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 的这个数学研究项目重点关注变分法和偏微分方程中的几个问题。其中包括最优传输问题、函数不等式的稳定性问题以及 Mumford-Shah 函数。最优运输问题包括找到将质量分布从一个地方运输到另一个地方的最便宜的方式。它除了是变分法中的一个自然问题之外，还与偏微分方程、黎曼几何和概率有关。函数不等式的稳定性问题包括理解某些不等式的最小化是否在某种适当的意义上是稳定的。为了理解和/或预测物理现象随时间的演变，这是一个重要的问题。例如，定量稳定性结果用于量化给定物理系统收敛到其稳态的速率，也可用于了解系统在外部因素（例如外部因素）的影响下发生变化的程度。力）。 Mumford-Shah 函数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（对应于对象的边缘、阴影和重叠对象）。 Mumford-Shah 泛函极小值的正则性质还远未被理解，而理解界面的光滑性及其拓扑性质是一个重要且具有挑战性的问题。Alessio Figalli 在这个数学研究项目中研究的所有问题都具有重要意义其他科学领域的应用。例如，最优运输问题是经济学中的一个基本问题，进一步应用于气象学、生物学和人口动态学；本项目中研究的 Mumford-Shah 泛函可应用于图像处理（它允许从模糊图像中提取出好的图像）。 本项目的部分问题将用于本科生、研究生和博士后的培养。菲加利的几名博士生和博士后研究员将从事这些领域的研究，所获得的成果将通过发表研究论文和讲义以及开发课程和研讨会来广泛传播。