Regularity Problems in the Calculus of Variations and Elliptic Partial Differential Equations

这个项目集中在经典领域的中心问题，如变分和自由边界问题。一些问题在应用科学中找到了动力，因此为数学家和其他研究人员之间的合作提供了机会。例如，所谓的“单相问题”描述了流体的运动，其中存在一个空腔，该空腔是蒸汽和气体的混合物，并且空腔上的压力是恒定的。正在调查的其他问题与最佳运输和资源分配问题有关。所有这些问题都需要发展新的尖端技术，并将向科学界传播进展，以推动该理论的进步。PI将研究“一相”自由边界问题的低维锥体分类。作为分析自由边界的工具，PI将研究一些边界Harnack类型的结果。在最优传输的背景下，PI感兴趣的是一类退化的Monge-Ampere方程的高正则性，其中右侧在边界上消失。最后，本项目的部分内容涉及变分中极小元的局部正则性这一基本问题。特别是，在标量情况下，PI将调查被积函数在某个紧集上变得高度退化的情况。在向量情况下，PI计划构造一些有趣的低维奇异极小化，并将探索某些类型泛函的正则性的粘性方法。

This project focuses on central problems in classical fields such as the calculus of variations and free boundary problems. Some problems find their motivation in the applied sciences and therefore offer opportunities for collaborations among mathematicians and other researchers. For example, the so-called "one-phase problem" describes the motion of a fluid in which a cavity is present that is a mixture of vapor and gas and the pressure on the cavity is constant. Other problems under investigation are related to issues of optimal transportation and allocation of resources. All such problems require the development of new sophisticated techniques, and progress will be disseminated to the scientific community to invigorate the advancement of the theory.The PI will study the classification of cones in low dimensions for the "one-phase" free boundary problem. As a tool in the analysis of free boundaries, the PI will investigate some boundary Harnack type results. In the context of optimal transportation, the PI is interested in the higher regularity for a class of degenerate Monge-Ampere equations in which the right hand side vanishes on the boundary. Finally, parts of this project deal with the fundamental question of local regularity of minimizers in the calculus of variations. In particular, in the scalar case the PI will investigate situations in which the integrand becomes highly degenerate on some compact set. In the vectorial case, the PI plans to construct some interesting singular minimizers in low dimensions and will also explore a viscosity approach to regularity for certain types of functionals.