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

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