On the extension problem for semiconcave functions with fractional modulus. (March 2022)

• Record Type: Journal Article
• Authors: Albano, Paolo; Basco, Vincenzo; Cannarsa, Piermarco
• Is Part Of: Nonlinear analysis. Volume 216(2022)
• Publication Date: 2022-03
• Subjects: 26A27 -- 26B25 -- 49J52 -- 49L20; Semiconcave functions -- Extension -- Approximation -- Singularities; 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.2021.112669
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Consider a locally Lipschitz function u on the closure of a possibly unbounded open subset Ω of R n with nonempty boundary. Suppose u is (locally) semiconcave on Ω ¯ with a fractional semiconcavity modulus. Is it possible to extend u in a neighborhood of any boundary point retaining the same semiconcavity modulus? We show that this is indeed the case and we give two applications of this extension property. First, we derive an approximation result for semiconcave functions on closed domains. Then, we use the above extension property to study the propagation of singularities of semiconcave functions at boundary points.