For more information about this meeting, contact Leonid Berlyand, Mark Levi, Alexei Novikov.

Title: | Sufficient conditions for strong local minima in vectorial variational problems. |

Seminar: | Applied Analysis Seminar |

Speaker: | Yuri Grabovsky, Temple University |

Abstract: |

The vectorial variational problem refers to the variational functional
involving multiple integrals, where the unknown is a vector field. Such
problems arise naturally in the context of non-linear elasticity and modeling
of shape memory materials. The field theory of Weierstrass and Caratheodori
provides a method by which one can prove that a given solution of the
Euler-Lagrange equation is a strong local minimizer. At some point it became
clear that the field theory approach cannot possibly provide the vectorial
analog of the Weierstrass sufficiency theorem for vectorial variational
problems. We prove such analog by studying the effect of an arbitrary strong
variation on the value of the functional. The key tool is the recently
developed Decomposition Lemma, first proposed by Jan Kristensen (1994), that permits
us to split the variation into the purely strong and weak part. We show that
the two parts of the variation act on the functional independently
(orthogonality principle). Positivity of second variation ensures that the
weak part cannot decrease the functional, while the quasiconvexity conditions
(the vectorial analog of the Weierstrass convexity condition) ensure that the
strong part is unable to decrease the functional either. The latter part is
accomplished by means of the localization principle. The use of the three key components
of our analysis: the Decomposition Lemma, the orthogonality and localization
principles were inspired by the work of Fonseca, Mueller and Pedregal (1998). |

### Room Reservation Information

Room Number: | MB106 |

Date: | 03 / 02 / 2010 |

Time: | 04:00pm - 04:55pm |