[2] Important results in the theory of Monge–Ampère equations have been obtained by Sergei Bernstein, Aleksei Pogorelov, Charles Fefferman, and Louis Nirenberg.
More recently, Alessio Figalli and Luis Caffarelli were recognized for their work on the regularity of the Monge–Ampère equation, with the former winning the Fields Medal in 2018 and the latter the Abel Prize in 2023.
[5] Suppose now that x is a variable with values in a domain in Rn, and that f(x,u,Du) is a positive function.
Accordingly, the operator L satisfies versions of the maximum principle, and in particular solutions to the Dirichlet problem are unique, provided they exist.
[6] Suppose that a real-valued function K is specified on a domain Ω in Rn, the problem of prescribed Gauss curvature seeks to identify a hypersurface of Rn+1 as a graph z = u(x) over x ∈ Ω so that at each point of the surface the Gauss curvature is given by K(x).