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The most complete results so far have been obtained when the equation is elliptic. Monge–Ampère equations frequently arise in differential geometry, for example, in the Weyl and Minkowski problems in differential geometry of surfaces. They were first studied by Gaspard Monge in 1784 [1] and later by André-Marie Ampère in 1820. [2]
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
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 ...
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 ...
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
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 ...
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
As a consequence of their resolution of the Minkowski problem, Cheng and Yau were able to make progress on the understanding of the Monge–Ampère equation. [CY77a] The key observation is that the Legendre transform of a solution of the Monge–Ampère equation has its graph's Gaussian curvature prescribed by a simple formula depending on the ...