Abstract
In this paper, we study the boundary regularity for viscosity solutions of fully nonlinear elliptic equations. We use a unified, simple method to prove that if the domain \(\Omega \) satisfies the exterior \(C^{1,\textrm{Dini}}\) condition at \(x_0\in \partial \Omega \), the solution is Lipschitz continuous at \(x_0\); if \(\Omega \) satisfies the interior \(C^{1,\textrm{Dini}}\) condition at \(x_0\), the Hopf lemma holds at \(x_0\). The key idea is that the curved boundaries are regarded as perturbations of a hyperplane. Moreover, we show that the \(C^{1,\textrm{Dini}}\) conditions are optimal.
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Acknowledgements
This research is supported by the China Postdoctoral Science Foundation (Grant No.2021M692086 and 2022M712081), the National Natural Science Foundation of China (Grant No.12031012, 11831003 and 12171299) and the Institute of Modern Analysis-A Frontier Research Center of Shanghai.
Funding
This research is supported by the China Postdoctoral Science Foundation (Grant No. 2021M692086 and 2022M712081), the National Natural Science Foundation of China (Grant No. 12031012, 11831003 and 12171299) and the Institute of Modern Analysis-A Frontier Research Center of Shanghai.
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Lian and Zhang discussed the problem many times and the main idea originated from their discussions. Lian wrote the main manuscript text and Zhang made some revisions.
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Lian, Y., Zhang, K. Boundary Lipschitz Regularity and the Hopf Lemma for Fully Nonlinear Elliptic Equations. Potential Anal 60, 1231–1247 (2024). https://doi.org/10.1007/s11118-023-10085-6
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DOI: https://doi.org/10.1007/s11118-023-10085-6