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Mean Field Equations and Inverse Wave Problems

平均场方程和反波问题

基本信息

批准号：
1953620
负责人：
Amir Moradifam
金额：
$ 19.89万
依托单位：
University of California-Riverside
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953620&HistoricalAwards=false
关键词：
Mean Field Equations Inverse Wave

项目摘要

This project focuses on the study of mean field equations and inverse wave equations, two important types of partial differential equations. Mean field equations arise in the study of several important phenomena in mathematics, mathematical physics, and biology such as Chern-Simons gauge field theories, rigidity of Hawking mass in general relativity, and mathematical models for cell mobility and turbulence. Inverse wave problems investigated in this project have immediate applications in photoacoustic and thermoacoustic tomography and will contribute to the advancement of new medical imaging methods with significant potential in clinical applications. Medical imaging methods are crucial for early detection, diagnosis, and treatment of diseases. This project provides research training opportunities for graduate students. Undergraduate students, and high school students from underrepresented groups and disadvantaged backgrounds will benefit from the proposed research, educational, and outreach plans. The principal investigator (PI) aims to study symmetry and uniqueness of solutions of mean field type equations by developing new functional inequalities in the spirit of the Sphere Covering Inequality. The Sphere Covering Inequality was recently discovered by the PI and his collaborator and has since led to solutions for several open problems. The PI will develop singular versions of the sphere covering inequality as well as inequalities with improved constants on non-simply connected regions, and extend such inequalities to higher dimensions. The project has important applications to various problems including Nirenberg's problem of prescribing Gaussian curvature on the sphere, Moser-Trudinger inequalities, self-dual and Chern-Simons gauge field theories, rigidity of Hawking mass in general relativity, Keller-Segal type free boundary models for cell mobility, Onsager's vortex theory, Toda systems, and cosmic string equations. In the second part of the project, the PI will investigate the inverse problem of determining both the sources of a wave and its speed inside a medium from the measurements of the solutions of the wave equation on the boundary, which is a long-standing open problem and has important applications in medical imaging.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

本计画主要研究平均场方程与逆波方程这两类重要的偏微分方程。平均场方程出现在数学、数学物理和生物学中的几个重要现象的研究中，如陈-西蒙斯规范场论、广义相对论中霍金质量的刚性以及细胞流动和湍流的数学模型。逆波问题的研究在这个项目中有直接的应用在光声和热声层析成像，并将有助于新的医学成像方法的进步，具有显着的潜力，在临床应用。医学成像方法对于疾病的早期检测、诊断和治疗至关重要。该项目为研究生提供了研究培训机会。来自代表性不足群体和弱势背景的本科生和高中生将从拟议的研究、教育和推广计划中受益。主要研究员（PI）的目的是研究对称性和唯一性的平均场型方程的解决方案，通过开发新的功能不等式的精神，球覆盖不等式。球面覆盖不等式是最近由PI和他的合作者发现的，并且已经导致了几个开放问题的解决方案。PI将开发奇异版本的球覆盖不等式以及非单连通区域上的改进常数不等式，并将这些不等式扩展到更高的维度。该项目有重要的应用到各种问题，包括尼伦伯格的问题规定高斯曲率的领域，莫泽Trudinger不等式，自对偶和陈-西蒙斯规范场理论，刚性霍金质量在广义相对论，凯勒-西格尔型自由边界模型的细胞流动性，昂萨格的涡旋理论，户田系统，和宇宙弦方程。在该项目的第二部分，PI将研究通过测量边界上波动方程的解来确定介质内的波源及其速度的逆问题，这是一个很长的-该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准。