In this talk, we investigate some regularity results for non-uniformly elliptic problems. We first present uniformly elliptic problems and the definition of non-uniform ellipticity. We then introduce a double phase problem which is characterized by the fact that its ellipticity rate and growth radically change with the position. We show gradient Hölder continuity and Calderón-Zygmund type estimates for distributional solutions to double phase problems in divergence form. We next introduce a general class of degenerate/singular fully nonlinear elliptic equations which covers the problems of double phase type. We provide C^1 regularity under minimal assumptions on associated operators whose ellipticity may degenerate or blow up along a region where the gradient of a viscosity solution vanishes.
