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 $\mathbb{R}^2$ 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 H\"older spaces, we prove short-time existence for initial data in $C^{1+\alpha}(\overline\Omega)$ and, under a smallness assumption, also for Lipschitz data in $C^{0,1}(\overline\Omega)$, even when the initial Willmore energy is not defined. In the H\"older 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.