Mean oscillation conditions for nonlinear equation and regularity results
Peter Hästö, Mikyoung Lee, Jihoon Ok
TL;DR
This work addresses regularity for nonlinear elliptic equations with nonstandard, non-autonomous growth by introducing sharp mean oscillation conditions on the coefficient map $A(x,\xi)$. The authors develop a robust framework based on quasi-isotropic $$(p,q)$$-growth and generalized Orlicz spaces, defining DMA and VMA mean oscillation notions and linking them to continuity and higher integrability via a two-step comparison-iteration method. Central contributions include two main results: Calderón–Zygmund type $W^{1,s}$ estimates and $C^{1}$-regularity under Dini mean continuity, with far-reaching implications for $p(x)$-growth, double-phase problems, and variable-exponent energies. The approach unifies pointwise and mean-continuity regimes, extends prior results, and provides new regularity criteria applicable to a broad class of nonuniformly elliptic problems, including those with Lorentz-type and log-modified growth conditions; the framework yields sharp, quantitative control of gradient regularity in terms of the mean oscillation of $A$ in $x$.
Abstract
We consider general nonlinear elliptic equations of the form \[ \mathrm{div}\, A(x,Du) = 0 \quad \text{in } Ω, \] where $ A:Ω\times \Rn \to \Rn $ 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 $A(x,ξ)$ 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 $÷(a(x)|Du|^{p-2}Du)=0$. Also included are variable exponent growth with and without perturbation as well as borderline double-phase growth and double-phase growth with a coefficient.