In this talk, we provide various regularity results for nonlinear measure data problems. The results obtained are part of a program devoted to nonlinear Calderón-Zygmund theory and nonlinear potential theory.
Firstly, we consider elliptic obstacle problems with measure data. We obtain gradient potential estimates and fractional differentiability results by using linearization techniques. In particular, we develop a new method to obtain potential estimates for irregular obstacle problems. For the case of single obstacle problems with L¹-data, we further obtain uniqueness results and comparison principles in order to improve such regularity results.
Secondly, we consider mixed local and nonlocal equations with measure data. We prove existence, regularity and potential estimates for solutions. As a consequence, we also obtain Calderón-Zygmund type estimates and continuity criteria for solutions.
Lastly, we consider nonlocal double phase problems. As a first step to the regularity theory for such anisotropic nonlocal problems, we establish local boundedness and Hölder regularity results by identifying sharp assumptions analogous to those for local double phase problems.