Abstract. In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend to many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.

In this note we consider regularity theory for a fractional $p$-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural analogue to the classical $p$-Laplacian situation, namely $C^{s+\alpha}_{loc}$-regularity for the homogeneous equation

In this paper, we prove L p estimates for the fractional derivatives of solutions to elliptic fractional partial differential equations whose coefficients are V M O. In particular, our work extends the optimal regularity known in the second order elliptic setting to a spectrum of fractional order elliptic equations. * armin.schikorra@unibas.ch, A.S. supported by SNF

No abstract