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. We also consider double phase problems with two modulating coefficients.

In this talk, we first review nonlinear elliptic equations when the right-hand side is a finite measure. We discuss global gradient estimates of a solution for such measure data problems in bounded nonsmooth domains. We provide proper solutions and conditions which guarantee the regularity results. If time permits, we will consider parabolic problems with measure data.