(C^{1,\alpha}\) regularity for degenerate fully nonlinear elliptic equations with oblique boundary conditions on \(C^1\) domains

We provide a sharp \(C^{1,\alpha}\) estimate up to the boundary for a viscosity solution of a degenerate fully nonlinear elliptic equation with the oblique boundary condition on a \(C^1\) domain. To this end, we first obtain a uniform boundary H{\"o}lder estimate with the oblique boundary condi...

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Bibliographic Details
Published inarXiv.org
Main Authors Sun-Sig Byun, Kim, Hongsoo, Oh, Jehan
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 01.07.2024
Subjects
Boundary conditions
Elliptic functions
Regularity
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Summary:We provide a sharp \(C^{1,\alpha}\) estimate up to the boundary for a viscosity solution of a degenerate fully nonlinear elliptic equation with the oblique boundary condition on a \(C^1\) domain. To this end, we first obtain a uniform boundary H{\"o}lder estimate with the oblique boundary condition in an "almost \(C^1\)-flat" domain for the equations which is uniformly elliptic only where the gradient is far from some point, and then we establish a desired \(C^{1,\alpha}\) regularity based on perturbation and compactness arguments.
ISSN:2331-8422