[Paper Review] On the microscopic spacetime convexity principle for fully nonlinear parabolic equations I: Spacetime convex solutions

Chuanqiang Chen|arXiv (Cornell University)|Mar 20, 2014

Geometric Analysis and Curvature Flows35 references3 citations

TL;DR

This paper establishes a microscopic spacetime convexity principle for fully nonlinear parabolic equations by providing a new, simplified proof of the constant rank theorem for spacetime Hessian matrices of solutions. It extends prior results by Chen-Hu [11] and Hu-Ma [18], proving that under a structural convexity condition on the operator $ F $, the rank of the spacetime Hessian remains constant in space and time, with its null space evolving smoothly, ensuring persistent convexity properties in solutions to fully nonlinear parabolic PDEs.

ABSTRACT

Spacetime convexity is a basic geometric property of the solutions of parabolic equations. In this paper, we study microscopic convexity properties of spacetime convex solutions of fully nonlinear parabolic partial differential equations and give a new simple proof of the constant rank theorem in \cite{CH13}.

Motivation & Objective

• To establish a microscopic spacetime convexity principle for solutions of fully nonlinear parabolic PDEs.
• To provide a simplified proof of the constant rank theorem for the spacetime Hessian of solutions, extending prior results by Chen-Hu [11] and Hu-Ma [18].
• To analyze the persistence and regularity of the null space of the spacetime Hessian under the structural condition on $ F $.
• To demonstrate that the rank of the spacetime Hessian remains constant over time and space under the given convexity condition on $ F $.

Proposed method

• Utilizes a coordinate-invariant approach to analyze the spacetime Hessian of solutions to fully nonlinear parabolic equations.
• Applies the constant rank theorem technique to the spacetime Hessian, leveraging the structural condition that $ F(A^{-1}, p, u, x, t) $ is locally convex in $ (A, u, x) $ for fixed $ (p, t) $.
• Employs a limiting argument with regularization $ D_y^2 u + \varepsilon I $ to handle degeneracy and derive inequalities for the Hessian components.
• Uses the spacetime coordinate transformation and the second-order structure set $ G $ to isolate and estimate relevant Hessian terms.
• Applies the maximum principle to a carefully constructed auxiliary function $ \phi $, showing $ \sum F^{ij} \phi_{ij} - \phi_t \leq C(\phi + |\nabla_x \phi|) $, which implies the desired estimate.
• Relies on the ellipticity of $ F $ and the inverse convexity condition to control curvature terms and establish non-negativity of the quadratic form $ Q $.

Experimental results

Research questions

• RQ1 Under what conditions does the spacetime Hessian of a solution to a fully nonlinear parabolic PDE maintain constant rank over time and space?
• RQ2 How can the constant rank theorem for the spacetime Hessian be proven in a simplified and coordinate-invariant manner?
• RQ3 What structural conditions on the operator $ F $ ensure the persistence of spacetime convexity in solutions?
• RQ4 How does the null space of the spacetime Hessian evolve smoothly in time and space under the given assumptions?
• RQ5 Can the microscopic convexity principle be extended to fully nonlinear parabolic equations beyond the heat equation?

Key findings

• The spacetime Hessian of a solution to the fully nonlinear parabolic equation maintains constant rank $ l(t) $ in space for each fixed $ t $, with $ l(s) \leq l(t) $ for $ s \leq t $.
• For each $ t \in (0, T) $, there exists a neighborhood where $ n - l(t) $ fixed directions span the null space of the spacetime Hessian.
• The null space of the spacetime Hessian evolves smoothly in time, remaining parallel in $ (x,t) $ over a short time interval $ (t_0, t_0 + \delta) $.
• The quadratic form $ Q $ derived from the second-order structure is non-negative under the inverse convexity condition, which is essential for the maximum principle argument.
• The proof establishes that $ \sum F^{ij} \phi_{ij} - \phi_t \leq C(\phi + |\nabla_x \phi|) $, confirming the validity of the maximum principle for the auxiliary function $ \phi $.
• The result holds for a broad class of operators, including linear operators, Hessian operators $ \sigma_k^{1/k} $, and compositions of convex, non-decreasing functions of such operators.