Eigenvalue problems in L∞: optimality conditions, duality, and relations with optimal transport

In this article we characterize the L ∞ L^∞ eigenvalue problem associated to the Rayleigh quotient ‖ ∇ u ‖ L ∞ / ‖ u ‖ ∞ .{\|∇ u\|L^∞} /{\|u\|_∞ } . and relate it to a divergence-form PDE, similarly to what is known for L p L^p eigenvalue problems and the p p -Laplacian for p > ∞ p>∞ . Contrary to existing methods, which study L ∞ L^∞ -problems as limits of L p L^p -problems for p → ∞ 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 u ↦ ‖