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

Domain Variation Solutions for degenerate two phase free boundary problems

We discuss the domain variation solutions notion for some degenerate elliptic two-phase free boundary problems as well as the viscosity definition of the problem when the operator is degenerate

Enhanced boundary regularity of planar nonlocal minimal graphs and a butterfly effect

In this note, we showcase some recent results obtained in [DSV19] concerning the stickiness properties of nonlocal minimal graphs in the plane. To start with, the nonlocal minimal graphs in the planeenjoy an enhanced boundary regularity, since boundary continuity with respect to the external datum is sufficient to ensure differentiability across the boundary of the domain. As a matter of fact, the Hoelder exponent of the derivative is in this situation sufficiently high to provide the validity of the Euler-Lagrange equation at boundary points as well. From this, using a sliding method, one also deduces that the stickiness phenomenon is generic for nonlocal minimal graphs in the plane, since an arbitrarily small perturbation of continuous nonlocal minimal graphs can produce boundary discontinuities (making the continuous case somehow ``exceptional'' in this framework.In questa nota, presentiamo alcuni risultati recenti ottenuti in [DSV19] relativi alla proprietà di ``appiccicosità'' dei grafici minimi nonlocali nel piano. I grafici minimi non locali nel piano godono di una regolarità ``accresciuta'' al bordo, in quanto la continuità al bordo rispetto al dato esterno è sufficiente a garantire la differenziabilità attraverso il bordo del dominio. Inoltre, l'esponente di Hoelder della derivata è sufficientemente grande da garantire la validità dell'equazione di Eulero-Lagrange anche ai punti di bordo del dominio. Da ciò, usando un metodo di scivolamento, si ottiene anche cheil fenomeno di appiccicosità è generico per grafici minimi non locali nel piano, nel senso che una perturbazione arbitrariamente piccola di i grafici minimi nonlocali continui produce discontinuità al bordo (rendendo quindi il caso continuo in qualche modo ``eccezionale'')

Something about nonlinear problems

In the volume can be found the proceedings of the workshop \u201cSomething about nonlinear problems\u201d, held at the Department of Mathematics of the University of Bologna, Italy, in June 13-14, 201