On a rigidity result for Kolmogorov-type operators

Abstract

Let D be a bounded open subset of ℝ^N and let z₀ be a point of D. Assume that the Newtonian potential of D is proportional outside D to the potential of a mass concentrated at z₀. Then D is a Euclidean ball centred at z₀. This theorem, proved by Aharonov, Schiffer and Zalcman in 1981, was extended to the caloric setting by Suzuki and Watson in 2001. In this note, we extend the Suzuki–Watson Theorem to a class of hypoellliptic operators of Kolmogorov-type.