Fully nonlinear free boundary problems: optimal boundary regularity beyond convexity

2024/12/22 by Damião J. Araújo, Araújo, Damião J., Andreas Minne +3
Computer Science · Mathematics · #35B65 #35D40 #35R35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2412.17079

openalex publication_date 2024/12/22 · openalex created_date 2024/12/25 · openalex updated_date 2026/07/28

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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