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Eigenvalue Problems in
L
β
\mathrm{L}^\infty
L
β
: Optimality Conditions, Duality, and Relations with Optimal Transport
Authors
Leon Bungert
Yury Korolev
Publication date
26 July 2021
Publisher
View
on
arXiv
Abstract
In this article we characterize the
L
β
\mathrm{L}^\infty
L
β
eigenvalue problem associated to the Rayleigh quotient
β₯
β
u
β₯
L
β
/
β₯
u
β₯
β
\left.{\|\nabla u\|_{\mathrm{L}^\infty}}\middle/{\|u\|_\infty}\right.
β₯β
u
β₯
L
β
β
/
β₯
u
β₯
β
β
, defined on Lipschitz functions with homogeneous boundary conditions on a domain
Ξ©
\Omega
Ξ©
. For this, we derive a novel fine characterization of the subdifferential of the functional
u
β¦
β₯
β
u
β₯
L
β
u\mapsto\|\nabla u\|_{\mathrm{L}^\infty}
u
β¦
β₯β
u
β₯
L
β
β
. Using the concept of tangential gradients we show that it consists of Radon measures of the form
β
div
β‘
Ο
-\operatorname{div}\sigma
β
div
Ο
where
Ο
\sigma
Ο
is a normalized divergence-measure field, in a suitable sense parallel to the gradient
β
u
\nabla u
β
u
and concentrated where it is maximal. We also investigate geometric relations between general minimizers of the Rayleigh quotient and suitable distance functions. Lastly, we investigate a "dual" Rayleigh quotient whose minimizers are subgradients of minimizers of the original quotient and solve an optimal transport problem associated to a generalized Kantorovich-Rubinstein norm. Our results apply to all minimizers of the Rayleigh quotient, including infinity ground states, infinity harmonic potentials, distance functions, etc., and generalize known results in the literature
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oai:arXiv.org:2107.12117
Last time updated on 14/08/2021