The Fractional Half Dirac Operator Over Sobolev Spaces

Abstract

Let a bounded and smooth domain in the n dimensional Euclidean space be given and consider a boundary value problem of the half order Dirac operator and its power, with f be the value of the half derivative of the solution in the domain and, g be the trace value of the solution over the boundary. We will show that the solution will be a generalized function u in the half order Hilbert space which is the orthogonal sum of the solution that emerged as a domain integral from the domain value of the fractional half Dirac operator and the one that is the boundary integrals that emerges from the boundary or trace value of the solution. We use two different symbols to designate orthogonal sums for subspaces and functions in orthogonal subspaces of the Sobolev space.