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Abstract

We establish existence and regularity results for boundary value problems arising from the first variation of the Willmore energy in the graphical setting. Our focus lies on two-dimensional surfaces with fixed clamped boundary conditions, embedded in three-dimensional Euclidean space, and represented as graphs of height functions over domains with non-smooth boundaries. Our approach involves constructing solutions through linearization and a fixed-point argument, requiring small boundary data in suitable functional spaces. Building on the results of Koch and Lamm \cite{koch2012geometric}, we rewrite the Willmore equation for graphs in a divergence form that allows the application of weighted second-order Sobolev spaces. This reformulation significantly weakens the regularity assumptions on both the boundary and the Dirichlet data, reducing them to the $C^{1+\alpha}$-class, while the solution remains smooth in the interior. Moreover, we extend the existence theory to domains with merely Lipschitz boundaries within a purely weighted Sobolev framework. Our approach is also applicable to other higher-order geometric PDEs, including the graphical Helfrich and surface diffusion equations.