Departamento de Matemáticas UAM

Seminario de análisis y aplicaciones

Título: Studying nonlinear eigenvalue problems in L-infinity with convex analysis

Ponente: Yury Korolev, University of Bath

Resumen: We study a nonlinear eigenvalue problem associated with the Rayleigh quotient |u|_Lip/|u|_C, where |u|_Lip is the Lipschitz constant of a function u defined on a bounded domain in ℝⁿ  and |u|_C is its supremum norm. The problem of minimising this Rayleigh quotient is closely related to the infinity Laplacian: minimisers include infinity-harmonic potentials and so-called infinity ground states defined as solutions of a certain limiting PDE obtained by taking the limit p → ∞ in the p-Laplace eigenvalue problem. Another notable minimiser is the distance function to the boundary of the domain. Unlike existing literature that studies L-infinity problems as limits of L^p problems, we study the limiting problem directly using tools from convex analysis. This allows us to obtain results that hold for _all_ minimisers of the Rayleigh quotient. We obtain optimality conditions in form of a divergence PDE using a novel characterisation of the subdifferential of the Lipschitz seminorm u ↦ |u|_Lip as a functional on C. We also study a minimisation problem for the dual Rayleigh quotient involving Radon measures and a variant of the Kantorovich-Rubinstein norm, and relate minimisers of the L-infinity Rayleigh quotient to solutions of an optimal transport problem.

This is joint work with Leon Bungert, University of Bonn.

Lugar y hora: Aula 520, Módulo 17; 11:00--12:00. 

Café: Habrá un coffee break entre el seminario de análisis y el de EDPs ☕☕