Higher differentiability and integrability for some nonlinear elliptic systems with growth coefficients in BMO.

We consider local solutions u of nonlinear elliptic systems of the type div A (x , D u) = div F in Ω ⊂ R n , where u : Ω → R N is in a weighted W loc 1 , p space, with p ≥ 2 , F is in a weighted W loc 1 , 2 space and x → A (x , ξ) has growth coefficients in the space of functions with bounded mean oscillation. We prove higher differentiability of u in the sense that the nonlinear expression of its gradient V μ (D u) : = (μ 2 + | D u | 2 ) p - 2 4 D u , with 0 < μ ≤ 1 , is weakly differentiable with D (V μ (D u)) in a weighted L loc 2 space. Moreover we derive some local Calderón–Zygmund estimates when the source term is not necessarily differentiable. Global estimates for a suitable Dirichlet problem are also available. [ABSTRACT FROM AUTHOR]