Open Access
August, 2007 On a Parabolic Symmetry Problem
John L. Lewis, Kaj Nyström
Rev. Mat. Iberoamericana 23(2): 513-536 (August, 2007).


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 $\Omega$.


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John L. Lewis. Kaj Nyström. "On a Parabolic Symmetry Problem." Rev. Mat. Iberoamericana 23 (2) 513 - 536, August, 2007.


Published: August, 2007
First available in Project Euclid: 26 September 2007

zbMATH: 1242.35130
MathSciNet: MR2371436

Primary: 35K05

Keywords: caloric measure , free boundary , Green's function , heat equation , symmetry theorem

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.23 • No. 2 • August, 2007
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