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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.
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 ...
The Monge cone at a given point (x 0, ..., x n) is the zero locus of the equation in the tangent space at the point. The Monge equation is unrelated to the (second-order) Monge–Ampère equation . References
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]
In 2023, he was awarded the Abel Prize "for his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the Monge–Ampère equation". [16] [17] [18]
"Made contributions in differential equations, also to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge–Ampère equations." [63] 1986 Berkeley, US Simon Donaldson: University of Oxford, UK Imperial College London, UK [66] Stony Brook University, US [67]
Monge's formulation of the optimal transportation problem can be ill-posed, because sometimes there is no satisfying () =: this happens, for example, when is a Dirac measure but is not. We can improve on this by adopting Kantorovich's formulation of the optimal transportation problem, which is to find a probability measure γ {\displaystyle ...
Gaspard Monge, Comte de Péluse (French pronunciation: [ɡaspaʁ mɔ̃ʒ kɔ̃t də pelyz]; 9 May 1746 [2] – 28 July 1818) [3] was a French mathematician, commonly presented as the inventor of descriptive geometry, [4] [5] (the mathematical basis of) technical drawing, and the father of differential geometry. [6]
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