# On a Parabolic Symmetry Problem

### John L. Lewis

University of Kentucky, Lexington, USA### Kaj Nyström

Uppsala University, Sweden

## 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 $Ω⊂R_{n+1}$. For $T>0$, let $Ω_{T}=Ω∩[R_{n}×(0,T)]$ and let $G$ be the Green function of $Ω_{T}$ with pole at $(0,0)∈∂_{p}Ω_{T}$. Assume that the adjoint caloric measure in $Ω_{T}$ defined with respect to $(0,0)$, $ω^$, is absolutely continuous with respect to a certain surface measure, $σ$, on $∂_{p}Ω_{T}$. Our main result states that if

for all $(X,t)∈∂_{p}Ω_{T}∖{(X,t):t=0}$ and for some $λ>0$, then $∂_{p}Ω_{T}⊆{(X,t):W(X,t)=λ}$ where $W(X,t)$ is the heat kernel and $G=W−λ$ in $Ω_{T}$. This result has previously been proven by Lewis and Vogel under stronger assumptions on $Ω$.

## Cite this article

John L. Lewis, Kaj Nyström, On a Parabolic Symmetry Problem. Rev. Mat. Iberoam. 23 (2007), no. 2, pp. 513–536

DOI 10.4171/RMI/504