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On space-time quasiconcave solutions of the heat equation

Chuanqiang Chen, Xi-Nan Ma, Paolo Salani

Abstract

In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.

On space-time quasiconcave solutions of the heat equation

Abstract

In this paper we first obtain a constant rank theorem for the second fundamental form of the space-time level sets of a space-time quasiconcave solution of the heat equation. Utilizing this constant rank theorem, we can obtain some strictly convexity results of the spatial and space-time level sets of the space-time quasiconcave solution of the heat equation in a convex ring. To explain our ideas and for completeness, we also review the constant rank theorem technique for the space-time Hessian of space-time convex solution of heat equation and for the second fundamental form of the convex level sets for harmonic function.

Paper Structure

This paper contains 30 sections, 25 theorems, 351 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and the Constant Rank Theorem technique
  3. Preliminaries
  4. The curvature matrix of the level sets of $u(x)$
  5. The curvature matrix of the spatial level sets of $u(x,t)$
  6. The curvature matrix of the space-time level sets of $u(x,t)$
  7. Elementary symmetric functions
  8. A constant rank theorem for the space-time convex solution of heat equation
  9. The constant rank properties of the spatial Hessian $\nabla^2 u$
  10. A constant rank theorem for the space-time Hessian: CASE 1
  11. A constant rank theorem for the space-time Hessian: CASE 2
  12. The strict convexity of the level sets of harmonic functions in convex rings
  13. A constant rank theorem for the second fundamental form of the level sets of harmonic functions
  14. The strict convexity of the level sets of $u(x)$
  15. A microscopic space-time Convexity Principle for space-time level sets
  16. ...and 15 more sections

Key Result

Theorem \oldthetheorem

Let $u$ be a solution to problem 1.3-1.4. Then the space-time superlevel sets $\Sigma^c_{x,t}$ of $u$ are convex for every $c\in[0,1]$.

Theorems & Definitions (48)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem: Bo82
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 38 more