Mathematical Sciences: Nonlinear Elliptic and Parabolic Problems from Physical Models in Several Space Dimensions

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WPC 2 B P Z Courier 12cpi 3| d d 6 X @ 8 ; @ HP LaserJet IIID - BACK HPIIIDB.PRS d 6 X @ 8 ; , \ , j 5{ @ 2 # X _ F ` Courier 12cpi Courier 12cpi (Bold) . s 4 d d d , d 6 X @ 8 ; @ r 5 d d d , d ` J ; ies of materials which are magnetically saturated. Certain configurations of the m tic field have been observed experimentally, and we wish to investigate whether st s 4 This project comprises three components uniqueness of minimizers for polyconvex energy functionals that arise in nonlinear elasticity. Criteria will be develop to distinguish those deformations that are diffeomorphisms; ii) stability and interaction of vortices (or defects) in time-dependent Ginzburg-Landau or Landau- Stuart models describing superfluids or superconductors. Various conjectures by physicists and applied mathematicians will be examined; iii) minimization problems concerning ferro-magnetic materials. The goal is to determine whether configurations that have been experimentally observed occur as limits of minimizers to the corresponding magnetic energy functionals. The above project concerns three mathematical models formulated by physicists and material scientists in order to describe the behavior of certain materials. The goal of this proposal is to determine rigorously whether solutions to the given mathematical models do, indeed, exhibit the expected behavior in each case. The first problem concerns deformations of certain hyper-elastic materials in two or three space dimensions. The presence or absence of singularities (such as sharp edges or interfaces) will be examined. The second problem concerns defects in a superconducting material. It is proposed to determine how such defects interact and stabilize in time. In particular, conjectures by physicists about when defects split apart or disappear will be investigated. The third problem concerns properties of materials which are magnetically saturated. Certain configurations of the magnetic field have been observed experimentally, and we wish to investigate whether stable solutions to the mathematical model exhibit these configurations.