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Unexpected norms on BMO and the Dirichlet problem

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Auteurs : Egert, Moritz (Auteur de la conférence)
CIRM (Editeur )

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Résumé : One of the many meaningful equivalent norms on BMO uses a Carleson-measure condition on the gradient of the Poisson extension. This is closely related to the Dirichlet problem for the Laplacian in the upper half-space with boundary data in BMO. The Poisson semigroup provides the unique solution in appropriate classes, and it is bounded on BMO, that is, it propagates the space boundary space in the transversal direction. If the tangential Laplacian is replaced by a general elliptic operator in divergence form, boundedness of the Poisson semigroup on BMO can fail in any dimension n ≥ 3. Somewhat unexpectedly, its gradient persists to give rise to a Carleson measure with norm equivalent to the BMO-norm at the boundary in dimensions n = 3, 4 and hence a unique solution to the corresponding Dirichlet problem. In my talk, I will try to explain the broader context behind this phenomenon and why we still do not know if the result is sharp.
Based on joint work with (of course) Pascal. It is Chapter 18 of our book but you will not have to read the seventeen preceding chapters to follow.

Mots-Clés : Second-order divergence-form operator; elliptic equations and systems; boundary value problems; solvability; uniqueness; wellposedness; BMO; Poisson semigroup

Codes MSC :
35J25 - Boundary value problems for second-order elliptic equations
35J67 - Boundary values of solutions to elliptic PDE
42B25 - Maximal functions, Littlewood-Paley theory
42B30 - $H^p$-spaces
42B35 - Function spaces arising in harmonic analysis
46E35 - Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47A60 - Functional calculus
47D06 - One-parameter semigroups and linear evolution equations
35J57 - Boundary value problems for second-order elliptic systems
42B37 - Harmonic analysis and PDE
35J46 - First-order elliptic systems

    Informations sur la Vidéo

    Réalisateur : Recanzone, Luca
    Langue : Anglais
    Date de Publication : 10/07/2024
    Date de Captation : 11/06/2024
    Sous Collection : Research talks
    Catégorie arXiv : Analysis of PDEs
    Domaine(s) : EDP
    Format : MP4 (.mp4) - HD
    Durée : 00:56:26
    Audience : Chercheurs ; Etudiants Science Cycle 2 ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2024-06-11_Egert.mp4

Informations sur la Rencontre

Nom de la Rencontre : Harmonic analysis and partial differential equations / Analyse harmonique et équations aux dérivées partielles
Organisateurs de la Rencontre : Bernicot, Frédéric ; Martell, José Maria ; Monniaux, Sylvie ; Portal, Pierre
Dates : 10/06/2024 - 14/06/2024
Année de la rencontre : 2024
URL de la Rencontre : https://conferences.cirm-math.fr/2979.html

Données de citation

DOI : 10.24350/CIRM.V.20189103
Citer cette vidéo: Egert, Moritz (2024). Unexpected norms on BMO and the Dirichlet problem. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20189103
URI : http://dx.doi.org/10.24350/CIRM.V.20189103

Voir Aussi

Bibliographie

  • AUSCHER, Pascal et EGERT, Moritz. Identification of Adapted Hardy Spaces. In : Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure. Cham : Springer International Publishing, 2023. p. 111-140. - http://dx.doi.org/10.1007/978-3-031-29973-5_9



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