The classical nonlinear potential theory has recently been extended to nonlocal nonlinear potential theory, which studies harmonic functions associated with nonlocal nonlinear operators. In this talk, we focus on the harmonic functions solving the nonlocal Dirichlet problem. As in the study of classical Dirichlet problem, the nonlocal Dirichlet problem can be solved by using Sobolev and Perron solutions. We provide several properties of such solutions. This talk is based on joint works with Anders Björn, Jana Björn, Ki-Ahm Lee and Se-Chan Lee.

This talk presents the C^{1,\alpha}-regularity for viscosity solutions of degenerate/singular fully nonlinear parabolic equations. For this purpose, we develop a new type of Bernstein technique in view of the difference quotient to obtain a priori estimates of the regularized equations. Also, we establish the well-posedness and the uniform C^{1,\alpha}-estimates for the regularized Cauchy-Dirichlet problem.

In this talk, we prove the Aleksandrov--Bakelman--Pucci estimate for non-uniformly elliptic equations in non-divergence form. Moreover, we investigate local behaviors of solutions of such equations by developing local boundedness and weak Harnack inequality. Here we impose an integrability assumption on ellipticity representing degeneracy or singularity, instead of specifying the particular structure of ellipticity.