Leon Bungert: Tackling L∞ eigenvalue problems with convex analysis

Time: Thu 2022-09-22 15.00 - 16.00

Location: Institut Mittag-Leffler, Seminar Hall Kuskvillan and Zoom

Video link: Meeting ID: 921 756 1880

Participating: Leon Bungert (University of Bonn)

Abstract:

I will characterize the L∞ eigenvalue problem which is solved by stationary points of the Rayleigh quotient $$∥∇u∥_{L^{\infty}}/∥u∥_∞$$and relate it to a divergence-form PDE, similarly to what is known for $$L^p$$ eigenvalue problems and the p-Laplacian for p < ∞. Contrary to most existing methods, which study $$L^{\infty}$$-problems as limits of $$L^p$$-problems for large values of p, I shall present a novel framework for analyzing the limiting problem directly using convex analysis and measure theory. Our results rely on a novel fine characterization of the subdifferential of the Lipschitz-constant-functional. I also study a dual Rayleigh quotient whose minimizers solve an optimal transport problem associated to a generalized Kantorovich--Rubinstein norm. This is joint work with Yury Korolev and based on the article (https://arxiv.org/abs/2107.12117).