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Non-Compact Problems for Elliptic Solutions of Monge-Ampere Equations

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Abstract

In this chapter we are concerned with elliptic generalized solutions of Monge-Ampere equations

$$ \det \left( {{u_{{ij}}}} \right) = f\left( {x,u,Du} \right) $$

in the entire n-dimensional Euclidean space En. Generalized elliptic solutions of equation (*) are convex or concave functions and their graphs are complete convex hypersurfaces in the space \( {E^{{n + 1}}} = {E^{n}} \times R \).These hypersurfaces project one-to-one on En. Clearly it is sufficient to investigate only convex generalized solutions of equation (*) and confine oneself only to nonnegative functionsf (x,u,p)for all xEn, uR, pRn. As we know, any convex generalized solution u(x) of equation (*) satisfies this equation almost everywhere in any compact subset of En and the set function

$$ w\left( {l,u,e} \right) = meas {X_{u}}\left( e \right), $$

generated by u(x), is absolutely continuous on the family of Borel subsets of En.

Keywords

  • Convex Function
  • Convex Cone
  • Normal Image
  • Borel Subset
  • Convex Polyhedron

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1994 Springer-Verlag Berlin Heidelberg

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Bakelman, I.J. (1994). Non-Compact Problems for 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_5

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  • DOI: https://doi.org/10.1007/978-3-642-69881-1_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-69883-5

  • Online ISBN: 978-3-642-69881-1

  • eBook Packages: Springer Book Archive