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arXiv · 2607.06387

Solvability of divergence equation in Lipschitz spaces

María Eugenia Cejas·Ricardo G. Durán
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Abstract

We study the solvability of the divergence equation $$ \operatorname{div} \u = f $$ in bounded $C^2$ domains under homogeneous Dirichlet boundary conditions for data $f\in C^{0,\alpha}(\Omega)$ satisfying the compatibility condition $ \int_\Omega f =0. $ We construct a solution $\u$ such that for every $0<\beta<\alpha$ $$ \u\in C^{1,\beta}(\Omega)^n $$ satisfies $$ \|\u\|_{C^{1,\beta}(\Omega)} \le C\|f\|_{C^{0,\alpha}(\Omega)}. $$ The proof combines localization techniques with a boundary flattening procedure reducing the problem to a model half-cube.

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María Eugenia Cejas, Ricardo G. Durán. 2026-07-07. Solvability of divergence equation in Lipschitz spaces. https://arxiv.org/abs/2607.06387

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