On the stability of the cut locus. (May 2016)

• Record Type: Journal Article
• Authors: Albano, Paolo
• Is Part Of: Nonlinear analysis. Volume 136(2016)
• Page Start: 51
• Page End: 61
• Publication Date: 2016-05
• Subjects: 35F21 -- 26B25 -- 49J52; Distance function -- Eikonal equation -- Cut locus -- Singular set; 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.2016.02.008
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In R d we consider a Riemannian metric, g, and an open bounded subset, Ω . We study the stability of the cut locus associated with Ω and g w.r.t. perturbations both of the set Ω and of the metric g . In order to have the stability of the cut locus, we assume C 2 regularity of the data, the metrics and the sets (in the case of sets with C 1, 1 boundaries, the cut locus may be unstable). We prove that to C 2 perturbations both of the set and of the metric correspond small changes of the cut locus w.r.t. the Hausdorff distance, i.e. the cut locus is stable in the C 2 category. We give some examples showing that C 1 perturbations may lead to large variations of the cut locus.