Classical solutions of singular Monge–Ampère equations in a ball

• Published in: Journal of mathematical analysis and applications 2005-05, Vol.305 (1), p.240-252
• Main Authors: Goncalves, J.V.A., Santos, C.A.P.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Existence of solutions; Fixed points; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Partial differential equations; Radially symmetric solutions; Sciences and techniques of general use; Shooting method; Singular Monge–Ampère equations; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2004.11.019

Our concern is on existence, uniqueness and regularity of convex, negative, radially symmetric classical solutions to det ( D 2 u ) = ψ ( x , − u ) in B , u = 0 on ∂ B , where ( D 2 u ) is the Hessian of u, B ⊂ R N , N ⩾ 1 , is the unit ball with boundary ∂ B, ψ : B × ( 0 , ∞ ) → [ 0 , ∞ ) is continuous and ψ ( x , t ) = ψ ( | x | , t ) , where | x | is the euclidean norm of x. The main interest is in the case ψ is singular at | x | = 1 and/or u = 0 , although several nonsingular cases are covered by the main result. Our approach to show existence, exploits fixed point arguments and the shooting method. Uniqueness and regularity are achieved through suitable estimates.