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

WPC 2 B P Z Courier 12cpi 3|d d 6 X@8；@HP LaserJet IIID-Back HPIIPRS d 6 X@8；，\，j 5{@2#X_F`快递12cpi快递12cpi(粗体)。S 4d，d6X@8；@r 5d，d‘J；几种磁饱和材料。实验上观察到了m场的某些构型，我们希望研究这个项目是否包含了非线性弹性力学中出现的多凸能量泛函极小值的三个分量唯一性。标准将被用来区分那些是微分同形的变形；ii)描述超流或超导体的含时Ginzburg-Landau或Landau-Stuart模型中涡旋(或缺陷)的稳定性和相互作用。物理学家和应用数学家的各种猜想将被检验；iii)关于铁磁材料的最小化问题。目标是确定实验观察到的构型是否作为相应磁能泛函的极小化极限出现。上述项目涉及物理学家和材料科学家为描述某些材料的行为而建立的三个数学模型。这项建议的目标是严格地确定给定数学模型的解是否确实在每种情况下都表现出预期的行为。第一个问题涉及某些超弹性材料在二维或三维空间中的变形。将检查奇点的存在或不存在(例如尖锐的边缘或界面)。第二个问题与超导材料中的缺陷有关。建议确定这些缺陷如何相互作用并及时稳定下来。特别是，物理学家关于缺陷何时分裂或消失的猜测将被调查。第三个问题涉及磁饱和材料的性质。在实验上观察到了磁场的某些构型，我们希望研究数学模型的稳定解是否表现出这些构型。

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.