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

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 $Omega subset mathbb{R}^n$.