Generalised vectorial<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si15.svg"><mml:mi>∞</mml:mi></mml:math>-eigenvalue nonlinear problems for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e27" altimg="si16.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msup></mml:math>functionals

Katzourakis, Nikos

doi:10.1016/j.na.2022.112806

Cited by 2 publications

(3 citation statements)

References 29 publications

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“…In fact, this paper is an extension of [23] to the second order case, and generalizes part of the results corresponding to the existence of minimizers and the satisfaction of PDEs from [25]. In turn, the paper [23] generalized results on the scalar case of eigenvalue problems for the ∞-Laplacian ( [18,19]). For various interesting results, see for instance [2,3,6,8,[28][29][30][31].…”

Section: And N Katzourakismentioning

confidence: 73%

“…Higher order problems and problems involving constraints present additional difficulties and have been studied even more sparsely, see e.g. [3,4,9,10,[20][21][22][23][24]26]. In fact, this paper is an extension of [23] to the second order case, and generalizes part of the results corresponding to the existence of minimizers and the satisfaction of PDEs from [25].…”

Section: And N Katzourakismentioning

confidence: 79%

“…However, it is possible to obtain further information about the underlying structure of these parametric measure coefficients. This requires techniques such as measure function pairs and mollifications up to the boundary as in [10,17,23], but to keep the presentation as simple as possible, we refrain from pursuing this -considerably more technical-endeavour, which also requires stronger assumptions.…”

Section: And N Katzourakismentioning

confidence: 99%

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Generalized second order vectorial ∞-eigenvalue problems

Clark,

Katzourakis

2024

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We consider the problem of minimizing the $L^\infty$ norm of a function of the hessian over a class of maps, subject to a mass constraint involving the $L^\infty$ norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the “hinged” and the “clamped” cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^\infty$ minimizer, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem. Furthermore, we establish upper and lower bounds for the eigenvalue.

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Section: And N Katzourakismentioning

confidence: 73%

Section: And N Katzourakismentioning

confidence: 79%

Section: And N Katzourakismentioning

confidence: 99%

See 1 more Smart Citation

Generalized second order vectorial ∞-eigenvalue problems

Clark,

Katzourakis

2024

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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

Bungert

Korolev

2022

Comm. Amer. Math. Soc.

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In this article we characterize the L ∞ \mathrm {L}^\infty eigenvalue problem associated to the Rayleigh quotient ‖ ∇ u ‖ L ∞ / ‖ u ‖ ∞ \left .{\|\nabla u\|_{\mathrm {L}^\infty }}\middle /{\|u\|_\infty }\right . and relate it to a divergence-form PDE, similarly to what is known for L p \mathrm {L}^p eigenvalue problems and the p p -Laplacian for p > ∞ p>\infty . Contrary to existing methods, which study L ∞ \mathrm {L}^\infty -problems as limits of L p \mathrm {L}^p -problems for p → ∞ 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 ↦ ‖ ∇ u ‖ L ∞ u\mapsto \|\nabla u\|_{\mathrm {L}^\infty } . We show that the eigenvalue problem takes the form λ ν u = − div ⁡ ( τ ∇ τ u ) \lambda \nu u =-\operatorname {div}(\tau \nabla _\tau u) , where ν \nu and τ \tau are non-negative measures concentrated where | u | |u| respectively | ∇ u | |\nabla u| are maximal, and ∇ τ u \nabla _\tau u is the tangential gradient of u u with respect to τ \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.

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Generalised vectorial∞-eigenvalue nonlinear problems forL∞functionals

Cited by 2 publications

References 29 publications

Generalized second order vectorial ∞-eigenvalue problems

Generalized second order vectorial ∞-eigenvalue problems

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

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