Mathematical Sciences: Nonconvex Variational Problems and Nonlinear Partial Differential Equations

本项目涉及非凸变分问题的研究 奇异摄动和非线性偏微分方程 微分方程 非凸函数之间的联系 问题和“放松问题”通过引入一个 凸扰动将被分析，以及连接 在变分问题和梯度流之间， 生成. 要强调的特殊情况是变化的 依赖于梯度的局部面积最小化问题 分区、由平均曲率驱动的流和最优设计 最小梯度 该项目的结果将导致更好的 理解与材料相关的物理问题 不稳定性，这反过来将有助于发展新的 材料. ***

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. ***