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Variational Problems and Generalized Elliptic Solutions of Monge-Ampere Equations

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Abstract

There is a profound connection between n-dimensional variational problems and the Dirichlet problem for n-dimensional Monge-Ampere equations. The absolute minimum of these variational problems turns out to be a generalized solution of the corresponding Monge-Ampere equations. In this chapter we study explicitly the main variational problem connected with the Monge- Ampere equation

$$ \det \left( {{u_{{ij}}}} \right) = f\left( {{x_{1}},{x_{2}}, \ldots ,{x_{n}}} \right) $$
(7.1)

and also consider generalizations of it.

Keywords

  • Convex Function
  • Variational Problem
  • Convex Body
  • Absolute Minimum
  • Supporting Hyperplane

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  • DOI: 10.1007/978-3-642-69881-1_4
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© 1994 Springer-Verlag Berlin Heidelberg

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Bakelman, I.J. (1994). Variational Problems and Generalized Elliptic Solutions of Monge-Ampere Equations. In: Convex Analysis and Nonlinear Geometric Elliptic Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-69881-1_4

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  • DOI: https://doi.org/10.1007/978-3-642-69881-1_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-69883-5

  • Online ISBN: 978-3-642-69881-1

  • eBook Packages: Springer Book Archive