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Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains. (English) Zbl 1436.35127

Authors’ abstract: Let \(n\geq2\) and \(\Omega\) be a bounded Lipschitz domain in \(\mathbb{R}^n.\) In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in \(\Omega.\) More precisely, for any given \(p\in(2,\infty),\) two necessary and sufficient conditions for \(W^{1,p}\) estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent \(p\) or weighted \(W^{1,q}\) estimates of solutions with \(q\in[2,p]\) and some Muckenhoupt weights, are obtained. As applications, for any given \(p\in(1,\infty)\) and \(\omega\in A_p(\mathbb{R}^n)\) (the class of Muckenhoupt weights), the authors establish weighted \(W_\omega^{1,p}\) estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz-)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J15 Second-order elliptic equations
42B35 Function spaces arising in harmonic analysis
42B37 Harmonic analysis and PDEs
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