Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition

2025/07/18 by Gatti, Michele, Scheuer, Julian, Weth, Tobias
#35B35 (Secondary) #35N25 #35R11 (Primary) 35B06 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.13715

Abstract

We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set Ω⊆ ℝⁿ. Specifically, we show that if Ω has positive reach and the nonlocal normal derivative introduced in (Dipierro, Ros-Oton, Valdinoci, Rev. Mat. Iberoam. 33 (2017), no. 2, 377-416) is constant on an external surface parallel and sufficiently close to ∂ Ω, then Ω must be a ball. Remarkably, this conclusion remains valid under the sole assumption that Ω is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on Ω. Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian.

Citations

• Serrin's overdetermined problem in rough domains
• Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates
• Quantitative stability for the nonlocal overdetermined Serrin problem
• Quantitative stability for overdetermined nonlocal problems with parallel surfaces and investigation of the stability exponents
• The role of antisymmetric functions in nonlocal equations
• Symmetry and quantitative stability for the parallel surface fractional torsion problem
• Uniform ball property and existence of optimal shapes for a wide class of geometric functionals
• Rigidity and sharp stability estimates for hypersurfaces with constant\n and almost-constant nonlocal mean curvature
• Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay
• Overdetermined problems for the fractional Laplacian in exterior and\n annular sets
• Hölder stability for Serrin's overdetermined problem
• Nonlocal problems with Neumann boundary conditions
• Overdetermined problems with fractional Laplacian
• Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration
• The Dirichlet problem for the fractional Laplacian: regularity up to the\n boundary
• The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary
• A symmetry problem in potential theory
• Remark on the preceding paper of Serrin

Related

• Symmetry and quantitative stability for the parallel surface fractional torsion problem
• Shape derivative approach to fractional overdetermined problems
• Overdetermined problems with fractional Laplacian
• Overdetermined problems for the fractional Laplacian in exterior and\n annular sets
• Stability results for nonlocal Serrin-type problems, antisymmetric Harnack inequalities, and geometric estimates