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Boundary H\"older Regularity for Elliptic Equations on Reifenberg Flat Domains

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arxiv 1812.11354 v2 pith:K3LYXXX7 submitted 2018-12-29 math.AP

Yuanyuan Lian , Kai Zhang This is my paper · ORCID
classification math.AP
keywords alphaequationsomegaellipticflatpartialreifenbergboundary
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abstract

In this paper, we investigate the boundary H\"{o}lder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic equations) on Reifenberg flat domains. We prove that for any $0<\alpha<1$, there exists $\delta>0$ such that the solution is $C^{\alpha}$ at $x_0\in \partial \Omega$ provided that $\Omega$ is $\delta$-Reifenberg flat at $x_0$ (see Definition 1.1). In particular, for any $0 < \alpha < 1$, if $\partial \Omega$ is $C^1$ and $u=g$ on $\partial \Omega$ with $g\in C^{\alpha}(x_0)$, then $u\in C^{\alpha}(x_0)$. A similar result for the Poisson equation has been proved by Lemenant and Sire, where the Alt-Caffarelli-Friedman's monotonicity formula is used.

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Cited by 3 Pith papers

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

  1. Boundary regularity for the polyharmonic Dirichlet problem

    math.AP 2025-02 conditional novelty 7.0 of 10

    Weak solutions of the m-polyharmonic Dirichlet problem in Reifenberg-flat domains are C^{m-1,α} up to the boundary, with an a priori estimate in terms of the L^q norm of the source.

  2. 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.

  3. Boundary H\"older regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle

    math.AP 2025-01 reject novelty 6.0 of 10

    Boundary Hölder regularity for the fractional Laplacian over Reifenberg flat domains is claimed for all 0<s<1, but the proof relies on a barrier estimate that contradicts the boundedness of the barrier.

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