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Optimal regularity for nonlocal elliptic equations and free boundary problems

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arxiv 2403.07793 v1 pith:IYWG3OQI submitted 2024-03-12 math.AP

Xavier Ros-Oton , Marvin Weidner This is my paper · ORCID
classification math.AP
keywords nonlocalregularityboundaryestablishkernelsoptimalasympelliptic
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abstract

In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates.

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Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Higher regularity in nonlocal free boundary problems

    math.AP 2025-07 accept novelty 8.0 of 10

    A C^{2,alpha} free boundary in the nonlocal one-phase problem is proved to be C^infinity for general integro-differential operators of order 2s.

  2. Optimal regularity for kinetic Fokker-Planck equations in domains

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    Solutions to kinetic Fokker-Planck equations with specular reflection are C^{4,1} in the kinetic Hölder sense up to the grazing set, and this regularity is optimal.

  3. Capacitary estimates for solutions to nonlocal Dirichlet problems

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    For nonlocal p-Laplace type equations with measurable coefficients, boundary Hölder regularity holds exactly when the exterior capacity density condition holds, with a quantitative modulus estimate.

  4. Bernoulli problem for the fractional $p$-Laplacian

    math.AP 2026-07 conditional novelty 7.0 of 10

    Minimizers of the fractional p-Laplacian one-phase Bernoulli problem exist, are locally Hölder continuous, solve the homogeneous equation in their positivity set, and satisfy the optimal free-boundary growth u(x) ≲ |x−x0|^s.

  5. $C^{\infty}$ Regularity for the free boundary of one-phase Fractional Laplacian problem

    math.AP 2025-08 conditional novelty 7.0 of 10

    Flat free boundaries in the one-phase fractional Laplacian problem are C∞, not merely C^{1,α}.

  6. Approximate factorizations for non-symmetric jump processes

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    Sharp two-sided heat kernel and Green function estimates are proved for killed and censored alpha-stable processes in Dini-smooth domains and for non-symmetric stable processes in double-Dini domains via approximate f...

  7. Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains

    math.AP 2025-02 conditional novelty 6.0 of 10

    For operators comparable to the fractional Laplacian of order 2s, solutions with zero exterior data on Reifenberg flat domains are C^{s-ε} up to the boundary.

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