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In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function u of two variables x,y is of Monge–Ampère type if it is linear in the determinant of the Hessian matrix of u and in the second-order partial derivatives of u.
Pogorelov's method of a priori estimates was used by S.-T. Yau to obtain a priori estimates for solutions of complex Monge-Ampere equations. This was the main step in the proof of the existence of Calabi-Yau manifolds, which play an important role in theoretical physics. A Monge-Ampère equation has the form
Name Dim Equation Applications Landau–Lifshitz model: 1+n = + Magnetic field in solids Lin–Tsien equation: 1+2 + = Liouville equation: any + = Liouville–Bratu–Gelfand equation
He studied at Karlsruhe (1949–51) and Göttingen (1951–54) where he received his doctorate in 1954 under Franz Rellich, with a thesis on the Monge-Ampere equation.From 1954-1958 he was at the Max Planck Institute for Physics and Astrophysics at Göttingen, with an interim stay at New York University (1956–57).
Calabi transformed the Calabi conjecture into a non-linear partial differential equation of complex Monge–Ampère type, and showed that this equation has at most one solution, thus establishing the uniqueness of the required Kähler metric. Yau proved the Calabi conjecture by constructing a solution of this equation using the continuity ...
For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's solution of the Calabi conjecture was the proof of existence for a Monge–Ampere equation. The open problem of existence (and smoothness) of solutions to the Navier–Stokes equations is one of the seven Millennium Prize problems in ...
This partial differential equation is similar to a real Monge–Ampere equation, and is known as a complex Monge–Ampere equation, and subsequently can be studied using tools from convex analysis. Its behaviour is highly sensitive to the sign of the topological constant λ = − 1 , 0 , 1 {\displaystyle \lambda =-1,0,1} .
By the 1970s, higher-dimensional understanding of the Monge–Ampère equation was still lacking. In 1976, Shiu-Yuen Cheng and Yau resolved the Minkowski problem in general dimensions via the method of continuity, making use of fully geometric estimates instead of the theory of the Monge–Ampère equation. [66]