = \lambda_k \phi_k$ with Dirichlet conditions on smooth domains $\Omega \subset \mathbb{R}^n$.