Nonlinear Partial Differential Equations Methods in the Study of Interfaces

这个项目研究了在纯数学和应用数学的几个领域中出现的偏微分方程（PDE）。的主要目标之一是正式推导的物理模型，其数学描述是基于偏微分方程的非本地类型，描述晶体材料中的缺陷线的动态。在另一个领域，PI将研究非线性椭圆偏微分方程模型分离现象的解的定性行为。该项目将通过弥合数学分析和物理之间的差距，从根本上促进这些领域，并将使物理现象的进一步数学理解。该项目为研究生和博士后研究人员提供研究培训机会。在技术层面上，该项目的目标是通过PDE方法形成界面表面的研究。重点专题领域是：㈠非局部反应扩散方程，其主要目标是通过研究Peierls-Nabarro模型的渐近极限，推导出描述晶体中缺陷线（位错）动态的中尺度和宏观尺度模型，并证明分数算子驱动的方程系统的异宿轨道和多凸轨道的存在; ㈡分离模型和非线性本征值问题。在这里，主要目标是分析分离的固定配置的节点集及其与完全非线性算子的特征值问题的联系。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project investigates partial differential equations (PDE) that arise in several areas of pure and applied mathematics. One of the main objectives is the formal derivation of physical models, whose mathematical description is based on PDE of nonlocal type, that describe the dynamics of defect lines in crystalline materials. In a different area, the PI will investigate the qualitative behavior of solutions to nonlinear elliptic PDE that model segregation phenomena. The project will fundamentally contribute to these fields by bridging the gap between mathematical analysis and physics and will enable further mathematical understanding of physical phenomena. The project provides research training opportunities for graduate students and postdoctoral researchers.At a technical level, the objective of this project is the study of forming interface surfaces through PDE methods. Thematic areas of focus are: (i) Nonlocal reaction-diffusion equations, where the main goals are to derive mesoscale and macroscopic scale models describing the dynamics of defect lines in crystals (dislocations) through the study of the asymptotic limits of Peierls-Nabarro models and to prove existence of heteroclinic and multibump orbits for systems of equations driven by fractional operators; and (ii) Segregation models and nonlinear eigenvalue problems. Here, the main objective is the analysis of nodal sets of segregated stationary configurations and their connection with eigenvalue problems for fully nonlinear operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.