Mass transport, Geometric inequalities and partial differential systems

偏微分方程与系统是数学与基础科学、应用科学和工程中许多其他学科之间的主要桥梁。这个数学的中心领域被用来模拟物理现象，生物系统，量子材料和固态器件，仅举几个应用领域。我的研究计划集中于开发新颖的，最好是通用的，包罗万象的方法来处理这类系统的分析，包括进化方程。目标是将这些方法应用于不能适用于标准技术的问题，或者展示通用特性，这有助于理解和部署通常需要的硬分析。这些方法通常用于研究稳态或动态微分方程和系统解的存在性、多重性、规律性和其他定性性质。它们包括：***在线性和弯曲空间上的自对偶变分微积分，用于处理非欧拉-拉格朗日型方程，例如那些涉及非自伴随项或/和非线性算子的方程。***质量传递理论处理非线性弹性理论中仍然难以捉摸的变分问题。*** Monge-Kantorovich理论解耦某些偏微分系统，如De giorgi型系统作为描述多态玻色-爱因斯坦凝聚相分离的极限椭圆系统。****它的等变形式适用于经济学中的匹配问题，以及密度泛函理论，这是一种解决多体系统量子力学的成熟方法。****无紧致边界问题的无限维临界点理论。这通常需要对所涉及的线性算子进行精细的分析，例如二阶和四阶Hardy-Schrodinger算子，分数阶拉普拉斯算子，以及对它们的适当扰动。奇点既可以在内部，也可以在所考虑的域的边界上。后者可以是欧几里得的，也可以是弯曲的。****用于识别最佳常数和/或建立稳态和其他极端存在的各种功能和几何不等式的新方法，如卡法雷利-科恩-尼伦伯格不等式，Moser-Aubin-Onofri不等式和gagliardo -尼伦伯格不等式。****解决各种非线性特征值问题（自伴随或非自伴随）中的规则性，稳定性和关键维度问题的新见解，特别是那些处理双调和算子和分数阶拉普拉斯算子的问题。**

Partial Differential Equations and Systems provide major bridges between mathematics and many other disciplines in basic and applied sciences and engineering. This central field of Mathematics is used to model physical phenomena, biological systems, quantum materials and solid state devices, to name only a few areas of applications. My research program is focused on developing novel, preferably general and encompassing methods to deal with the analysis of classes of such systems, including evolution equations. The goal is to either apply these methods to problems that are not amenable to standard techniques, or to exhibit universal features, which could aid in the understanding and the deployment of the often-required hard analysis. The methods normally address questions of existence, multiplicity, regularity and other qualitative properties of solutions of stationary or dynamic differential equations and systems. They include:*** Self-dual variational calculus on linear as well as curved spaces, to deal with equations, which are not of Euler-Lagrange type, such as those involving non self-adjoint terms or/and non-linear operators.*** Mass transport theory to handle the still elusive variational problems arising in non-linear elasticity theory. *** Monge-Kantorovich theory to decouple certain partial differential systems, such as those of De Giorgi-type arising as a limiting elliptic system describing phase separation of multiple state Bose-Einstein condensates.**** Its equivariant form for applications to matching problems in economics, and to Density Functional Theory, a well-established method for tackling the quantum mechanics of many-body systems.**** Infinite dimensional critical point theory for borderline problems that lack compactness. These often require a fine analysis of the linear operators involved, such as the second and fourth-order Hardy-Schrodinger operators, the fractional Laplacians, and appropriate perturbations of them. The singularities could lie either in the interior, or on the boundary of the domain under consideration. The latter can be Euclidean or curved.**** Novel approaches for the identification of best constants and/or the establishment of the existence of steady states and other extremals for various functional and geometric inequalities, such as those of Caffarelli-Kohn-Nirenberg, Moser-Aubin-Onofri, and Gagliardo-Nirenberg inequalities.**** New insight for addressing issues of regularity, stability, and critical dimensions in various non-linear eigenvalue problems (self-adjoint or not), especially those dealing with the biharmonic operator, and the fractional Laplacians.  **