Analysis and applications of nonlinear problems with lack of compactness

缺乏紧性的非线性问题的分析与应用

• 批准号: RGPIN-2022-04213
• 负责人: Vétois, Jérôme
• 金额: $ 1.53万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759159
• 关键词: Analysis; applications; nonlinear; problems; lack

This research program is concerned with the study of nonlinear partial differential equations and systems connected to various problems in Nonlinear Analysis, Riemannian Geometry and Mathematical Physics. The main issue that we propose to investigate comes from potential lack of compactness of solutions (or approximating solutions) of the equations in energy spaces. The lack of compactness usually generates concentration phenomena, which take the form of families of spike solutions (also called blowing-up solutions). These concentration phenomena can be seen for example as a lack of stability when looking at problems from Mathematical Physics. These phenomena arise in several important equations in pure and applied Mathematics. The equations that we will consider in this program include, among others, curvature prescription equations in Conformal Geometry, nonlinear Schrödinger-type equations and systems in Mathematical Physics (more specifically, in nonlinear optics and the Hartree-Fock theory for Bose-Einstein condensates) and Euler-Lagrange equations of extremal functions to Sobolev-type inequalities in Nonlinear Analysis. In all these problems, analyzing the potential lack of compactness of solutions or approximating solutions will be fundamental. In some cases, we will be able to prove that all families of solutions are compact, namely that there does not exist any families of blowing-up solutions. This usually requires performing a very fine pointwise analysis of solutions. In other cases, on the contrary, we will be able to find existence of families of blowing-up solutions. In such cases, several other questions emerge about the profiles of such solutions and the locations of the points where they concentrate. These questions are usually answered by using pointwise analysis on the one hand and constructive methods based on the so-called Lyapunov-Schmidt reduction method on the other hand. These different cases, questions and methods will all be investigated in this program. We will also explore questions of existence and multiplicity of solutions by using variational methods, where, here again, the analysis of compactness will play a crucial role. In summary, this program will aim to impact the many aspects of problems with lack of compactness, by investigating equations and systems from different areas of pure and applied mathematics and exploring new types of solutions (such as for example sign-changing and high-energy solutions as discussed in Section 1 of my proposal), systems with complex structures (see Section 2), and equations with more complex operators (such as the higher-order operators in Section 3 and the quasilinear operators in Section 4). Each area of the program will be crucially supported by the recruitment of several HQP (postdoctoral fellows and students of every level).

该研究项目涉及非线性偏微分方程和与非线性分析、黎曼几何和数学物理中的各种问题相关的系统。我们建议研究的主要问题来自解（或近似解）的潜在缺乏紧性。能量空间中方程的缺乏通常会产生集中现象，其形式为尖峰解族（也称为爆炸解），例如，当观察时，这些集中现象可以被视为缺乏稳定性。这些现象出现在纯数学和应用数学中的几个重要方程中，我们将在本课程中考虑的方程包括共形几何中的曲率规定方程、数学物理中的非线性薛定谔方程和系统。 （更具体地说，在非线性光学和玻色-爱因斯坦凝聚体的哈特里-福克理论中）和极值函数的欧拉-拉格朗日方程非线性分析中的索博列夫型不等式在所有这些问题中，分析解或近似解的潜在缺乏紧性将是基础，在某些情况下，我们将能够证明所有解族都是紧的，即确实存在。不存在任何爆炸解族，这通常需要对解进行非常精细的逐点分析，相反，在这种情况下，我们将能够找到多个爆炸解族。其他问题也随之出现这些问题通常一方面使用逐点分析，另一方面使用基于所谓的 Lyapunov-Schmidt 约简方法的构造方法来回答。 ，问题和方法都将在这个程序中进行研究，我们还将使用变分方法探索解的存在性和多重性问题，其中，紧性分析将再次发挥至关重要的作用。 总之，本程序的目标是。影响问题的许多方面由于缺乏紧凑性，通过方程研究和来自纯数学和应用数学不同领域的系统，并探索新型解决方案（例如我的提案第 1 节中讨论的符号改变和高能解决方案），具有复杂的系统结构（参见第 2 节），以及具有更复杂算子的方程（例如第 3 节中的高阶算子和第 4 节中的拟线性算子）。该计划的每个领域都将得到几个 HQP 的招募的关键支持。 （博士后研究员和学生每个级别）。