Calder\'on-Zygmund gradient estimates for $p$-Laplace systems with BMO complex coefficients

Abstract

This work is concerned with global gradient bounds for a class of divergence-form degenerate elliptic systems with complex-valued coefficients. Notably, the leading coefficients are merely required to be sufficiently small in BMO, which is strictly weaker than the VMO condition. In the complex setting, the well-posedness of this problem was recently investigated in [W. Kim, M. Vestberg, Existence, uniqueness and regularity for elliptic $p$-Laplace systems with complex coefficients,arXiv:2503.18932], where the authors established a strong accretivity condition on the leading coefficients, and this structural condition allows them to derive Schauder-type estimates for weak solutions. In our study, it has already been observed that gaining existence and uniqueness of weak solutions is possible under a natural and less restrictive assumption on the complex-valued coefficients. Following this direction, we prove a global Cader\'on-Zygmund-type estimate for weak solutions, from which the Morrey-space regularity follows as a consequence. This paper is a contribution to the better understanding of solution behavior and may be viewed as part of a series of works aimed at extending regularity theory in the complex-valued setting.