This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how these can be approximated using $\Gamma$-convergence. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to normalized gradient flows. Furthermore, we show that the implicit Euler discretization of gradient flows gives rise to a nonlinear power method of the proximal operator and prove their convergence to nonlinear eigenfunctions. Finally, we prove that $\Gamma$-convergence of functionals implies convergence of their ground states.

翻译：本章描述了Banach空间的梯度流和非线性电功率方法如何用于解决非线性电子元值依赖性电子元值问题,以及如何用$\Gamma$-converging来比较这些问题。我们审查了文献中的几种流,这些流旨在计算非线性电子元件,并表明这些流都与正常的梯度流有关。此外,我们表明,梯度流的隐性电极分解导致预产体操作者采用非线性电源方法,并证明它们与非线性电子元的趋同。最后,我们证明,美元-功能的趋同意味着其地面状态的趋同。