CNA 2001 Summer School
ABSTRACT: Higher order variational theories are relevant in the study of elasto-plastic materials (with micro cracks) and image segmentation in computer vision (the Blake Zisserman model). Although several issues involving lower semicontinuity and relaxation can be recasted under the notion of A-quasiconvexity, there are still many questions for variational problems with higher derivatives that, being completely understood in the case of first order gradients, still defy the analysts. In order to illustrate some of the difficulties encountered, in the first lecture we will address lower semicontinuity (jointly with Giovanni Leoni, Jan Maly, and Roberto Paroni) and phase transitions (jointly with Sergio Conti and Giovanni Leoni) for multiple integrals involving second order derivatives. The second lecture will be devoted to the study of exact minima of the energy for large magnetic bodies with vanishing induced magnetic field (with Bernard Dacorogna), followed by the characterization of the integral representation of the relaxed energy in the absence of exchange energy (in collaboration with Giovanni Leoni).
Carnegie Mellon University
Department of Mathematical Sciences
Pittsburgh, PA 15213