Mathematical Sciences: Nonconvex Variational Problems and Nonlinear Partial Differential Equations

本项目涉及非凸变分问题、奇异摄动和非线性偏微分方程的研究。本文将分析非凸问题与通过引入凸扰动得到的“松弛问题”之间的联系，以及变分问题与它们产生的梯度流之间的联系。需要强调的特殊情况是依赖于梯度的变分问题，局部面积最小化分区，平均曲率驱动的流动，以及通过最小梯度的优化设计。该项目的结果将导致更好地理解与材料不稳定性相关的物理问题，这反过来将有助于新材料的开发。＊＊＊

This project involves research on nonconvex variational problems, singular perturbation, and nonlinear partial differential equations. The connections between nonconvex problems and the "relaxed problems" obtained by introducing a convex perturbation will be analyzed, as well as the connections between variational problems and the gradient flows they generate. Particular cases to be emphasized are variational problems depending on a gradient, locally area-minimizing partitions, flows driven by mean curvature, and optimal design via least gradient. The results of this project will lead to better understanding of physical problems related to material instabilities, which in turn will help in the development of new materials. ***