Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Willmore Flow of Graphs with Boundary Data: Low-Regularity Initial Data and Global Convergence

Boris Gulyak

Abstract

We study the Willmore flow for graphs over a bounded domain in $\mathbb{R}^2$ with Dirichlet (clamped) boundary conditions, a still little-studied setting that also serves as a prototype for higher-order flows with fixed boundary data. We develop a low-regularity theory that avoids the classical fourth-order compatibility condition at $t=0$. Combining a reformulation of the graphical equation, which isolates the quasilinear fourth-order principal part from the lower-order terms, with time-weighted parabolic Hölder spaces, we prove short-time existence for initial data in $C^{1+α}(\overlineΩ)$ and, under a smallness assumption, also for Lipschitz data in $C^{0,1}(\overlineΩ)$, even when the initial Willmore energy is not defined. In the Hölder regime, uniqueness is obtained. In the small-data Lipschitz regime, we also prove global existence, uniform gradient bounds, and exponential convergence to a stationary solution of the elliptic Willmore equation with the prescribed boundary data. A key ingredient is an $L^2$-smallness criterion for graphical surfaces with small Willmore energy and small boundary values. The approach is mainly analytic and extends naturally to related higher-order geometric flows and other boundary conditions.

The Willmore Flow of Graphs with Boundary Data: Low-Regularity Initial Data and Global Convergence

Abstract

We study the Willmore flow for graphs over a bounded domain in with Dirichlet (clamped) boundary conditions, a still little-studied setting that also serves as a prototype for higher-order flows with fixed boundary data. We develop a low-regularity theory that avoids the classical fourth-order compatibility condition at . Combining a reformulation of the graphical equation, which isolates the quasilinear fourth-order principal part from the lower-order terms, with time-weighted parabolic Hölder spaces, we prove short-time existence for initial data in and, under a smallness assumption, also for Lipschitz data in , even when the initial Willmore energy is not defined. In the Hölder regime, uniqueness is obtained. In the small-data Lipschitz regime, we also prove global existence, uniform gradient bounds, and exponential convergence to a stationary solution of the elliptic Willmore equation with the prescribed boundary data. A key ingredient is an -smallness criterion for graphical surfaces with small Willmore energy and small boundary values. The approach is mainly analytic and extends naturally to related higher-order geometric flows and other boundary conditions.

Paper Structure

This paper contains 15 sections, 31 theorems, 365 equations.

Table of Contents

  1. Introduction
  2. The Initial-Boundary Value Problem
  3. Goals and Methods
  4. Main Results
  5. Outline
  6. Graphical Willmore flow
  7. Supremum and L2-Smallness Estimates for Graphs
  8. Parabolic Hölder spaces
  9. Local Existence
  10. Hölder Initial Data
  11. Lipschitz initial data
  12. Global Existence
  13. Main Global Existence Theorem
  14. Subconvergence/Convergence to a Critical Point
  15. Appendix

Key Result

Theorem 1

Let $\Omega \subset \mathbbm {R}^2$ be a bounded domain with $C^{4+\alpha}$ boundary, $\alpha\in(0,1)$, and let $g_0\in C^{4+\alpha}(\partial\Omega),g_1\in C^{3+\alpha}(\partial\Omega)$.

Theorems & Definitions (61)

  • Theorem 1: Local Existence
  • proof
  • Theorem 2: Global Existence
  • proof
  • Theorem 3: $L^2$-Smallness
  • proof
  • Lemma 4: Ellipticity
  • proof
  • Lemma 5
  • proof
  • ...and 51 more