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

TL;DR

This paper establishes a Lewy-Stampacchia inequality for the fractional Laplacian and applies it to prove well-posedness of anomalous unidirectional diffusion equations, extending previous results for the standard Laplacian.

Contribution

It introduces a Lewy-Stampacchia inequality for the fractional Laplacian and demonstrates its use in analyzing fractional diffusion equations, overcoming nonlocality challenges.

Findings

Proved a Lewy-Stampacchia inequality for the fractional Laplacian.

Established well-posedness of fractional anomalous diffusion equations.

Extended classical results to nonlocal fractional operators.

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.