A variable diffusivity fractional Laplacian

TL;DR

This paper investigates the mathematical properties of a variable diffusivity fractional Laplacian equation on the unit disk, establishing conditions for existence, uniqueness, and regularity of solutions based on the operator's order and diffusivity matrix.

Contribution

It provides new theoretical results on the existence, uniqueness, and regularity of solutions to variable diffusivity fractional Laplacian equations, extending understanding in this mathematical area.

Findings

Unique solutions exist under specific bounds on the diffusivity matrix.

Solution regularity depends on the regularity of the input function and diffusivity matrix.

Conditions on the diffusivity matrix ensure well-posedness of the problem.

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 $K(\mathbf{x})$.