On a Parabolic Symmetry Problem

Abstract

In this paper we prove a symmetry theorem for the Green function associated to the heat equation in a certain class of bounded domains $\Omega \subset {\mathbb{R}}^{n+1}$. For $T>0$, let ${\Omega }_T=\Omega \cap [{\mathbb{R}}^n\times (0,T)]$ and let $G$ be the Green function of ${\Omega }_T$ with pole at $(0,0)\in {\partial }_p{\Omega }_T$. Assume that the adjoint caloric measure in ${\Omega }_T$ defined with respect to $(0,0)$, $\hat{\omega }$, is absolutely continuous with respect to a certain surface measure, $\sigma$, on ${\partial }_p{\Omega }_T$. Our main result states that if

for all $(X,t)\in {\partial }_p{\Omega }_T\setminus \{(X,t):t=0\}$ and for some $\lambda >0$, then ${\partial }_p{\Omega }_T\subseteq \{(X,t):W(X,t)=\lambda \}$ where $W(X,t)$ is the heat kernel and $G=W-\lambda$ in ${\Omega }_T$. This result has previously been proven by Lewis and Vogel under stronger assumptions on $\Omega$.