Calderón-Zygmund estimates for general elliptic operators with double phase (Q2309737)

The interesting paper under review deals with regularity issues for the distributional solutions \(u\in W^{1,H}(\Omega)\) of the double-phase type equation \[ \operatorname{div}\left[ b(x,u) \big( |Du|^{p-2}Du + a(x)|Du|^{q-2} Du\big) \right]=\operatorname{div}\big( |F|^{p-2}F + a(x)|F|^{q-2}F \big) \] over a bounded domain \(\Omega\subset\mathbb{R}^n,\) \(n\geq 2,\) where \(F\in L^H(\Omega;\mathbb{R}^n)\), \(1 < p < q\), \(0\leq a(\cdot)\in C^\alpha\) and the function \(b(x,z)\) is small-BMO with respect to \(x\), uniformly continuous in \(z\) and satisfies \(0<\nu\leq b(x,z)\leq \Lambda\). The author proves a Calderón-Zygmund type estimate for the gradient of the bounded solution, showing that \[ \big(|F|^{p}+a(x)|F|^{q}\big)\in L^\gamma_{\mathrm{loc}}\text{ implies }\big(|Du|^{p}+a(x)|Du|^{q}\big)\in L^\gamma_{\mathrm{loc}} \] for all \(\gamma>1\).