The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations

Pu-Zhao Kow; Masato Kimura

doi:10.3934/dcdsb.2021167

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2022, Volume 27, Issue 6: 2935-2957. Doi: 10.3934/dcdsb.2021167

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The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations

Pu-Zhao Kow^1, , and
Masato Kimura^2,

1.
Department of Mathematics, National Taiwan University, Taipei 106
2.
Faculty of Mathematics and Physics, Kanazawa University, Kakuma, Kanazawa 920-1192

^* Corresponding author: Pu-Zhao Kow
^* Corresponding author: Pu-Zhao Kow

Received: December 31, 2020

Revised: April 30, 2021

Published: June 2022

This work was partially supported by JSPS KAKENHI JP20H01812, JP20H00117, JP20KK0058, and MOST 108-2115-M-002-002-MY3

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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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Mathematics Subject Classification: 35R11, 35K61, 35K86.

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