In this talk we will survey on recent results related to the

existence and regularity of solutions to nonlocal differential equations

of different type. The topic is intrinsically connected to probability

and Markov processes with jumps and was revitalized a few years ago by

Luis Caffarelly and his many collaborators. In many cases, the

integro-differential operator is modeled by the fractional Laplacian.

We will concentrate in three examples that show similarities and

differences with respect to standard local PDE: the regularity of

solutions for the fractional porous medium flow; well posedness for

positive solutions to the heat equation with nonlocal diffusion, and an

eigenvalue nonlocal mixed problem in bounded domains. The results

coming from these examples can be found in the following publications:

[1] L. Caffarelli; F. Soria; JL Vazquez; "Regularity of solutions of

the fractional porous medium flow". J. Eur. Math. Soc. (JEMS) 15 (2013),

no. 5

[2] B. Barrios; I. Peral; F. Soria; E. Valdinoci; "A Widder's type

theorem for the heat equation with nonlocal diffusion". Arch. Ration.

Mech. Anal. 213 (2014)

[3] T. Leonori; M. Medina; I. Peral; A. Primo; F.Soria; "Principal

eigenvalue of mixed problem for the fractional Laplacian: Moving the

boundary conditions". J. Differential Equations (2018)