Sub-potential lower bounds and Liouville’s type rigidity for parabolic De Giorgi classes

Abstract

We discuss several structural properties of functions belonging to a parabolic energy class, reminiscent of the elliptic De Giorgi class. In earlier works, sub-potential lower bounds, giving insight into the structural behavior of elements of these classes, were established for the linear case: Here, we extend these results to the nonlinear one. By showing that sub-potential lower bounds follow solely from the Harnack inequality, we show that positive solutions to Trudinger’s equation and elements of parabolic De Giorgi classes have a common lower bound. For both cases, we derive Liouville-type rigidity results in the parabolic setting.