Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

Abstract

Let n ≥ 2 and Ω be a bounded Lipschitz domain in 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 in Ω. More precisely, let p ∈ (n/(n − 1), ∞). Using a real-variable argument, the authors obtain two necessary and sufficient conditions for W 1,p estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1,q estimates of solutions with q ∈ (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 know result. 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.

Global Gradient Estimates for Dirichlet Problems of Elliptic Operators with a BMO Anti-Symmetric Part

“…In this section, we prove Theorem 1.7 via using a real-variable argument for (weighted) L p (Ω) estimates, which is inspired by the work of Caffarelli and Peral [18] (see also [58]). When Ω is a bounded Lipschitz domain in R n , the conclusion of Theorem 3.1 was essentially established in [54,Theorem 3.4] (see also [ [62,Theorem 3.1] are also valid in the case of bounded NTA domains. Thus, we omit the proof of Theorem 3.1 here.…”

“…We prove Theorem 1.7 via using a (weighted) real-variable argument (see Theorem 3.1 below), which was essentially established in [54,Theorem 3.4] (see also [28,29,53,55,62]) and inspired by [18,58]. It is worth pointing out that a similar real-variable argument with the different motivation was used in [3,4].…”

“…It is well known that the research of global regularity estimates in various function spaces for (non-)linear elliptic equations (or systems) in non-smooth domains is one of the most interesting and important topics in partial differential equations (see, for instance, [1,5,12,16,21,23,25,28,35,55] for the linear case, and [2,8,9,17,18,44,45,63,66] for the non-linear case). Moreover, the global regularity estimates for solutions to elliptic boundary problems depend not only on the structure of equations, the integrability of the right-hand side datum, and the properties of coefficients appearing in equations, but also on the smooth property or the geometric property of the boundary of domains (see, for instance, [1,5,16,24,25,28,36,41,44,62]).…”

“…Motivated by [28,42,51,55,62], in this article, we study global gradient estimates in various function spaces for Dirichlet boundary value problems of second order elliptic equations of divergence form with an elliptic symmetric part and a BMO anti-symmetric part on non-smooth domains of R n . More precisely, let n ≥ 2 and Ω ⊂ R n be a bounded non-tangentially accessible domain (for short, NTA domain).…”

John--Nirenberg-$Q$ Spaces via Congruent Cubes

“…Another well-known space appeared in John and Nirenberg [23] is BMO (R n ), the space containing functions of bounded mean oscillation, which can be regarded as the limit space of JN con p,q (R n ) as p → ∞; see [19,Proposition 2.21] and also [31,Proposition 2.6]. The space BMO (R n ) has wide applications in harmonic analysis and partial differential equations; see, for instance, [2,11,12,14,15,16,17,25,28,34]. In particular, we refer the reader to [29] for a systematic survey on function spaces of John-Nirenberg type.…”