Regularity results for segregated configurations involving fractional Laplacian. (April 2020)

• Record Type: Journal Article
• Authors: Tortone, Giorgio; Zilio, Alessandro
• Is Part Of: Nonlinear analysis. Volume 193(2020)
• Publication Date: 2020-04
• Subjects: Free-boundary problem -- Optimal regularity -- Nonlocal diffusion -- Monotonicity formulas -- Segregation problems -- Variational methods; 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.2019.05.013
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the regularity of segregated profiles arising from competition–diffusion models, where the diffusion process is of nonlocal type and is driven by the fractional Laplacian of power s ∈ ( 0, 1 ) . Among others, our results apply to the regularity of the densities of an optimal partition problem involving the eigenvalues of the fractional Laplacian. More precisely, we show C 0, α ∗ regularity of the density, where the exponent α ∗ is explicit and is given by α ∗ = s for s ∈ ( 0, 1 ∕ 2 ] 2 s − 1 for s ∈ ( 1 ∕ 2, 1 ) . Under some additional assumptions, we then show that solutions are C 0, s . These results are optimal in the class of Hölder continuous functions. Thus, we find a complete correspondence with known results in case of the standard Laplacian.