Regularity and nondegeneracy for nonlocal Bernoulli problems with variable kernels

2024/03/18 by Stanley Snelson, Eduardo V. Teixeira, Snelson, Stanley +1
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2403.11937

openalex publication_date 2024/03/18 · openalex created_date 2024/03/20 · openalex updated_date 2026/07/28

Abstract

We consider a generalization of the Bernoulli free boundary problem where the underlying differential operator is a nonlocal, non-translation-invariant elliptic operator of order 2s∈ (0,2). Because of the lack of translation invariance, the Caffarelli-Silvestre extension is unavailable, and we must work with the nonlocal problem directly instead of transforming to a thin free boundary problem. We prove global Hölder continuity of minimizers for both the one- and two-phase problems. Next, for the one-phase problem, we show Hölder continuity at the free boundary with the optimal exponent s. We also prove matching nondegeneracy estimates. A key novelty of our work is that all our findings hold without requiring any regularity assumptions on the kernel of the nonlocal operator. This characteristic makes them crucial in the development of a universal regularity theory for nonlocal free boundary problems.

Related

• Higher regularity in nonlocal free boundary problems
• Optimal regularity for nonlocal elliptic equations and free boundary problems
• Boundary regularity for nonlocal operators with kernels of variable orders
• The Dirichlet problem for nonlocal elliptic operators with C^0,α exterior data
• Optimal boundary regularity and Green function estimates for nonlocal equations in divergence form