Absence of the Lavrentiev phenomenon for generalized double-phase functionals with variable matrix weights

We investigate a wide class of degenerate/singular variational integrals exhibiting generalized double-phase behavior governed by matrix weights. We establish absence of the Lavrentiev phenomenon under sharp structural assumptions on the pair of Orlicz functions and the associated modulating function. This is achieved by imposing a small log log -BMO condition on the matrix weight, which ensures sufficient control over its local oscillation and degeneracy/singularity. As a consequence, we show that the associated functional admits minimizers that can be stably approximated within the corresponding weighted Orlicz-Sobolev space. Our results extend the classical double-phase theory developed for constant matrix coefficients to a general framework with fully variable, anisotropic matrix weights.