Fractional Diffusion in the full space: decay and regularity

Abstract

We consider fractional partial differential equations posed on the full space R d . Using the well-known Caffarelli-Silvestre extension to R d × R + as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on R d × (0, Y) converge to the solution of the original problem as Y → ∞. Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of numerical methods for the full space problem, such as FEM-BEM coupling techniques.