Geodesic
On very weak solutions of a class of nonlinear elliptic systems
Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 3, pp. 493-508.

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In this paper we prove a regularity result for very weak solutions of equations of the type $- \operatorname{div} A(x,u,Du)=B(x, u,Du)$, where $A$, $B$ grow in the gradient like $t^{p-1}$ and $B(x, u, Du)$ is not in divergence form. Namely we prove that a very weak solution $u\in W^{1,r}$ of our equation belongs to $W^{1,p}$. We also prove global higher integrability for a very weak solution for the Dirichlet problem $$ \cases -\operatorname{div} A(x,u,Du)\,=B(x, u,Du) \quad \text{in } \Omega , \ u-u_o\in W^{1,r}(\Omega,\Bbb R^m). \endcases $$
Classification : 35B65, 35D05, 35J50, 35J55, 35J60, 35J99, 46E30
Keywords: nonlinear elliptic systems; maximal operator theory 
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     title = {On very weak solutions of a class of nonlinear elliptic systems},
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Carozza, Menita; Passarelli di Napoli, Antonia. On very weak solutions of a class of nonlinear elliptic systems. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 3, pp. 493-508. http://geodesic.mathdoc.fr/item/CMUC_2000__41_3_a7/