A variable diffusivity fractional Laplacian

Abstract: 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 $\mathbf{v} \in \mathbb{R}^{2}$, $\mathbf{x} \in \Omega$, 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 $K(\mathbf{x})$.