Uniqueness, existence and regularity of solutions of integro-PDE in domains of R^n

Abstract

The main goal of the thesis is to study integro-differential equations. Integro-differential equations arise naturally in the study of stochastic processes with jumps. These types of processes are of particular interest in finance, physics and ecology. In the first part of my thesis, we study interior regularity for the regional fractional Laplacian operator. We first obtain the integer order differentiability of the regional fractional Laplacian. We further extend the integer order differentiability to the fractional order of the regional fractional Laplacian. Schauder estimates for the regional fractional Laplacian are also provided. In the second and third parts of my thesis, we consider uniqueness and existence of viscosity solutions for a class of nonlocal equations. This class of equations includes Bellman-Isaacs equations containing operators of L\'evy type with measures depending on $x$ and control parameters, as well as elliptic nonlocal equations that are not strictly monotone in the $u$ variable. In the fourth part of my thesis, we obtain semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in $\mathbb R^n$. This class of equations includes Bellman equations containing operators of L\'evy-It\^o type. H\"{o}lder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also proved. In the last part of my thesis, we study interior regularity of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations with Dini continuous terms. We obtain $C^{\sigma}$ regularity estimates for the nonlocal equations by perturbative methods and a version of a recursive Evans-Krylov theorem.