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Optimal Regularity for a Dirichlet-Conormal Problem in Reifenberg Flat Domain

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Abstract

We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique \(W^{1,p}\) solvability is obtained with p being in the optimal range (4 / 3, 4). The leading coefficients are assumed to have small mean oscillations and the boundary of domain is Reifenberg flat. We also assume that the two boundary conditions are separated by some Reifenberg flat set of co-dimension 2 on the boundary.

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Authors and Affiliations

  1. School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Republic of Korea

    Jongkeun Choi

  2. Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA

    Hongjie Dong & Zongyuan Li

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  1. Jongkeun Choi
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  2. Hongjie Dong
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  3. Zongyuan Li
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Correspondence to Hongjie Dong.

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J. Choi was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2019R1F1A1058826). H. Dong and Z. Li were partially supported by the NSF under agreement DMS-1600593.

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Choi, J., Dong, H. & Li, Z. Optimal Regularity for a Dirichlet-Conormal Problem in Reifenberg Flat Domain. Appl Math Optim 83, 1547–1583 (2021). https://doi.org/10.1007/s00245-019-09600-2

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