Eigenvalue Problems in L^∞: Optimality Conditions, Duality, and Relations with Optimal Transport

In this article we characterize the L^∞ eigenvalue problem associated to the Rayleigh quotient left.|nabla u|L^∞middle/|u|_∞right. and relate it to a divergence-form PDE, similarly to what is known for Lp eigenvalue problems and the p-Laplacian for p<∞. Contrary to existing methods, which study L^∞-problems as limits of Lp-problems for p→∞, 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 umapsto|nabla u|L^∞. We show that the eigenvalue problem takes the form λ nu u =-operatornamediv(τnablaτ u), where nu and τ are non-negative measures concentrated where |u| respectively |nabla u| are maximal, and nablaτ u is the tangential gradient of u with respect to τ. 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.