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 ${\mathbb{R}}^2$. For $\alpha$ the order of the differential operator, our results show that for the symmetric, positive definite, diffusivity matrix, $K(\mathbf{x})$, satisfying ${\lambda }_m{\mathbf{v}}^T\mathbf{v}\le {\mathbf{v}}^{T}K(\mathbf{x})\mathbf{v}\le {\lambda }_M{\mathbf{v}}^T\mathbf{v}$, for all $\mathbf{v}\in {\mathbb{R}}^2$, $\mathbf{x}\in \mathrm{\Omega }$, with ${\lambda }_M<\frac{\sqrt{\alpha (2+\alpha )}}{(2-\alpha )}{\lambda }_m$, 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 $K(\mathbf{x})$.