On a Parabolic Symmetry Problem

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 ⊂ Rn+1. For T > 0, let ΩT = Ω ∩ [Rn × (0, T )] and let G be the Green function of ΩT with pole at (0, 0) ∈ ∂pΩT . Assume that the adjoint caloric measure in ΩT defined with respect to (0, 0), ω̂, is absolutely continuous with respect to a certain surface measure, σ, on ∂pΩT . Our main result states that if dω̂ dσ (X, t) = λ |X| 2t for all (X, t) ∈ ∂pΩT \ {(X, t) : t = 0} and for some λ > 0, then ∂pΩT ⊆ {(X, t) : W (X, t) = λ} where W (X, t) is the heat kernel and G = W − λ in ΩT . This result has previously been proven by Lewis and Vogel under stronger assumptions on Ω.