The Willmore Flow of Graphs with Boundary Data: Low-Regularity Initial Data and Global Convergence
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2026
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| author | Gulyak, Boris |
| author_facet | Gulyak, Boris |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_27848 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Willmore Flow of Graphs with Boundary Data: Low-Regularity Initial Data and Global Convergence Gulyak, Boris Analysis of PDEs Differential Geometry 35K55, 53E40, 35B40, 35K59, 35B65, 35G31, 35K25 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. |
| title | The Willmore Flow of Graphs with Boundary Data: Low-Regularity Initial Data and Global Convergence |
| topic | Analysis of PDEs Differential Geometry 35K55, 53E40, 35B40, 35K59, 35B65, 35G31, 35K25 |
| url | https://arxiv.org/abs/2603.27848 |