- I. Bakelman, Convex analysis and nonlinear geometric elliptic equations. Springer-Verlag, Germany (1994).
- G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271–283. [MathSciNet]
- K. Böhmer, On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46 (2008) 1212–1249. [CrossRef] [MathSciNet]
- L.A. Caffarelli and M. Milman, Eds., Monge Ampère equation: applications to geometry and optimization, Contemporary Mathematics 226. American Mathematical Society, Providence, USA (1999).
- L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (1984) 369–402. [CrossRef] [MathSciNet]
- M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. [CrossRef] [MathSciNet]
- E.J. Dean and R. Glowinski, An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions. Electron. Trans. Numer. Anal. 22 (2006) 71–96. [MathSciNet]
- E.J. Dean and R. Glowinski, Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1344–1386. [CrossRef] [MathSciNet]
- E.J. Dean and R. Glowinski, On the numerical solution of the elliptic Monge-Ampère equation in dimension two: a least-squares approach, in Partial differential equations, Comput. Methods Appl. Sci. 16, Springer, Dordrecht, The Netherlands (2008) 43–63.
- E.J. Dean, R. Glowinski and T.-W. Pan, Operator-splitting methods and applications to the direct numerical simulation of particulate flow and to the solution of the elliptic Monge-Ampère equation, in Control and boundary analysis, Lect. Notes Pure Appl. Math. 240, Chapman & Hall/CRC, Boca Raton, USA (2005) 1–27.
- L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, in Current developments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, USA (1999) 65–126.
- X. Feng and M. Neilan, Galerkin methods for the fully nonlinear Monge-Ampère equation. http://arxiv.org/abs/0712.1240v1 (2007).
- X. Feng and M. Neilan, Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal. 47 (2009) 1226–1250. [CrossRef] [MathSciNet]
- X. Feng and M. Neilan, Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations. J. Sci. Comput. 38 (2009) 74–98. [CrossRef] [MathSciNet]
- R. Glowinski, Numerical methods for fully nonlinear elliptic equations, in 6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures, R. Jeltsch and G. Wanner Eds. (2009) 155–192.
- R. Glowinski, E.J. Dean, G. Guidoboni, L.H. Juárez and T.-W. Pan, Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic Monge-Ampère equation. Japan J. Indust. Appl. Math. 25 (2008) 1–63. [CrossRef] [MathSciNet]
- C.E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications 44. Birkhäuser Boston Inc., Boston, USA (2001).
- G. Loeper and F. Rapetti, Numerical solution of the Monge-Ampère equation by a Newton's algorithm. C. R. Math. Acad. Sci. Paris 340 (2005) 319–324. [CrossRef] [MathSciNet]
- A.M. Oberman, Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44 (2006) 879–895. [CrossRef] [MathSciNet]
- A.M. Oberman, Computing the convex envelope using a nonlinear partial differential equation. Math. Models Methods Appl. Sci. 18 (2008) 759–780. [CrossRef] [MathSciNet]
- A.M. Oberman, Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete Contin. Dyn. Syst. Ser. B 10 (2008) 221–238. [CrossRef] [MathSciNet]
- V.I. Oliker and L.D. Prussner, On the numerical solution of the equation (∂2z/∂x2)(∂2z/∂y2) - (∂2z/∂x∂y)2 = f and its discretizations, I. Numer. Math. 54 (1988) 271–293. [CrossRef] [MathSciNet]
Volume 44, Number 4, July-August 2010
|Page(s)||737 - 758|
|Published online||23 February 2010|