Singularity Behavior in Some Geometric Variational Problems

这个项目位于几何变分领域，研究各种可能受约束的最优或静态函数、场、测量或几何结构的奇点行为和能量集中。第一类具体的项目包括与T.Riviere继续研究黎曼流形与其同伦类之间映射的p能量之间的关系。在非挠同伦的情况下，我们以前关于气泡的几何和拓扑结构的结果允许攻击关于Sobolev映射的弱光滑逼近的公开问题以及关于p平稳映射和热流中的气泡的精化问题。我们还攻击高阶Soblev空间，它对于某些同伦类似乎更自然，但其基本逼近结果和构造以前从未被研究过，第二类项目涉及与Thierry de Pauw继续合作，推广几何测度论中的概念，并在具有一般群系数的度量空间中的链的背景下解决Platform型问题。我们考虑了各种质量型泛函和扫描的概念，它推广了Ambrosio-Kirchheim的有限质量度规空间流和White的可直且平坦的欧几里德空间G链。在欧氏空间中，我们逼近了Almgren的尺寸泛函，并证明了这种逼近泛函的极小点的最优正则性。在纯数学和应用数学中，许多变分问题的解常常被迫具有奇点，即涉及发生大振荡的区域。例如，球形容器中的向列相液晶材料，其光轴被迫指向容器外，其内部必然会有奇点(通过交叉偏振器或x射线衍射可以观察到)。在该示例中，光轴具有能量密度，该能量密度测量其局部变化率，并且其积分在所有可能的构型中倾向于具有最小值。我们的研究建议理解这类变分问题中的能量之间的关系，以及这些问题的物理所施加的拓扑障碍。我们已经得到了新的概念，这些概念允许处理和精确描述从肥皂膜及其高维推广到各种复杂介质中的最佳传输路径的各种问题。在许多物理问题中自然出现的几何约束导致了新的数学和计算问题。具体地说，我们正在研究的两个约束是在植物结构中表现出的恒定横截面积约束和在某些晶体材料的微结构形成中的梯度约束。

This project lies in the area of geometric variational calculus, treating the behavior of singularities and energy concentration for various optimal or stationary functions, fields, measures, or geometric structures, possibly subject to constraints. The first specific class of projects involves continuing work with T. Riviere, on relations between the p energy of a map between Riemannian manifolds and its homotopy class. In case of nontorsion homotopy, our previous results on the geometric and topological structure of bubbles allows attack on open questions about the weak smooth approximability of Sobolev maps as well as refined questions about bubbling in p stationary maps and heat flows. We also are attacking higher order Sobolev spaces which seem more natural for certain homotopy classes, but for which basic approximation results and constructions have not been previously studied, A second class of projects involves continuing work with Thierry De Pauw on extending notions from geometric measure theory and solutions of Plateau-type problems in the context of chains in a metric space with coefficients in a general group. We consider a variety of mass-type functionals and the notion of a scan which generalizes the finite mass metric-space currents of Ambrosio-Kirchheim and the rectifiable and flat Euclidean-space G-chains of White. In the Euclidean space context we approximate the size functional of Almgren and prove optimal regularity for the minimizers of such approximate functionals. Solutions to many variational problems in both pure and applied mathematics often are forced to have singularities, that is, to involve regions where large oscillations occur. For example a nematic liquid crystal material in a spherical container whose optical axis is forced to point outward on the container necessarily will have singularities inside (observable through cross-polarizers or x-ray diffraction). In this example the optical axis has an energy density, which measures its local rate of change and whose integral tends to have a minimum value among all possible configurations. Our research proposes to understand the relationship between energies in such variational problems and the topological barriers imposed by the physics of these problems. We have derived new notions which allow the treatment and precise description of a wide variety of problems from soap films and their higher dimensional generalizations to optimal transport paths in various complex media. Geometric constraints which occur naturally in many physical problems have led to new mathematical and computational issues. In particular, two that we are studying are the constant cross-sectional area constraint exhibited in plant structure and gradient constraints in the microstructure formation in certain crystalline materials.