Unexpected norms on BMO and the Dirichlet problem | Vidéo | Carmin.tv

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Unexpected norms on BMO and the Dirichlet problem

By Moritz Egert

Appears in collection : Harmonic analysis and partial differential equations / Analyse harmonique et équations aux dérivées partielles

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.

Information about the video

Date of recording 11/06/2024
Date of publication 10/07/2024
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Luca Recanzone
Format MP4

Citation data

DOI 10.24350/CIRM.V.20189103
Cite this video Egert, Moritz (11/06/2024). Unexpected norms on BMO and the Dirichlet problem. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20189103
URL https://dx.doi.org/10.24350/CIRM.V.20189103

Domain(s)

Analysis of PDEs

Bibliography

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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