Quantitative and Qualitative properties of solutions of partial differential equations

偏微分方程解的定量和定性性质

• 批准号: 1656845
• 负责人: Jiuyi Zhu
• 金额: $ 9.29万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1656845&HistoricalAwards=false
• 关键词: Quantitative; Qualitative; properties; solutions; partial

The proposal is concerned with the analysis and applications of nonlinear partial differential equations. The model problems in this proposal arise from the study of various nonlinear phenomena and other scientific disciplines, including condensed matter physics, elasticity, inverse problem, electrodynamics, quantum mechanics, fluid mechanics, mathematics biology, differential geometry, etc. The focus of the proposal research is the investigation of the quantitative and qualitative properties of solutions for partial differential equations. Providing quantitative and qualitative information for the solutions is fundamental and essential in the study of partial differential equations, which lies in the core of mathematical analysis. It is often the case that the most effective and economical way in scientific research is to explore properties of solutions and then to develop algorithm in accordance. Besides being very useful in applied science, the investigation of various kinds of structures and properties of solutions for various types of equations absolutely leads to new theories in mathematics.The proposed projects include quantitative uniqueness, eigenfunction and eigenvalue estimates, as well as Liouville-type theorems. Techniques and ideas from analysis area, such as elliptic estimates and Fourier analysis, will be combined and applied into this project. The proposed research should enhance the understanding of classical and Steklov eigenvalue problems, semilinear and higher order elliptic equations, wave equations, fractional Laplacians, fully nonlinear equations, etc. Further research will be devoted to the study of quantitative uniqueness of parabolic differential equations and other important equations from mathematical physics. Another related direction is the study of phase separations phenomenon in Bose-Einstein condensate. Emphasis will be placed on the two components Gross-Pitaevskki system. An important part of proposed research is on eigenfunction and eigenvalue estimates. Techniques and insights in the various areas cross-fertilize each other in a fruitful way in this area. The topics consist of measure of nodal sets (zero level sets), asymptotic behavior of eigenvalues, Lebesgue norm estimates, as well as doubling estimates of Steklov eigenfunctions and classical eigenfunctions. Much effort will be made towards Yau's conjecture asserting that the size of nodal sets is comparable to its frequency. The principal investigator will also continue the previous investigation on Liouville-type theorems on nonexistence of solutions for fractional Laplacian equations and fully nonlinear partial differential equations.

该提案涉及非线性偏微分方程的分析和应用。本提案中的模型问题源于对各种非线性现象和其他科学学科的研究，包括凝聚态物理、弹性、反问题、电动力学、量子力学、流体力学、数学生物学、微分几何等。本提案的重点研究是对偏微分方程解的定量和定性特性的研究。为解提供定量和定性信息是偏微分方程研究的基础和关键，它是数学分析的核心。通常情况下，科学研究中最有效和最经济的方法是探索解决方案的性质，然后开发相应的算法。除了在应用科学中非常有用之外，对各种类型方程解的各种结构和性质的研究绝对会带来新的数学理论。所提出的项目包括定量唯一性、本征函数和本征值估计以及刘维尔型定理。椭圆估计和傅里叶分析等分析领域的技术和思想将被结合并应用到该项目中。拟议的研究应增强对经典和Steklov特征值问题、半线性和高阶椭圆方程、波动方程、分数拉普拉斯、完全非线性方程等的理解。进一步的研究将致力于抛物型微分方程和其他方程的定量唯一性研究数学物理中的重要方程。 另一个相关方向是玻色-爱因斯坦凝聚相分离现象的研究。重点将放在 Gross-Pitaevskki 系统的两个组成部分上。 拟议研究的一个重要部分是特征函数和特征值估计。各个领域的技术和见解在该领域相互交叉，取得了丰硕的成果。主题包括节点集（零水平集）的测量、特征值的渐近行为、勒贝格范数估计以及 Steklov 特征函数和经典特征函数的加倍估计。 丘的猜想将得到很大的努力，该猜想断言节点集的大小与其频率相当。首席研究员还将继续之前对分数拉普拉斯方程和全非线性偏微分方程不存在解的刘维尔型定理的研究。