On a Parabolic Symmetry Problem

  • John L. Lewis

    University of Kentucky, Lexington, USA
  • Kaj Nyström

    Uppsala University, Sweden


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

dω^dσ(X,t)=λX2t\frac {d\hat\omega}{d\sigma}(X,t)=\lambda\frac {|X|}{2t}

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

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