On the microscopic spacetime convexity principle of 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.