Introduction to Nonlinear Spectral Analysis

TL;DR

This paper introduces nonlinear spectral theory, covering eigenvalue problems, gradient flows, and applications to nonlinear PDEs, with a focus on variational methods, convergence, and numerical algorithms.

Contribution

It provides a comprehensive introduction to nonlinear spectral analysis, connecting variational eigenvalue problems with gradient flows and numerical methods, based on convex analysis and calculus of variations.

Findings

Conditions for finite time extinction in gradient flows

Convergence rates for nonlinear eigenfunctions

Numerical methods for solving nonlinear eigenvalue problems

Abstract

These notes are meant as an introduction to the theory of nonlinear spectral theory. We will discuss the variational form of nonlninear eigenvalue problems and the corresponding non-linear Euler--Lagrange equations, as well as connections with gradient flows. For the latter ones, we will give precise conditions for finite time extinction and discuss convergence rates. We will use this theory to study asymptotic behaviour of nonlinear PDEs and present applications in $L^\infty$ variational problems. Finally we will discuss numerical methods for solving gradient flows and computing nonlinear eigenfunctions based on a nonlinear power method. Our main tools are convex analysis and calculus of variations, necessary background on which will be provided. It is expected that the reader is familiar with Hilbert spaces; familiarity with Banach spaces is beneficial but not strictly necessary. The notes are based on the lectures taught by the authors at the universities of Bonn and Cambridge in 2022.