Analysis and applications of nonlinear problems with lack of compactness

这项研究计划涉及研究与非线性分析、黎曼几何和数学物理中的各种问题有关的非线性偏微分方程组和系统。我们建议研究的主要问题来自于能量空间中方程的解(或近似解)的紧性的潜在缺乏。缺乏紧凑性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，这些集中的现象可以被视为在看数学物理的问题时缺乏稳定性。这些现象出现在纯数学和应用数学中的几个重要方程式中。我们将在本程序中考虑的方程包括，保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-爱因斯坦凝聚的哈特里-福克理论中)，以及非线性分析中的极值函数的欧拉-拉格朗日方程。在所有这些问题中，分析解的紧致性或近似解的潜在不足将是基本的。在某些情况下，我们将能够证明所有解族都是紧的，即不存在任何爆破解族。这通常需要对解决方案执行非常精细的逐点分析。相反，在其他情况下，我们将能够找到爆破解族的存在性。在这种情况下，出现了关于这种解决方案的概况及其集中点的位置的其他几个问题。这些问题的答案通常一方面是逐点分析，另一方面是基于Lyapunov-Schmidt约化方法的构造性方法。这些不同的案例、问题和方法都将在这个节目中进行调查。我们还将使用变分方法探索解的存在性和多解性问题，在这里，紧致性分析将发挥关键作用。总而言之，本程序旨在通过研究纯数学和应用数学的不同领域的方程和系统，并探索新类型的解(例如，如我的提案第1节所讨论的变号和高能解)、具有复杂结构的系统(见第2节)以及具有更复杂运算符的方程(如第3节中的高阶算子和第4节中的拟线性算子)，来影响缺乏紧致性的问题的许多方面。该计划的每个领域都将由几名HQP(博士后研究员和各级学生)的招聘提供关键支持。

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).