Eigenvalue problems in $\mathrm {L}^\infty$: optimality conditions, duality, and relations with optimal transport
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- by Leon Bungert and Yury Korolev;
- Comm. Amer. Math. Soc. 2 (2022), 345-373
- DOI: https://doi.org/10.1090/cams/11
- Published electronically: October 14, 2022
- HTML | PDF
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.References
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Bibliographic Information
- Leon Bungert
- Affiliation: Hausdorff Center for Mathematics, University of Bonn, Endenicher Allee 62, Villa Maria, 53115 Bonn, Germany
- MR Author ID: 1261337
- ORCID: 0000-0002-6554-9892
- Email: leon.bungert@hcm.uni-bonn.de
- Yury Korolev
- Affiliation: Department of Mathematical Sciences, University of Bath, Claverton Down BA2 7AY, United Kingdom
- MR Author ID: 996474
- ORCID: 0000-0002-6339-652X
- Email: ymk30@bath.ac.uk
- Received by editor(s): September 21, 2021
- Received by editor(s) in revised form: June 28, 2022, and July 18, 2022
- Published electronically: October 14, 2022
- Additional Notes: This work was funded by the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 777826 (NoMADS). The first author acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - GZ 2047/1, Projekt-ID 390685813. The second author was financially supported by the EPSRC (Fellowship EP/V003615/1), the Cantab Capital Institute for the Mathematics of Information at the University of Cambridge and the National Physical Laboratory.
- © Copyright 2022 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Comm. Amer. Math. Soc. 2 (2022), 345-373
- MSC (2020): Primary 26A16, 35P30, 46N10, 47J10, 49R05
- DOI: https://doi.org/10.1090/cams/11
- MathSciNet review: 4496957