Globally Existing Solutions to the Problem of Dirichlet for the Fractional 3<i>D</i> Poisson Equation

Abstract

A general approach to solving the Dirichlet problem, both for bounded 3D domains and for their unbounded complements, in terms of the fractional (3D) Poisson equation, is presented. Lauren Schwartz class solutions are sought for tempered distributions. The solutions found are represented by a formula that contains the volume Riesz potential and the one-layer potential, the latter depending on the boundary data. Infinite regularity of fractional harmonic functions, analogous to the infinite smoothness of the classical harmonic functions, is also proved in the respective domain, no matter what the boundary conditions are. Other properties of the solutions, that are presumably of interest to mathematical physics, are also investigated. In particular, an intrinsic decay property, valid far from the common boundary, is shown.

Keywords

• fractional laplacian
• Riesz potentials
• integral equations
• unbounded domains
• explicit solutions
• regularity