Rigidity for the hyperbolic Monge-Ampère equation

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Abstract

Some properties of nonlinear partial differential equations are naturally associated with the geometry of sets in the space of matrices. In this paper we consider the model case when the compact set K is contained in the hyperboloid H−1, where H−1 ⊂ M2 sym ×2, the set of symmetric 2 × 2 matrices. The hyperboloid H−1 is generated by two families of rank-one lines and related to the hyperbolic Monge-Ampère equation det ∇2u = −1. For some compact subsets K ⊂ H−1 containing a rank-one connection, we show the rigidity property of K by imposing proper topology in the convergence of approximate solutions and affine boundary conditions.

Original languageEnglish
Pages (from-to)609-623
Number of pages15
JournalAnnali della Scuola Normale Superiore di Pisa - Classe di Scienze
Volume3
Issue number3
Publication statusPublished - 2004 Jan 1
Externally publishedYes

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Mathematics (miscellaneous)

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