On a Generalization of Sobolev-Slobodecki Spaces and an Associated Fractional Derivative

Abstract

In this article, we present a new kind of fractional derivative D and fractional integral I, which are related to a generalization W s,q(Ω; ∂Ω) of Sobolev-Slobodecki spaces and satisfy a kind of fractional divergence theorem ∫∂Ω(Iv)(y) dS (y) = ∫Ω(Dν)(x) dx for functions ν∈ W s,q(Ω; ∂Ω). Further, we show how this space occurs in a discussion of linear elliptic equations with singular Neumann boundary data.