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 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 Ω ⊂ 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 −∆φ k = λ k φ k with Dirichlet conditions on smooth domains Ω ⊂ R n .