Observations on $W^{1,p}$ estimates for divergence elliptic equations with VMO coefficients

Auscher, P. ; Qafsaoui, M.

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 487-509

Résumé

In this paper, we make some observations on the work of Di Fazio concerning $W^{1,p}$ estimates, $1 p\infty$, for solutions of elliptic equations $\text{div} \, A \nabla u = \text{div} f$ , on a domain $\Omega$ with Dirichlet data $0$ whenever $A \in VMO(\Omega)$ and $f \in L^{p} (\Omega)$. We weaken the assumptions allowing real and complex non-symmetric operators and $C^1$ boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove the 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.