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Observations on $$W^{1,p}$$ estimates for divergence elliptic equations with VMO coefficients. (English) Zbl 1173.35419
Summary: In this paper we make some observations on the work of G. Di Fazio [Boll. Un. Mat. Ital. 7, 409–420 (1996; Zbl 0865.35048)] concerning $$W^{1,p}$$ estimates, $$1<p<\infty$$, for solutions of the elliptic equation $$\text{div}\,A\nabla u=\text{div}\,f$$ on a domain $$\Omega$$ with zero Dirichlet data when $$A\in \text{VMO}(\Omega)$$ and $$f\in L^p(\Omega)$$. We weaken the assumptions, which allows us to consider real and complex nonsymmetric operators and $$C^1$$ boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove a similar result. The main tool is an appropriate representation for the Green (and Neumann) function on the upper half space. We propose two such representations.

##### MSC:
 35J15 Second-order elliptic equations 35B45 A priori estimates in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs 35J25 Boundary value problems for second-order elliptic equations 35R10 Functional partial differential equations
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