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arXiv · 2412.17079

Fully nonlinear free boundary problems: optimal boundary regularity beyond convexity

Damião J. Araújo·Andreas Minne·Edgard A. Pimentel
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Abstract

We study a general class of elliptic free boundary problems equipped with a Dirichlet boundary condition. Our primary result establishes an optimal $C^{1,1}$-regularity estimate for $L^p$-strong solutions at points where the free and fixed boundaries intersect. A key novelty is that no convexity or concavity assumptions are imposed on the fully nonlinear operator governing the system. Our analysis derives BMO estimates in a universal neighbourhood of the fixed boundary. It relies solely on a differentiability assumption. Once those estimates are available, applying by now standard methods yields the optimal regularity.

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Damião J. Araújo, Andreas Minne, Edgard A. Pimentel. 2024-12-22. Fully nonlinear free boundary problems: optimal boundary regularity beyond convexity. https://arxiv.org/abs/2412.17079

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