### 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} ⊂ M^{2} _{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 ∇^{2}u = −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 language | English |
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Pages (from-to) | 609-623 |

Number of pages | 15 |

Journal | Annali della Scuola Normale Superiore di Pisa - Classe di Scienze |

Volume | 3 |

Issue number | 3 |

Publication status | Published - 2004 Jan 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Mathematics (miscellaneous)