Calculus of Variations and Geometric Measure Theory

L. Bungert - M. Burger - A. Chambolle - M. Novaga

Nonlinear Spectral Decompositions by Gradient Flows of One-Homogeneous Functionals

created by novaga on 23 Jan 2019
modified on 08 Jul 2022


Published Paper

Inserted: 23 jan 2019
Last Updated: 8 jul 2022

Journal: Anal. PDE
Volume: 14
Number: 3
Pages: 823-860
Year: 2021

ArXiv: 1901.06979 PDF


The aim of this paper is to establish a theory of nonlinear spectral decompositions in an infinite dimensional setting by considering the eigenvalue problem related to an absolutely one-homogeneous functional in a Hilbert space. This approach is motivated by works for the total variation and related functionals in $L^2$, where some interesting results on the eigenvalue problem and the relation to the total variation flow have been proven previously, and by recent results on finite-dimensional polyhedral semi-norms, where gradient flows can yield exact decompositions into eigenvectors. We provide a geometric characterization of eigenvectors via a dual unit ball, which applies to the infinite-dimensional setting and allows applications to several relevant examples. In particular our analysis highlights the role of subgradients of minimal norm and thus connects directly to gradient flows, whose time evolution can be interpreted as a decomposition of the initial condition into subgradients of minimal norm. We can show that if the gradient flow yields a spectral decomposition it is equivalent in an appropriate sense to other schemes such as variational methods with regularization parameter equal to time, and that the decomposition has an interesting orthogonality relation. Indeed we verify that all approaches where these equivalences were known before by other arguments -- such as one-dimensional TV flow and multidimensional generalizations to vector fields or the gradient flow of certain polyhedral semi-norms -- yield exact spectral decompositions, and we provide further examples. We also investigate extinction times and extinction profiles, which we characterize as eigenvectors in a very general setting, generalizing several results from literature.