Boundary Hölder Regularity for Elliptic Equations on Reifenberg Flat Domains

2018/12/29 by Yuanyuan Lian, Kai Zhang, Lian, Yuanyuan +1
Computer Science · Mathematics · #35B65 #35D30 #35D40 #35J25 #35J60 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1812.11354

openalex publication_date 2018/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate the boundary Hölder 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<α<1, there exists δ>0 such that the solution is C^α at x₀∈ ∂ Ω provided that Ω is δ-Reifenberg flat at x₀ (see Definition 1.1). In particular, for any 0 < α< 1, if ∂ Ω is C¹ and u=g on ∂ Ω with g∈ C^α(x₀), then u∈ C^α(x₀). A similar result for the Poisson equation has been proved by Lemenant and Sire, where the Alt-Caffarelli-Friedman's monotonicity formula is used.

Related

• Boundary regularity for the Poisson equation in reifenberg-flat domains
• Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains
• Boundary Lipschitz Regularity and the Hopf Lemma on Reifenberg Domains for Fully Nonlinear Elliptic Equations
• Boundary Hölder Regularity for Elliptic Equations
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