A variable diffusivity fractional Laplacian

In this paper we analyze the existence, uniqueness and regularity of the solution to the generalized, variable diffusivity, fractional Laplace equation on the unit disk in R^2. For the order of the differential operator, our results show that for the symmetric, positive definite, diffusivity matrix, K (x), satisfying ₌ v^T v v^T K (x) v ₌ v^T v, for all v R^2, x, with ₌ < (2 +) (2 -) ₌, the problem has a unique solution. The regularity of the solution is given in an appropriately weighted Sobolev space in terms of the regularity of the right hand side function and K (x).