Singularity Behavior in Some Geometric Variational Problems

项目编号：dms -0072486项目负责人：Robert M. hardt本项目属于几何变分微积分领域，研究受几何或解析约束的各种最优或平稳函数、场或几何结构的奇点和能量集中行为。具体的研究集中在流形、铁磁和液晶材料之间的映射中的能量和拓扑障碍之间的关系、多项式变量的不适当切片、卡诺群中电流的紧致性以及松弛能量最小化的规律性。我们将研究p-能量最小化映射在接近一个临界功率时的奇点极限沿面积最小化集的能量集中。在各种高维情况下，光滑映射的极限能量集中可能出现在无限测度集上，并且与球面的同伦非平凡映射有关。在静止和动态环境下，我们也将研究奇点、磁场和液晶材料。我们将研究代数几何中多项式零集的反常相交理论，以了解其在分析和偏微分方程中的相关行为。本文将研究电流群的理论，并着眼于其在变分问题上的应用。许多物理现象的基础是一个最小能量原理，根据这个原理，某些结构、场或几何形状的特征是它们比竞争物体具有更少的能量或面积。外部约束往往导致奇异点，奇异点是在很小的空间区域内发生结构快速变化的特殊点。例如，在应力作用下固体中的位错缺陷，磁化材料中的畴壁，超导材料中的涡流，肥皂膜中的液体边缘和角落，以及各种液晶材料中的点、曲线和表面缺陷。我们处理解释和预测这些现象所必需的新的数学结构和理论。在这些问题中，纯数学的理论研究、应用数学的数值计算研究和物理学的现象学研究相互受益，都具有至关重要的科学作用。

AbstractAward: DMS-0072486Principal Investigator: Robert M. HardtThis project lies in the area of geometric variational calculus,treating the behavior of singularities and energy concentrationfor various optimal or stationary functions, fields, or geometricstructures subject to geometric or analytic constraints. Specificinvestigations focus on the relation between energy andtopological obstruction in mappings between manifolds,ferromagnetic and liquid crystal materials, improper slicing ofpolynomial varieties, compactness of currents in Carnot groups,and the regularity of relaxed energy minimizers. We willinvestigate the energy concentration along area-minimizing setsfor limits of singularities of p-energy minimizing maps as papproaches a critical power. In various higher dimensionalcases, energy concentration of limits of smooth mappings mayoccur along sets of infinite measure and is related tohomotopically nontrivial mappings of spheres. Singularities inferromagnetic and liquid crystal materials will be also studiedin both stationary and dynamic contexts. The theories of improperintersections of polynomial zero sets from algebraic geometrywill be investigated to understand related behavior in analysisand partial differential equations. A theory of currents inCarnot groups will be studied with an eye on applications tovariational problems.Underlying many physical phenomena is a least-energy principlewhereby certain configurations or fields or geometric shapes aredistinguished by their property of having less energy or areathan competing objects. The external constraints often lead tosingularities, which are special points characterized by rapidchanges of structure occurring in very small spatial regions. Forexample, one observes dislocation faults in solids under stress,domain walls in magnetized materials, vortices in superconductingmaterials, liquid edges and corners in soap films, and point,curve, and surface defects in various liquid crystalmaterials. We deal with new mathematical structures and theoriesnecessary to explain and predict such phenomena. In theseproblems, the theoretical studies of pure mathematics, thenumerical computational studies of applied mathematics, and thephenomenological studies from physics all benefit each other andall have a crucial scientific role.