Advanced properties of some nonlocal operators.

In this thesis, we deal with problems, related to nonlocal operators. In particular, we introduce a suitable notion of integral operators acting on functions with minimal requirements at infinity. We also present results of stability under the appropriate notion of convergence and compatibility results between polynomials of different orders. The theory is developed not only in the pointwise sense, but also in viscosity setting. Moreover, we discover the main properties of extremal type operators, with some applications. Then using the notion of viscosity solutions and Ishii-Lions technique, we give a different proof of the regularity of the solutions to equations involving fully nonlinear nonlocal operators. In the last part of the thesis we deal with domain variation solutions and with notions of a viscosity solution to two phase free boundary problem. We are looking at minima of energy functionals, the latter involving p(x)-Laplace operator or a non-negative matrix. Apart from the Riemannian case, we also consider the related Bernoulli functional in noncommutative framework. Finally, we formulate the suitable definition of a viscosity solution in Carnot groups.