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Curved quantum nonlinear waveguides

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Auteurs : Baldelli, Laura (Auteur de la conférence)
CIRM (Editeur )

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Résumé : In the last decade, there has been an increasing interest in the p-Laplacian, which plays an important role in geometry and partial differential equations. The p-Laplacian is a natural generalization of the Laplacian. Although the Laplacian has been much studied, not much is known about the nonlinear case p >1. Motivated by these facts, the purpose of the present paper is to review recent developments in the spectral theory of a specific class of quantum waveguides modeled by the Dirichlet Laplacian, i.e. p = 2, in unbounded tubes of uniform cross-section rotating w.r.t. the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension. We discuss how the spectrum depends upon three geometric deformations: straightness, asymptotic straightness, and bending. Precisely, if the reference curve is straight or asymptotic straight, the essential spectrum is preserved. While dealing with bent tubes, such geometry produces a spectrum below the first eigenvalue. All the results confirm the literature for the Laplacian operator. The results are obtained via a very delicate analysis since the nonlinearity given by the p-Laplacian operator adds different types of difficulties with respect to the linear situation. These results are contained in a work written jointly with D. Krejčiřík.

Mots-Clés : p-Laplacian; quantum waveguides; curved tubes; twisting; bending; essential spectrum; discrete eigenvalues; Hardy inequalities

Codes MSC :
58C40 - Spectral theory; eigenvalue problems
58J50 - Spectral problems; spectral geometry; scattering theory
35J92 - Quasilinear elliptic equations with p-Laplacian

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de Publication : 26/06/2024
    Date de Captation : 04/06/2024
    Sous Collection : Research talks
    Catégorie arXiv : Analysis of PDEs
    Domaine(s) : EDP
    Format : MP4 (.mp4) - HD
    Durée : 00:22:07
    Audience : Chercheurs ; Etudiants Science Cycle 2 ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2024-06-04_Baldelli.mp4

Informations sur la Rencontre

Nom de la Rencontre : Mathematical aspects of the physics with non-self-adjoint operators / Les aspects mathématiques de la physique avec les opérateurs non-auto-adjoints
Organisateurs de la Rencontre : Boulton, Lyonell ; Cossetti, Lucrezia ; Krejcirik, David ; Siegl, Petr
Dates : 03/06/2024 - 07/06/2024
Année de la rencontre : 2024
URL de la Rencontre : https://conferences.cirm-math.fr/2971.html

Données de citation

DOI : 10.24350/CIRM.V.20187803
Citer cette vidéo: Baldelli, Laura (2024). Curved quantum nonlinear waveguides. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20187803
URI : http://dx.doi.org/10.24350/CIRM.V.20187803

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