Weighted global regularity estimates for elliptic problems with Robin boundary conditions in Lipschitz domains

[摘要] Let n >= 2 and Omega be a bounded Lipschitz domain of R-n. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second-order elliptic equations of divergence form with real-valued, bounded, measurable coefficients on Omega. More precisely, let p is an element of (n/(n - 1), infinity). Using a real-variable argument, the authors obtain two necessary and sufficient conditions for W-1,W-p estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Holder inequality with exponent por weighted W-1,W-q estimates of solutions with q is an element of (n/(n - 1), p] and some Muckenhoupt weights. As applications, the authors establish some global regularity estimates for solutions to Robin boundary value problems of second-order elliptic equations of divergence form with small BMO coefficients, respectively, on bounded Lipschitz domains, C-1 domains or (semi-)convex domains, in the scale of weighted Lebesgue spaces, via some quite subtle approach which is different from the existing ones and, even when n = 3 in case of bounded C-1 domains, also gives an alternative correct proof of some known result under an additional assumption. By this and some technique from harmonic analysis, the authors further obtain the global regularity estimates, respectively, in Morrey spaces, (Musielak-)Orlicz spaces, and variable Lebesgue spaces. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-09-25

[关键词] Elliptic equation;Robin boundary problem;Lipschitz or (Semi-)convex domain;Weak reverse Holder inequality;Gradient estimate;Muckenhoupt weight