Curvature estimates for the level sets of spatial quasiconcave solutions to a class of parabolic equations

Abstract

We prove a constant rank theorem for the second fundamental form of the spatial convex level surfaces of solutions to equations ut = F (∇ 2 u, ∇u, u, t) under a structural condition, and give a geometric lower bound of the principal curvature of the spatial level surfaces.

On Space-Time Quasiconcave Solutions of the Heat Equation

“…So the proof of Corollary 3.1.4 is complete. Please see the detail in Chen-Shi [18] or Ishige-Salani [29].…”

“…About quasiconcave solutions in convex rings, we already mentioned Korevaar [33] who got a constant rank theorem for the second fundamental form of the level sets of quasiconcave pharmonic functions; then Bian-Guan-Ma-Xu [5] and Guan-Xu [26] obtained a generalization to fully nonlinear elliptic equations, while Chen-Shi [18] got a parabolic version of [5,26] for the second fundamental form of spatial level sets.…”

The Concavity of the Gaussian Curvature of the Convex Level Sets of $$p$$ p -Harmonic Functions with Respect to the Height

“…Then, Ma-Ou-Zhang [19] and Chang-Ma-Yang [6] got the Gaussian curvature and principal curvature estimates of the convex level sets of higher-dimensional harmonic function (in [6], they also treated a class of semilinear elliptic equations) in terms of the Gaussian curvature or principal curvature of the boundary and the norm of the gradient on the boundary. For more recent results on curvature estimates, please see the papers [7,10,[25][26][27] and the references therein.…”

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

“…for fixed (θ, u) ∈ S n−1 × R. But the condition that u is spacetime quasiconcave is necessary, and it is the main difference between Theorem 1.3 and the result in [14]. That is, if u is spacetime quasiconcave, the constant rank theorem for spatial level sets holds for the parabolic equations with (5).…”

“…That is, if u is spacetime quasiconcave, the constant rank theorem for spatial level sets holds for the parabolic equations with (5). Otherwise, if u is spatial quasiconcave, Chen-Shi [14] established the constant rank theorem for spatial level sets holds for the parabolic equations with a totally different structural condition as follows…”