On a new class of fractional partial differential equations
  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 many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way. 
Tien-Tsan Shieh, Daniel Spector
math.AP