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 $\left .{\|\nabla u\|_{\mathrm {L}^\infty }}\middle /{\|u\|_\infty }\right .$ 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.