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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.

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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