Irene Fonseca, Department of Mathematical Sciences, Carnegie Mellon University
" Higher Order Variational Problems, II "
igher order variational problems appear often in the engineering literature and in connection with the so-called gradient theories of phase transitions within elasto-plastic regimes. The study of equilibria of micromagnetic materials asks for mastery of second order energies, and the Blake-Zisserman model for image segmentation in computer vision seats squarely among second-order free discontinuity models which may be recasted as higher order Griffiths' models for fracture mechanics. The energy functionals may include lower dimensional order terms to take into account interfacial energies and discontinuities of underlying fields. Here the role played by these terms will be neglected and the focus will be placed on the added difficulties inherent to the presence of derivatives of order two (or more). Talk I will address Lusin-type theorems, maximal functions techniques, and lowersemicontinuity results. Talk II will be on the Hessian theory of phase transitions, relevant to the study of higher order singular perturbations of nonconvex, multiple-well variational problems that may be found in gradient strain theories in plasticity, ferromagnetics, and other areas of materials science and engineering. This is work in collaboration with S. Conti, G. Leoni, J. Maly, and R. Paroni.
November 6, 2001