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

Abstract

In this article we characterize the $\mathrm{L}^\infty$ eigenvalue problem associated to the Rayleigh quotient $\left.{\|\nabla u\|_{\mathrm{L}^\infty}}\middle/{\|u\|_\infty}\right.$, defined on Lipschitz functions with homogeneous boundary conditions on a domain $\Omega$. For this, we derive a novel fine characterization of the subdifferential of the functional $u\mapsto\|\nabla u\|_{\mathrm{L}^\infty}$. Using the concept of tangential gradients we show that it consists of Radon measures of the form $-\operatorname{div}\sigma$ where $\sigma$ is a normalized divergence-measure field, in a suitable sense parallel to the gradient $\nabla u$ and concentrated where it is maximal. We also investigate geometric relations between general minimizers of the Rayleigh quotient and suitable distance functions. Lastly, we investigate a "dual" Rayleigh quotient whose minimizers are subgradients of minimizers of the original quotient and solve an optimal transport problem associated to a generalized Kantorovich-Rubinstein norm. Our results apply to all minimizers of the Rayleigh quotient, including infinity ground states, infinity harmonic potentials, distance functions, etc., and generalize known results in the literature