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Second boundary value problem for the Hessian curvature flow

Rongli Huang, Changzheng Qu, Zhizhang Wang, Weifeng Wo

TL;DR

This work studies the second boundary value problem for the $k$-Hessian curvature flow of strictly convex graphs. The authors first identify translating solutions and then prove global existence and convergence to translating solutions, employing an orthogonal invariance method to obtain boundary $C^2$ estimates. By relating primal and dual formulations via the Legendre transform, they extend Schnürer–Urbas type results from the Gauss curvature flow to general $k$-Hessian flows. The results provide a robust framework for long-time behavior and boundary regularity of nonlinear parabolic Hessian flows under prescribed gradient images. Overall, the paper advances the theory of curvature-driven flows with second boundary conditions and offers techniques applicable to broader nonlinear parabolic equations with similar boundary constraints.

Abstract

We investigate the evolution of strictly convex hypersurfaces driven by the $k$-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary $C^2$ estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schnürer-Smoczyk \cite{Schnurer2003} and Schnürer \cite{Schnurer2002} from the second boundary value problem of Gauss curvature flow to $k$-Hessian curvature flow.

Second boundary value problem for the Hessian curvature flow

TL;DR

This work studies the second boundary value problem for the -Hessian curvature flow of strictly convex graphs. The authors first identify translating solutions and then prove global existence and convergence to translating solutions, employing an orthogonal invariance method to obtain boundary estimates. By relating primal and dual formulations via the Legendre transform, they extend Schnürer–Urbas type results from the Gauss curvature flow to general -Hessian flows. The results provide a robust framework for long-time behavior and boundary regularity of nonlinear parabolic Hessian flows under prescribed gradient images. Overall, the paper advances the theory of curvature-driven flows with second boundary conditions and offers techniques applicable to broader nonlinear parabolic equations with similar boundary constraints.

Abstract

We investigate the evolution of strictly convex hypersurfaces driven by the -Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schnürer-Smoczyk \cite{Schnurer2003} and Schnürer \cite{Schnurer2002} from the second boundary value problem of Gauss curvature flow to -Hessian curvature flow.

Paper Structure

This paper contains 15 sections, 9 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric notations
  4. Support function
  5. Legendre transform
  6. Second boundary conditions
  7. Previous results
  8. Translating Solutions
  9. $C^0$ Estimates
  10. Estimate of $w^* (F^*)^{-1}$
  11. Estimates of Strict Obliqueness
  12. $C^2$ estimates
  13. Global $C^2$ estimates
  14. Boundary $C^2$ estimates
  15. Proof of Theorem 1.1

Key Result

Theorem 1.1

Let $\Omega$ and $\Omega^*$ be bounded strictly convex domains with smooth boundaries in $\mathbb{R}^n$. Then, there exists a unique smooth, strictly convex graphical solution $u(x,t)$ to uflow for all $t\geq 0$. Moreover, as $t \to \infty$, $u$ converges to a translating solution with a specified v

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 4.1
  • Proposition 5.1
  • Proposition 6.1
  • proof
  • Lemma 7.1
  • proof
  • Lemma 7.2
  • ...and 1 more