### Abstract

This article is concerned with interface problems for Lipschitz mappings f_{+} : ℝ_{+}^{n} → ℝ^{m} and f_{-} : ℝ_{-}^{n} → ℝ^{m} in the half spaces, which agree on the common boundary ℝ^{n-1} = ∂_{+}^{n} = ∂ℝ_{-}^{n}. These naturally occur in mathematical models for material microstructures and crystals. The task is to determine the relationship between the sets of values of the differentials Df_{+} and Df_{-}. For some time it has been thought that the polyconvex hulls [Df_{+}]^{pc} and [Df_{-}]^{pc} satisfy Hadamard's jump condition or are at least rank-one connected. Our examples here refute this idea. The estimates of the Jacobians we obtain in the course of solving the so-called Monge-Ampère inequalities seem also to be of independent interest. As an application, we construct uniformly elliptic systems of first order partial differential equations in the same homotopy class as the familiar Cauchy-Riemann equations, for which the unique continuation property fails.

Original language | English (US) |
---|---|

Pages (from-to) | 125-169 |

Number of pages | 45 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 163 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2002 |

### ASJC Scopus subject areas

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering