The planar Least Gradient problem in convex domains, the case of continuous datum. (January 2022)

• Record Type: Journal Article
• Authors: Rybka, Piotr; Sabra, Ahmad
• Is Part Of: Nonlinear analysis. Volume 214(2022)
• Publication Date: 2022-01
• Subjects: primary 49J10 -- secondary 49J52 49Q10 49Q20; Least gradient -- Trace solutions -- Convex but not strictly convex domains -- BV functions; 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.2021.112595
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the two dimensional least gradient problem in a convex polygonal set in the plane. We show existence of solutions when the boundary data are attained in the trace sense. Due to the lack of strict convexity, the classical results are not applicable. We state the admissibility conditions on the continuous boundary datum f that are sufficient for establishing an existence and uniqueness result. The solutions are constructed by a limiting process, which uses the well-known geometry of superlevel sets of least gradient functions.