Two-phase problems: Perron solutions and regularity of the Neumann problem in convex cones

2024/07/28 by Thomas Beck, Beck, Thomas, Daniela De Silva +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2407.19538

openalex publication_date 2024/07/28 · openalex created_date 2024/08/01 · openalex updated_date 2026/07/28

Abstract

We investigate a fully nonlinear two-phase free boundary problem with a Neumann boundary condition on the boundary of a general convex set K ⊂ ℝⁿ with corners. We show that the interior regularity theory developed by Caffarelli for the classical two-phase problem in his pioneer works \citeC1,C2, can be extended up to the boundary for the Neumann boundary condition under very mild regularity assumptions on the convex domain K. To start, we establish a general existence theorem for the Dirichlet two-phase problem driven by two different fully nonlinear operators, which is a result of independent interest.

Related

• Two-phase free boundary problems in convex domains
• Free boundary regularity for fully nonlinear non-homogeneous two-phase problems
• Existence and regularity in the fully nonlinear one-phase free boundary problem
• Perron's solutions for two-phase free boundary problems with distributed sources
• Solutions to a two-dimensional, Neumann free boundary problem