Gradient estimates for generalized double phase problems with two modulating coefficients

Abstract

We prove local Calder\'on-Zygmund estimates for distributional solutions to non-uniformly elliptic equations in divergence form modeled on the energy density $a(x)G(|z|)+b(x)H(|z|)$, where $G$ and $H$ are Young functions and $a(\cdot)$, $b(\cdot)$ are nonnegative coefficients, H\"older continuous with exponents $\alpha$ and $\beta$. Either coefficient may vanish, and only their sum is bounded away from zero, so that each phase can dominate or disappear. Under a gap condition that reduces to $q/p\le1+\min\{\alpha,\beta\}/n$ for $G(t)=t^p$ and $H(t)=t^q$, including the borderline case of equality, we show that the energy density of the gradient inherits the Orlicz integrability of the energy density of the datum. The proof combines a fractional differentiability estimate, which gives higher integrability up to the borderline growth, with comparison arguments that freeze the two coefficients in the appropriate order.