On comparison of fractional Laplacians. (May 2022)

• Record Type: Journal Article
• Authors: Nazarov, Alexander I.
• Is Part Of: Nonlinear analysis. Volume 218(2022)
• Publication Date: 2022-05
• Subjects: Restricted fractional Laplacian -- Spectral fractional Laplacians -- Nonlocal differential operators; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.112790
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: For s > − 1, s ∉ N 0, we compare two natural types of fractional Laplacians ( − Δ ) s, namely, the restricted Dirichlet and the spectral Neumann ones. We show that the quadratic form of their difference taken on the space H ˜ s ( Ω ) is positive or negative depending on whether the integer part of s is even or odd. For s ∈ ( 0, 1 ) and convex domains we prove also that the difference of these operators is positivity preserving on H ˜ s ( Ω ) . This paper complements (Musina and Nazarov, 2014; 2016) where similar statements were proved for the spectral Dirichlet and the restricted Dirichlet fractional Laplacians.