FRG: Collaborative Research: Vectorial and geometric problems in the calculus of variations

这个项目专注于解决物理和工程中自然出现的问题的数学方法，例如，非线性弹性、相变和相互作用能量。除了是需要新思想和新方法的引人入胜的数学问题外，它们的研究对于加深对物理现象本身的理解也是非常重要的。例如，理解弄皱纸张的最佳方法是弹性力学中的一个自然问题，它在材料科学中有几个应用，而非局域相互作用能出现在物理和化学中，作为描述原子核能量的一种方式。我们的数学研究将为原子核稳定的能量范围提供新的线索。该项目还将对人力资源开发产生重要影响。我们提出了两国院校之间的人员交流制度，旨在为学生和博士后提供丰富的培训经验。我们建议协调我们的活动，并每年在私人投资机构的其中一所本土大学举行重点活动。我们还建议举办暑期班，旨在将小组成员(包括研究生和博士后)聚集在一起，以刺激科学进步，同时向更广泛的受众展示我们的研究成果，并帮助教育和吸引新一代研究人员进入这个令人兴奋的新兴数学挑战和思想领域。该项目的目标包括开发新的通用方法，在矢量环境和非局部相互作用能量的情况下研究变分问题。我们计划研究的问题包括：与褶皱有关的非线性弹性力学中的变分问题；拟凸能量泛函的梯度流的存在性；以及涉及表面张力和非局部能量的变分问题。我们集中于变分中的中心问题，例如证明能量泛函极小值的存在性，以及理解它们(可能的)唯一性和正则性。虽然在标量情况下已知很多，但在矢量设置(静态和演化情况下)和涉及非局部能量的几何问题中可用的结果很少。通过共同努力，将研究人员和合作者聚集在一起，为这项重点研究小组拨款，我们预计将出现一些通用方法，使我们能够获得新的实质性结果。除了纯粹的数学兴趣外，这个项目还将提高对这些模型试图反映的现象的理解。

This project focuses on mathematical methods for problems that appear naturally in physics and engineering, for example, non-linear elasticity, phase transitions and interaction energies. Apart from being fascinating mathematical problems that require new ideas and methods, their study is also of great importance for a deeper understanding of the physical phenomena themselves. For instance, understanding the best way to crumple a paper is a natural problem in elasticity which has several applications in material sciences, while nonlocal interaction energies appear in physics and chemistry as a way to describe the energy of nuclei. Our mathematical research will shed new light on the range of energies under which nuclei are stable. The project will also have important impact on human resource development. We propose a system of personnel exchanges between our home institutions, designed to provide a rich training experience to students and postdocs. We propose to coordinate our activities and to have an emphasis month every year in one of the home universities of the PIs. We also propose summer schools designed to bring members of the group together (including graduate students and postdocs) to stimulate scientific progress, while exposing the fruits of our research to a broader audience and helping to educate and attract a new generation of researchers to this exciting area of emerging mathematical challenges and ideas.The goals of this project include developing new general methods to study variational problems both in a vectorial setting and in the case of non-local interaction energies. The problems we plan to study include: variational problems in nonlinear elasticity related to crumpling; existence of gradient flows for quasiconvex energy functionals and variational problems involving surface tension and nonlocal energies. We focus on central issues in the calculus of variations, such as proving existence of minimizers of energy functionals and understanding their (possible) uniqueness and regularity. While much is known in the scalar case, very few results are available in the vectorial setting (both in the static and evolutionary cases) and for geometric problems involving non-local energies. Through a common effort, bringing together the investigators and collaborators for this Focused Research Group Grant, we expect some general methods to emerge, allowing us to obtain new substantial results. In addition to its purely mathematical interest, this project will improve the understanding of the phenomena that these models attempt to reflect.