A generalized fractional Laplacian

Abstract

In this article we show that the fractional Laplacian in $R^{2}$ can be factored into a product of the divergence operator, a Riesz potential operator, and the gradient operator. Using this factored form we introduce a generalization of the fractional Laplacian, involving a matrix $K(x)$, suitable when the fractional Laplacian is applied in a non homogeneous medium. For the case of $K(x)$ a constant, symmetric positive definite matrix we show that the fractional Poisson equation is well posed, and determine the regularity of the solution in terms of the regularity of the right hand side function.