On the microscopic spacetime convexity principle of fully nonlinear parabolic equations I: Spacetime convex solutions

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.…”

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

“…In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations, and we give a more technical proof under the coordinate system such that the spacetime second fundamental form â(x, t) (see (2.7)) is diagonalized at each point. As in [17,12] and [11], the key difficulties of two calculations are the same, and the processes of the two proofs are also the same. So the corresponding proof holds for fully nonlinear equations based on the coordinate system such that the spatial second fundamental form a(x, t) is diagonalized at each point, and the calculations must be more complicate than [13].…”

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

“…Proof. For (x, t) ∈ O × (t 0 − δ, t 0 ) with the suitable coordinate (39), we have from (11) and (67)…”

“…In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations, and we give a more technical proof under the coordinate system such that the spacetime second fundamental formâ(x, t) (see (15)) is diagonalized at each point. As in [17,12] and [11], the key difficulties of two calculations are the same, and the processes of the two proofs are also the same. So the corresponding proof holds for fully nonlinear equations based on the coordinate system such that the spatial second fundamental form a(x, t) is diagonalized at each point, and the calculations must be more complicate than [13].…”

“…As it is well known, one needs to choose a suitable coordinate system to simplify the calculations in the proof of the constant rank theorem. In [13], the proof for the heat equation is based on a coordinate system such that the spatial second fundamental form a(x, t) (see (11)) is diagonalized at each point. In this paper, we generalize the constant rank theorem in [13] to fully nonlinear parabolic equations, and we give a more technical proof under the coordinate system such that the spacetime second fundamental formâ(x, t) (see (15)) is diagonalized at each point.…”