Gradient estimates for generalized double phase problems with two modulating coefficients

Abstract

We establish Calder\'on-Zygmund estimates for solutions to non-uniformly elliptic equations in divergence form modeled on the generalized double phase structure $$\Psi(x,z)=a(x)G(|z|)+b(x)H(|z|),$$ where $G$ and $H$ are Young functions and $a,b$ are non-negative, H\"older continuous coefficients satisfying a natural non-degeneracy condition $a(\cdot)+b(\cdot)\ge\mu>0$. Under natural assumptions on $G,H$ and the H\"older regularity of $a,b$, we prove that the gradient of any local solution inherits the same integrability as the datum. More precisely, if $\Psi(\cdot,F)\in L^\Theta_{\mathrm{loc}}$, then $\Psi(\cdot,Du)\in L^\Theta_{\mathrm{loc}}$ for every $\Theta\in\mathcal{N}$. Our results extend those of Baasandorj-Byun-Oh (\emph{J. Funct. Anal.} \textbf{279}(7), 2020) from the classical generalized double phase structure $G+a(x)H$ to the two modulating coefficient setting and extend the gradient estimates of Kim-Kim-Oh (\emph{Nonlinear Differ. Equ. Appl.} \textbf{33}, 2026) by establishing Calder\'on-Zygmund estimates for generalized double phase functionals in a borderline case within the two modulating coefficient framework.