Speaker: Dr. James Scott (Columbia University)

Title: Nonlocal Boundary Value Problems with Rough Data

Abstract: We describe and analyze nonlocal equations with classical local boundary conditions. The interaction kernel of the nonlocal operator has horizon parameter dependent on position in the domain, and vanishes as the boundary of the domain is approached. This heterogeneous localization allows for boundary values to be captured in the trace sense. We state and prove a nonlocal Green's identity for these nonlocal operators that involve local boundary terms. We use this identity to state and establish the well-posedness of variational formulations of the nonlocal problems with several types of classical boundary conditions. The consistency of these nonlocal boundary-value problems with their classical local counterparts in the vanishing horizon limit is demonstrated via the convergence of solutions. The Poisson data for the local boundary-value problem is permitted to be quite irregular, belonging to the dual of the classical Sobolev space.