In this talk, we provide comprehensive regularity results and optimal conditions for a general class of functionals involving Orlicz multi-phase, which exhibits non-standard growth conditions and non-uniformly elliptic properties.

First, we give a unified treatment to show various regularity results for minima of Orlicz multi-phase type functionals with coefficient functions not necessarily H"older continuous even for a lower level of the regularity.
Moreover, assuming that minima of such functionals belong to better spaces such as H"older spaces and Lebesgue spaces, we address optimal conditions on nonlinearity for each variant under which we build comprehensive regularity results.

Second, we discuss local Calder'on-Zygmund type estimates under the optimal conditions on the nonlinearity for distributional solutions to non-uniformly elliptic equations of Orlicz multi-phase type in divergence form with the coefficient functions not necessarily H"older continuous.

Lastly, we establish an optimal H"older continuity for the gradient of viscosity solutions of a class of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.