Published Paper

Inserted: 4 apr 2024
Last Updated: 4 apr 2024

Journal: Calc. Var.
Volume: 63
Number: 80
Year: 2024
Doi: 10.1007/s00526-024-02685-w

Abstract:

We consider local solutions $u$ of nonlinear elliptic systems of the type \[ \text{div} \,A(x, Du) = \text{div} \, F \qquad \text{in} \quad \Omega \subset \mathbb{R}^n, \] where $u : \Omega \to \mathbb{R}^N$ is in a weighted $W^{1, p}_{loc}$ space, with $p \ge 2$, $F$ is in a weighted $W^{1, 2}_{loc}$ space and $x$ $\to$ $A(x, \xi)$ 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 $V_\mu(Du):=(\mu^2 + \lvert Du\rvert^2)^{\frac{p - 2}{4}}Du$, with $0 < \mu \le 1$, is weakly differentiable with $D(V_\mu(Du))$ in a weighted $L^2_{loc}$ space. Moreover we derive some local Calderòn-Zygmund estimates when the source term is not necessarily differentiable. Global estimates for a suitable Dirichlet problem are also available.