Powers of discrete Laplacians and Hardy-type inequalities | Vidéo | Carmin.tv

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Powers of discrete Laplacians and Hardy-type inequalities

By Borbala Gerhat

Appears in collection : Mathematical aspects of the physics with non-self-adjoint operators / Les aspects mathématiques de la physique avec les opérateurs non-auto-adjoints

We study the existence of non-trivial lower bounds for positive powers of the discrete Dirichlet Laplacian on the half line. Unlike in the continuous setting where both $-\Delta$ and $(-\Delta)^2$ admit a Hardy-type inequality, their discrete analogues exhibit a different behaviour. While the discrete Laplacian is subcritical, its square is critical and the threshold where the criticality of $(-\Delta)^\alpha$ first appears turns out to be $\alpha=3 / 2$. We provide corresponding (non-optimal) Hardy-type inequalities in the subcritical regime. Moreover, for the critical exponent $\alpha=2$, we employ a remainder factorisation strategy to derive a discrete Rellich inequality on a suitable subspace (with a weight improving upon the classical Rellich weight). Based on joint work with D. Krejčiřík and F. Štampach.

Information about the video

Date of recording 04/06/2024
Date of publication 26/06/2024
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.20188203
Cite this video Gerhat, Borbala (04/06/2024). Powers of discrete Laplacians and Hardy-type inequalities. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20188203
URL https://dx.doi.org/10.24350/CIRM.V.20188203

Domain(s)

Spectral Theory

Bibliography

GERHAT, Borbala, KREJCIRIK, David, et STAMPACH, Frantisek. Criticality transition for positive powers of the discrete Laplacian on the half line. arXiv preprint arXiv:2307.09919, 2023. - https://doi.org/10.48550/arXiv.2307.09919

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