Fractional Diffusion in the full space: decay and regularity
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We consider fractional partial differential equations posed on the full space ^d. Using the well-known Caffarelli-Silvestre extension to ^d ×^+ as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on ^d × (0,) converge to the solution of the original problem as →∞. 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.
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