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.

Abstract (VO)

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.

Riassunto

In questo lavoro esponiamo alcune osservazioni circa il lavoro di Di Fazio riguardante le stime $W^{1,p}$ per $1 p\infty$ per soluzioni di equazioni ellittiche del tipo $\text{div} \, A \nabla u = \text{div} \, f$ su un dominio $\Omega$ con dati di Dirichlet nulli, $A$ nella classe $VMO$ ed $f$ in $L^{p}$. Si considera il caso in cui i coefficienti della parte principale sono complessi e la frontiera di $\Omega$ è di classe $C^{1}$. Si considera inoltre il caso del problema di Neumann non omogeneo e si dimostrano risultati analoghi. Il principale strumento utilizzato è una conveniente formula di rappresentazione per la funzione di Green e di Neumann.

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