The Lewy-Stampacchia Inequality for the Fractional Laplacian and Its Application to Anomalous Unidirectional Diffusion Equations

Abstract

In this paper, we consider a Lewy-Stampacchia-type inequality for the fractional Laplacian on a bounded domain in Euclidean space. Using this inequality, we can show the well-posedness of fractional-type anomalous unidirectional diffusion equations. This study is an extension of the work by Akagi-Kimura (2019) for the standard Laplacian. However, there exist several difficulties due to the nonlocal feature of the fractional Laplacian. We overcome those difficulties employing the Caffarelli-Silvestre extension of the fractional Laplacian.

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• Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains
• A comparative study on nonlocal diffusion operators related to the fractional Laplacian
• Sobolev spaces on non-Lipschitz subsets of ${\mathbb{R}}^n$ with application to boundary integral equations on fractal screens