Explicit self-similar solutions for a class of zero mean curvature equation and minimal surface equation. (August 2020)

• Record Type: Journal Article
• Authors: Li, Hengyan; Yan, Weiping
• Is Part Of: Nonlinear analysis. Volume 197(2020)
• Publication Date: 2020-08
• Subjects: 35L05 -- 35J60 -- 74H35 -- 83C15; Mean curvature flow -- Blowup solution -- Exact 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.2020.111814
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we consider the existence of self-similar solution for a class of zero mean curvature equations including the Born–Infeld equation, the membrane equation and maximal surface equation. By Calabi's correspondence, this also gives a family of explicit self-similar solutions for the minimal surface equation. Those models arise in string theory and geometric minimal surfaces theory. Moreover, we construct a family of instanton metric obtained from new exact singular solutions for minimal surfaces by noticing the correspondence between minimal surfaces in the three dimensional Euclidean space and gravitational instantons possessing two killing vectors.