The geometric Cauchy problem for the hyperbolic Hessian one equation. (September 2015)

• Record Type: Journal Article
• Authors: Martínez, Antonio; Milán, Francisco
• Is Part Of: Nonlinear analysis. Volume 125(2015)
• Page Start: 323
• Page End: 333
• Publication Date: 2015-09
• Subjects: 53A15; Cauchy problem -- Hyperbolic Hessian one equation -- Improper affine spheres; 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.2015.05.021
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We solve the problem of finding all indefinite improper affine spheres passing through a given regular curve of R 3 with a prescribed affine co-normal vector field along this curve. We prove the problem is well-posed when the initial data are non-characteristic and show that uniqueness of the solution can fail at characteristic directions. As application we classify the indefinite improper affine spheres admitting a geodesic planar curve.