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Nonlinear elliptic theory and the Monge-Ampère equation. (English) Zbl 1040.35018

Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. I: Plenary lectures and ceremonies. Beijing: Higher Education Press (ISBN 7-04-008690-5/3 vol. set). 179-187 (2002).
In a nontechnical way, the author presents a variety of ideas (energy estimates, de Giorgi’s theorem, invariances, Liouville theorems) connecting the calculus of variations and the theory of nonlinear equations. As a model problem, he discusses the Monge-Ampère equation \(D^2 u = f(\cdot,u,\nabla u)\) and in particular its relation to the problem of optimal transportation. Some current problems are described briefly.
For the entire collection see [Zbl 1011.00026].

MSC:

35J15 Second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35J70 Degenerate elliptic equations
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