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Global regularity estimates for Neumann problems of elliptic operators with coefficients having a BMO anti-symmetric part in NTA domains

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    *Corresponding author 

This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11871254, 12071431, 11971058 and 12071197), the National Key Research and Development Program of China (Grant No. 2020YFA0712900) and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2021-e18)

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  • Let $ n\ge2 $ and $ \Omega\subset\mathbb{R}^n $ be a bounded NTA domain. In this article, the authors study (weighted) global regularity estimates for Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in $ \Omega $. Precisely, for any given $ p\in(2,\infty) $, via a weak reverse Hölder inequality with the exponent $ p $, the authors give a sufficient condition for the global $ W^{1,p} $ estimate and the global weighted $ W^{1,q} $ estimate, with $ q\in[2,p] $ and some Muckenhoupt weights, of solutions to Neumann boundary value problems in $ \Omega $. As applications, the authors further obtain global regularity estimates for solutions to Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both a small $ \mathrm{BMO} $ symmetric part and a small $ \mathrm{BMO} $ anti-symmetric part, respectively, in bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, $ C^1 $ domains, or (semi-)convex domains, in weighted Lebesgue spaces. The results given in this article improve the known results by weakening the assumption on the coefficient matrix.

    Mathematics Subject Classification: Primary: 35J25; Secondary: 35J15, 42B35.


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