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 $$\frac {d\hat\omega}{d\sigma}(X,t)=\lambda\frac {|X|}{2t}$$ 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 $\Omega$.