An inverse mean value property for evolution equations. (English) Zbl 1297.35014

Kogoj, Alessia E.; Lanconelli, Ermanno; Tralli, Giulio

Adv. Differ. Equ. 19, No. 7-8, 783-804 (2014).

If \(u\) is a harmonic function then \(u\) satisfies the Gauss mean value property. It is well known that the Euclidean balls are the only open sets characterizing the harmonic functions. Suzuki and Watson proved this inverse property for caloric functions. The aim of this paper is to extend the Suzuki-Watson result, in particular, to the heat operator on stratified Lie groups and to Kolmogorov-Fokker-Planck-type operators. The authors prove that the open sets characterizing the solutions to the involved equations, in terms of suitable average operators, have to be the level sets of the fundamental solutions of the relevant operators. The technique adopted exploits the structure of the sets where the solutions to the involved equations attain their maximum. Such sets are known as propagation sets.

Reviewer: Vincenzo Vespri (Firenze)

MSC:

35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35Q84	Fokker-Planck equations
35K65	Degenerate parabolic equations
35H10	Hypoelliptic equations

Keywords:

Kolmogorov-Fokker-Planck operators; propagation sets