Boundary Lipschitz regularity and the Hopf lemma on Reifenberg domains for fully nonlinear elliptic equations
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Yuanyuan Lian · Wenxiu Xu · Kai Zhang
Boundary Lipschitz regularity and the Hopf lemma on Reifenberg domains for fully nonlinear elliptic equations Received: 3 January 2020 / Accepted: 9 September 2020 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 satisfies the exterior Reifenberg C 1,Dini condition at x0 ∈ ∂ (see Definition 1.3), the solution is Lipschitz continuous at x0 ; if satisfies the interior Reifenberg C 1,Dini condition at x0 (see Definition 1.4), the Hopf lemma holds at x0 . Our paper extends the results under the usual C 1,Dini condition.
1. Introduction In this paper, we intend to obtain the pointwise boundary Lipschitz regularity and prove the Hopf Lemma for the viscosity solutions of the following fully nonlinear elliptic equations u ∈ S(λ, , f ) in ; (1.1) u=g on ∂, where is a bounded domain and S (λ, , f ) denotes the Pucci class with uniform constants λ and (see Definition 1.8). It is well known that the exterior sphere condition and the interior sphere condition imply the boundary Lipschitz regularity and the Hopf lemma respectively. In recent decades, sphere condition has been extended to a more general geometrical condition, i.e., the C 1,Dini condition (see Remark 1.6). With aid of the boundary Harnack inequality, Safonov [13] proved the boundary Lipschitz regularity under the exterior C 1,Dini condition, and the Hopf lemma under the interior C 1,Dini condition for classical solutions of linear elliptic equations in nondivergence form. 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). W. Xu (B)· K. Zhang: Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710129, Shaanxi, People’s Republic of China e-mail: [email protected] K. Zhang: e-mail: [email protected]; [email protected] Y. Lian: School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China Y. Lian: e-mail: [email protected]; [email protected] Mathematics Subject Classification: 35B65 · 35J25 · 35J60 · 35D40
https://doi.org/10.1007/s00229-020-01246-7
Y. Lian et al.
Huang, Li and Wang [7] also obtained the boundary Lipschitz regularity for linear elliptic equations under the exterior C 1,Dini condition. They used an auxiliary barrier function and the iteration technique, without using the boundary Harnack inequality. Lieberman [12] proved the Hopf lemma for linear elliptic equations under the interior C 1,Dini condition by applying the regularized distance. Recently, Lian and Zhang [11] extended above results to fully nonlinear elliptic equations by a unified method. Moreover, the proof is simple. In that paper, curved boundaries were regarded as the perturbation of a hyperplane. Then t
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