Regularity and stability results in variational problems

规律性和稳定性导致变分问题

• 批准号: 1262411
• 负责人: Francesco Maggi
• 金额: $ 50.91万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1262411&HistoricalAwards=false
• 关键词: Regularity; stability; results; variational; problems

This mathematics research project by Alessio Figalli is focused on several problems in the calculus of variations and partial differential equations. These include the optimal transport problem, the issue of stability in functional inequalities, and the Mumford-Shah functional. The optimal transport problem consists of finding the least expensive way to transport a distribution of mass from one place to another. In addition to being a natural problem in the calculus of variations, it is also related to partial differential equations, Riemannian geometry, and probability. The issue of stability in functional inequalities consists of understanding whether a minimizer of some inequality is stable in some suitable sense. This is an important issue in order to understand and/or predict the evolution in time of a physical phenomenon. For instance, quantitative stability results are used to quantify the rate of convergence of a given physical system to its steady state, and they can also be used to understand the extent to which the system changes under the influence of external factors (for instance, external forces). The Mumford-Shah functional is a classical model in image segmentation which is used to extract from a blurry image the meaningful discontinuities (which correspond to edges of objects, shadows, and overlapping objects). The regularity properties of minimizers of the Mumford-Shah functional are still far from being understood, and understanding the smoothness of the interfaces and their topological properties is an important and challenging problem.All the problems investigated in this mathematics research project by Alessio Figalli have important applications in other areas of sciences. For instance, the optimal transport problem is a fundamental problem in economics, with further applications to meteorology, biology, and population dynamics; the Mumford-Shah functional, which is studied in this project, has applications to image processing (it allows to extract good images out of blurry ones). Some of the problems in this project will be used in the training of undergraduate students, graduate students and postdoctoral fellows. Several of Figalli's PhD students and postdoctoral fellows will engage in research in these areas, and the results obtained will be widely disseminated via the publication of research papers and lecture notes, as well as through the development of courses and seminars.

Alessio Figalli的该数学研究项目集中在变化和部分微分方程的计算中的几个问题上。这些包括最佳运输问题，功能不平等的稳定性问题以及芒福德 - 夏（Mumford-Shah）功能。最佳运输问题包括找到最便宜的方式将质量分布从一个地方传输到另一个地方。除了在变化的计算中是自然问题外，它还与部分微分方程，riemannian几何形状和概率有关。功能不平等的稳定性问题包括了解某些不平等的最小化在某种程度上是否稳定。这是一个重要的问题，以理解和/或预测物理现象的演变。例如，定量稳定性结果用于量化给定物理系统对其稳态的收敛速率，并且它们也可以用来了解系统在外部因素影响下（例如，外部力量）下发生变化的程度。 Mumford-Shah功能是图像分割中的经典模型，用于从模糊图像中提取有意义的不连续性（对应于对象，阴影和重叠对象的边缘）。 Mumford-Shah功能的最小化器的规律性尚未被理解，并且了解界面及其拓扑特性的平稳性是一个重要且具有挑战性的问题。Alessio Figalli在此数学研究项目中研究的所有问题在其他科学领域都具有重要的应用。例如，最佳运输问题是经济学中的一个基本问题，并在气象，生物学和人群动态中进一步应用。在该项目中研究的Mumford-Shah功能具有用于图像处理的应用（它允许从模糊的图像中提取良好的图像）。 该项目中的一些问题将用于培训本科生，研究生和博士后研究员。 Figalli的一些博士学位学生和博士后研究员将在这些领域进行研究，并且获得的结果将通过研究论文和讲义的发表以及课程和研讨会的开发来广泛传播。