Viscosity solutions to quaternionic Monge–Ampère equations. (July 2016)

• Record Type: Journal Article
• Authors: Wan, Dongrui; Wang, Wei
• Is Part Of: Nonlinear analysis. Volume 140(2016)
• Page Start: 69
• Page End: 81
• Publication Date: 2016-07
• Subjects: Viscosity solutions -- Quaternionic Monge–Ampère equations -- Comparison principle; 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.2016.03.011
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Quaternionic Monge–Ampère equations have recently been studied intensively using methods from pluripotential theory. We present an alternative approach by using the viscosity methods. We study the viscosity solutions to the Dirichlet problem for quaternionic Monge–Ampère equations d e t ( f ) = F ( q, f ) with boundary value f = g on ∂ Ω . Here Ω is a bounded domain on the quaternionic space H n, g ∈ C ( ∂ Ω ), and F ( q, t ) is a continuous function on Ω × R → R + which is non-decreasing in the second variable. We prove a viscosity comparison principle and a solvability theorem. Moreover, the equivalence between viscosity and pluripotential solutions is shown.