Mean oscillation conditions for nonlinear equation and regularity results

Abstract

We consider general nonlinear elliptic equations of the form \[ \operatorname{div}\, A(x,Du) = 0 \quad \text{in } \Omega, \] where $A:\Omega \times \mathbb R^n \to \mathbb R^n$ satisfies a quasi-isotropic $(p,q)$-growth condition, which is equivalent to the point-wise uniform ellipticity of $A$. We establish sharp and comprehensive mean oscillation conditions on $A(x,\xi)$ with respect to the $x$ variable to obtain $C^1$- and $W^{1,s}$-regularity results. The results provide new conditions even in the standard $p$-growth case with coefficient $\operatorname{div}(a(x)|Du|^{p-2}Du)=0$. Also included are variable exponent growth with and without perturbation as well as borderline double-phase growth and double-phase growth with a coefficient.