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arXiv · 2407.01061

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

Sun-Sig Byun·Hongsoo Kim·Jehan Oh
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Abstract

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.

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Sun-Sig Byun, Hongsoo Kim, Jehan Oh. 2024-07-01. $C^{1,\alpha}$ regularity for degenerate fully nonlinear elliptic equations with oblique boundary conditions on $C^1$ domains. https://arxiv.org/abs/2407.01061

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