2012/12/23 by A. Figalli, Figalli, A., H. Shahgholian +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1212.5809
openalex publication_date 2012/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the fully nonlinear free boundary problem arrayll F(D2u)=1 amp; a.e. inB1 ∩ Ω|D2 u| ≤ K amp; a.e. inB1∖Ω, array. where K>0, and Ω is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W2,n solutions are locally C1,1 inside B1. Under the extra condition that Ω⊃ \D u≠ 0 \, and a uniform thickness assumption on the coincidence set \D u = 0 \, we also show local regularity for the free boundary ∂Ω∩ B1.