On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions

Abstract

In [13], Chen-Ma-Salani established the strict convexity of spacetime level sets of solutions to heat equation in convex rings, using the constant rank theorem and a deformation method. In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations, that is, establish the corresponding microscopic spacetime convexity principles for spacetime level sets. In fact, the results hold for fully nonlinear parabolic equations under a general structural condition, including the p-Laplacian parabolic equations (p > 1) and some mean curvature type parabolic equations.

A note on parabolic power concavity

“…Furthermore, they developed in [18] the notion of a-parabolic q-concavity for nonnegative functions and study parabolic power concavity properties of solutions to parabolic boundary value problems in a convex cylinder with vanishing initial datum and a suitable source term. See also [6,7,8,13,31] for further results related to space-time concavity of solutions of parabolic problems.…”