Vol. 14, No. 3, 2021

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Nonlinear spectral decompositions by gradient flows of one-homogeneous functionals

Leon Bungert, Martin Burger, Antonin Chambolle and Matteo Novaga

Vol. 14 (2021), No. 3, 823–860
Abstract

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is motivated by works for the total variation, where 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 seminorms, where gradient flows can yield spectral decompositions into eigenvectors.

We provide a geometric characterization of eigenvectors via a dual unit ball and prove that they are subgradients of minimal norm. This establishes the connection to gradient flows, whose time evolution is a decomposition of the initial condition into subgradients of minimal norm. If these are eigenvectors, this implies an interesting orthogonality relation and the equivalence of the gradient flow to a variational regularization method and an inverse scale space flow. Indeed we verify that all scenarios where these equivalences were known before by other arguments — such as one-dimensional total variation, multidimensional generalizations to vector fields, or certain polyhedral seminorms — yield 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.

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Keywords
nonlinear spectral decompositions, gradient flows, nonlinear eigenvalue problems, one-homogeneous functionals, extinction profiles
Mathematical Subject Classification 2010
Primary: 35P10, 35P30, 47J10
Milestones
Received: 9 February 2019
Revised: 25 September 2019
Accepted: 2 December 2019
Published: 18 May 2021
Authors
Leon Bungert
Department Mathematik
Universität Erlangen-Nürnberg
Erlangen
Germany
Martin Burger
Department Mathematik
Universität Erlangen-Nürnberg
Erlangen
Germany
Antonin Chambolle
CMAP
École Polytechnique, CNRS
Palaiseau
France
Matteo Novaga
Dipartimento di Matematica
Università di Pisa
Pisa
Italy