Schwarz Reflection Principles for Solutions of Parabolic Equations

Abstract

A reflection principle is obtained for solutions of the heat equation defined in a cylindrical domain of the form $\Omega \times (0,T)$ where $\Omega$ is a ball in ${{\mathbf {R}}^n}$ and the solution vanishes on $\partial \Omega \times (0,T)$. It is shown that the domain of dependence of the solution at a point outside the cylinder $\Omega \times (0,T)$ is a line segment contained inside $\Omega \times (0,T)$. In the case $n = 2$ this result is generalized to the case of analytic solutions of parabolic equations with analytic coefficients defined in an arbitrary bounded simply connected cylinder $D \times (0,T)$ where the solution vanishes on a portion of $\partial D \times (0,T)$.