Quantitative and Qualitative properties of solutions of partial differential equations

本文讨论了非线性偏微分方程的分析与应用。本提案中的模型问题来自于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。本课题研究的重点是研究偏微分方程解的定性和定量性质。为解提供定量和定性信息是研究偏微分方程的基础和必要条件，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的性质，然后根据解的性质开发算法。除了在应用科学中非常有用外，对各种类型方程解的各种结构和性质的研究绝对会导致数学中的新理论。提出的项目包括定量唯一性，特征函数和特征值估计，以及liouville型定理。分析领域的技术和思想，如椭圆估计和傅立叶分析，将被结合并应用到这个项目中。建议的研究应加强对经典和Steklov特征值问题，半线性和高阶椭圆方程，波动方程，分数阶拉普拉斯方程，全非线性方程等的理解。进一步研究抛物型微分方程和其他重要数学物理方程的定量唯一性。另一个相关方向是研究玻色-爱因斯坦凝聚体中的相分离现象。重点将放在格罗斯-皮塔耶夫斯基系统的两个组成部分。本征函数和本征值估计是本征函数和本征值估计的重要组成部分。不同领域的技术和见解在这一领域中以富有成效的方式相互促进。主题包括节点集（零水平集）的度量，特征值的渐近行为，Lebesgue范数估计，以及Steklov特征函数和经典特征函数的加倍估计。我们将对Yau的猜想进行大量的研究，该猜想认为节点集的大小与其频率相当。主要研究者还将继续先前关于分数阶拉普拉斯方程和完全非线性偏微分方程解不存在性的liouville型定理的研究。

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.