Analytical and geometrical problems in calculus of variations and partial differential equations

变分法和偏微分方程中的分析和几何问题

• 批准号: 0969962
• 负责人: Alessio Figalli
• 金额: $ 16.8万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0969962&HistoricalAwards=false
• 关键词: Analytical; geometrical; problems; calculus; variations

The research of the principal investigator will be focused on different mathematical areas. First of all, he plans to work on optimal transport theory. This problem, which consists in finding the cheapest way to transport a distribution of mass from one place to another, has recently found many applications in meteorology, biology and populations dynamics, or for instance to study the antenna design problem. The investigator has already worked on this subject for several years. He also intends to study Euler equations for incompressible fluids and to work on problems of semiclassical limit coming from quantum physics. Finally, he wishes to apply some of his recent results on some "improved" version of the classical isoperimetric inequalities to study the stability of the shapes of crystals under the action of an external potential.The research described above has applications in many different areas: the optimal transport problem has obvious application to economics, but it has also shown to be very interdisciplinary, with links with other areas of mathematics like geometry, probability and partial differential equations, and also with physics and biology. Euler equations and semiclassical limits are classical problem in physics and quantum physics, and their mathematical study may increase deeper understanding of the physical phenomena themselves. Finally, the study of the rigidity of crystals under exterior potential should help to explain many phenomena which are currently observed in experiments but not yet completely understood.

首席研究员的研究将集中在不同的数学领域。首先，他计划研究最佳运输理论。这个问题，其中包括寻找最便宜的方式运输的质量分布从一个地方到另一个地方，最近发现了许多应用在气象学，生物学和人口动力学，或例如研究天线设计问题。研究人员已经研究这个问题好几年了。他还打算研究欧拉方程的不可压缩流体和工作的问题，半经典极限来自量子物理。最后，他希望应用他最近的一些结果对一些“改进”版本的经典等周不等式，以研究稳定性的形状下的行动，一个外部potential.The研究上述应用在许多不同的领域：最优运输问题在经济学中有明显的应用，但它也显示出非常跨学科的性质，与其他数学领域的联系，如几何，概率和偏微分方程，也与物理学和生物学。欧拉方程和半经典极限是物理学和量子物理学中的经典问题，对它们的数学研究可以加深对物理现象本身的理解。最后，研究晶体在外力作用下的刚性，有助于解释许多目前在实验中观察到但尚未完全理解的现象。