A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation. (April 2019)

• Record Type: Journal Article
• Authors: Ohnuma, Masaki; Sakaguchi, Shigeru
• Is Part Of: Nonlinear analysis. Volume 181(2019)
• Page Start: 180
• Page End: 188
• Publication Date: 2019-04
• Subjects: primary 35J93 -- secondary 35D40 35B50 35B51; Prescribed mean curvature equation -- Strong comparison principle -- Semicontinuous viscosity solution; 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.2018.11.010
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.