Boundary Value Problem Governed by Symmetric Nonlocal Operators

Abstract

When natural phenomena and data-based relations are driven by dynamics which are not purely local, they cannot be described satisfactorily by partial differential equations. As a consequence, mathematical models governed by nonlocal operators are of interest. This thesis is concerned with nonlocal operators of the form $\mathcal{L}u(x) = PV \int_{\mathbb{R}^d} (u(x)-u(y)) K(x,dy), x \in \mathbb{R}^d$, which are determined through a family of Borel measures $K=(K(x, \cdot))_{x \in \mathbb{R}^d}$ on $\mathbb{R}^d$ and which act on the vector space of Borel measurable functions $u: \mathbb{R}^d \rightarrow \mathbb{R}$. For a large class of families $K$, namely those where $K$ is a symmetric transition kernel satisfying a specific non-degeneracy condition, a variational theory for nonlocal equations of the type $\mathcal{L}u=f$ is established which builds upon gadgets from both measure theory and classical analysis. While measure theory is used to provide a nonlocal integration by parts formula that allows to set up a reasonable variational formulation of the above equation in dependency of the particular boundary condition (Dirichlet, Robin, Neumann) considered, Hilbert space theory and fixed-point approaches are utilized to develop sufficient conditions for the existence of variational solutions. This theory is then applied to two specific realizations of $\mathcal{L}$ of interest before a weak maximum principle is established which is finally used to study overlapping domain decomposition methods for the nonlocal and homogeneous Dirichlet problem