Eigenvalue Problems in $\mathrm{L}^\infty$: Optimality Conditions, Duality, and Relations with Optimal Transport

TL;DR

This paper characterizes the $ ext{L}^ ext{infty}$ eigenvalue problem using convex analysis and geometric measure theory, relating it to divergence PDEs and optimal transport, and extends known results to a broader class of stationary points.

Contribution

It introduces a direct framework for $ ext{L}^ ext{infty}$ eigenproblems, deriving a novel subdifferential characterization and linking the problem to optimal transport theory.

Findings

Characterization of $ ext{L}^ ext{infty}$ eigenvalue problem via divergence PDEs.

Development of a new subdifferential analysis for Lipschitz functionals.

Connection of eigenproblems to optimal transport and Kantorovich--Rubinstein norms.

Abstract

In this article we characterize the $\mathrm{L}^\infty$ eigenvalue problem associated to the Rayleigh quotient $\left.{\|\nabla u\|_{\mathrm{L}^\infty}}\middle/{\|u\|_\infty}\right.$ and relate it to a divergence-form PDE, similarly to what is known for $\mathrm{L}^p$ eigenvalue problems and the $p$-Laplacian for $p<\infty$. Contrary to existing methods, which study $\mathrm{L}^\infty$-problems as limits of $\mathrm{L}^p$-problems for $p\to\infty$, we develop a novel framework for analyzing the limiting problem directly using convex analysis and geometric measure theory. For this, we derive a novel fine characterization of the subdifferential of the Lipschitz-constant-functional $u\mapsto\|\nabla u\|_{\mathrm{L}^\infty}$. We show that the eigenvalue problem takes the form $\lambda \nu u =-\operatorname{div}(\tau\nabla_\tau u)$, where $\nu$ and $\tau$ are non-negative measures concentrated where $|u|$ respectively $|\nabla u|$ are maximal, and $\nabla_\tau u$ is the tangential gradient of $u$ with respect to $\tau$. Lastly, we investigate a dual Rayleigh quotient whose minimizers solve an optimal transport problem associated to a generalized Kantorovich--Rubinstein norm. Our results apply to all stationary points of the Rayleigh quotient, including infinity ground states, infinity harmonic potentials, distance functions, etc., and generalize known results in the literature.