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 $\Omega \subset \mathbb{R}^n$.