Abstract
In this paper, we prove the boundary Lipschitz regularity and the Hopf Lemma by a unified method on Reifenberg domains for fully nonlinear elliptic equations. Precisely, if the domain \(\Omega \) satisfies the exterior Reifenberg \(C^{1,\mathrm {Dini}}\) condition at \(x_0\in \partial \Omega \) (see Definition 1.3), the solution is Lipschitz continuous at \(x_0\); if \(\Omega \) satisfies the interior Reifenberg \(C^{1,\mathrm {Dini}}\) condition at \(x_0\) (see Definition 1.4), the Hopf lemma holds at \(x_0\). Our paper extends the results under the usual \(C^{1,\mathrm {Dini}}\) condition.
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This research is supported by the National Natural Science Foundation of China (Grant No. 11701454) and the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JQ1039).
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Lian, Y., Xu, W. & Zhang, K. Boundary Lipschitz regularity and the Hopf lemma on Reifenberg domains for fully nonlinear elliptic equations. manuscripta math. 166, 343–357 (2021). https://doi.org/10.1007/s00229-020-01246-7
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DOI: https://doi.org/10.1007/s00229-020-01246-7