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

Abstract

AbstractWe consider local solutions u of nonlinear elliptic systems of the type $$\begin{aligned} \text {div} \,A(x, Du) = \text {div} \, F \qquad \text {in} \quad \Omega \subset \mathbb {R}^n, \end{aligned}$$ div A ( x , D u ) = div F in Ω ⊂ R n , where $$u: \Omega \rightarrow \mathbb {R}^N$$ u : Ω → R N is in a weighted $$W^{1, p}_{loc}$$ W loc 1 , p space, with $$p \ge 2$$ p ≥ 2 , F is in a weighted $$W^{1, 2}_{loc}$$ W loc 1 , 2 space and x$$\rightarrow $$ → $$A(x, \xi )$$ 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 $$V_\mu (Du):=(\mu ^2 + |Du|^2)^{\frac{p - 2}{4}}Du$$ V μ ( D u ) : = ( μ 2 + | D u | 2 ) p - 2 4 D u , with $$0 < \mu \le 1$$ 0 < μ ≤ 1 , is weakly differentiable with $$D(V_\mu (Du))$$ D ( V μ ( D u ) ) in a weighted $$L^2_{loc}$$ 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.