Concentration Phenomena in Nonlinear PDEs and Elasto-plasticity Theory

从偏微分方程组的补偿紧凑性理论、弹性材料的形状优化，或流体中的涡丝和晶体材料中的位错线等几何结构的传输等不同领域，分析中的许多重要开放问题的核心都有关于地图、测量或电流的“扩散集中”序列的深刻问题。随着典型的长度尺度接近于零，这类典型序列显示出越来越多的薄而重复的结构。挑战在于理解这个结构的“网络”可能表现出的渐近构型，这通常受到像无散度这样的(线性)PDE约束的存在的高度限制。尽管在过去十年中，有关测量奇点的相关研究取得了很大进展，但弥散浓度仍然笼罩在神秘的阴影中。基于PI最近在偏微分方程理论、几何测量理论和变分演算的交叉点取得的突破性进展，集中式建议旨在这一高度活跃和快速发展的研究领域取得变革性的进展。作为对理论研究的应用和指导，该项目将进一步解决由位错运动驱动的大应变弹塑性的微观到宏观的均匀化问题，从而提供一个严格而现实的塑性变形模型。通常被称为塑性理论的“圣杯”，这样的均化结果迄今被证明是难以捉摸的，尽管有很多集体努力，因为它需要对从离散的位错线到位错场时遇到的扩散浓度有很好的理解。PI在这些领域的研究领导地位使他处于独特的地位，可以通过开发新的数学工具和解决长期存在的纯粹和应用性质的猜想来实现这项建议的雄心勃勃的目标。

Numerous important open problems in Analysis, from such diverse areas as the compensated compactness theory of PDEs, the shape optimization of elastic materials, or the transport of geometric structures like vortex filaments in fluids and dislocation lines in crystalline materials, have at their core deep questions about "diffusely concentrating" sequences of maps, measures, or currents. Prototypical sequences of this kind display an increasing number of thin and repetitive structures as the typical length scale goes to zero. The challenge is to understand the asymptotic configurations that this "network" of structures can exhibit, which are usually highly restricted by the presence of a (linear) PDE constraint like divergence-freeness. Despite much progress in the related study of singularities in measures over the last decade, diffuse concentrations have remained shrouded in mystery. Building on the recent groundbreaking advances by the PI at the intersection of PDE Theory, Geometric Measure Theory, and the Calculus of Variations, the CONCENTRATE proposal aims at transformative progress in this highly active and rapidly evolving research area. As an application and guiding light to the theoretical investigation, the project will furthermore tackle the micro-to-macro homogenization of large-strain elasto-plasticity driven by the motion of dislocations, thus furnishing a rigorous and realistic model of plastic deformations. Often referred to as the "Holy Grail" of plasticity theory, such a homogenization result has so far proved elusive, despite much collective effort, since it requires a fine understanding of the diffuse concentrations encountered when passing from discrete dislocation lines to fields of dislocations. The PI's research leadership in these areas makes him uniquely placed to tackle the ambitious goals of this proposal through the development of novel mathematical tools and the solution of long-standing conjectures of both pure and applied character.