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H 1 The weak-$A_\infty$ condition for harmonic measure

Auteurs : Tolsa, Xavier (Auteur de la Conférence)
... (Editeur )

Résumé : The weak-$A_\infty$ condition is a variant of the usual $A_\infty$ condition which does not require any doubling assumption on the weights. A few years ago Hofmann and Le showed that, for an open set $\Omega\subset \mathbb{R}^{n+1}$ with $n$-AD-regular boundary, the BMO-solvability of the Dirichlet problem for the Laplace equation is equivalent to the fact that the harmonic measure satisfies the weak-$A_\infty$ condition. Aiming for a geometric description of the open sets whose associated harmonic measure satisfies the weak-$A_\infty$ condition, Hofmann and Martell showed in 2017 that if $\partial\Omega$ is uniformly $n$-rectifiable and a suitable connectivity condition holds (the so-called weak local John condition), then the harmonic measure satisfies the weak-$A_\infty$ condition, and they conjectured that the converse implication also holds.
In this talk I will discuss a recent work by Azzam, Mourgoglou and myself which completes the proof of the Hofman-Martell conjecture, by showing that the weak-$A_\infty$ condition for harmonic measure implies the weak local John condition.

Keywords : BMO; Dirichlet problem; harmonic measure; weak-$A_\infty$; John condition

Codes MSC :
28A75 - Length, area, volume, other geometric measure theory
28A78 - Hausdorff and packing measures
31B15 - Potentials and capacities, extremal length
35J15 - General theory of second-order, elliptic equations
35J08 - Green's functions

 Informations sur la Vidéo Langue : Anglais Date de publication : 25/04/2018 Date de captation : 24/04/2018 Collection : Research talks ; Analysis and its Applications ; Partial Differential Equations Format : MP4 Durée : 00:58:21 Domaine : Analysis and its Applications ; PDE Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2018-04-24_Tolsa.mp4 Informations sur la rencontre Nom de la rencontre : Harmonic analysis of elliptic and parabolic partial differential equations / Analyse harmonique des équations aux dérivées partielles elliptiques et paraboliquesOrganisateurs de la rencontre : Monniaux, Sylvie ; Portal, PierreDates : 23/04/2018 - 27/04/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1741.htmlCitation Data DOI : 10.24350/CIRM.V.19398003 Cite this video as: Tolsa, Xavier (2018). The weak-$A_\infty$ condition for harmonic measure. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19398003 URI : http://dx.doi.org/10.24350/CIRM.V.19398003

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Bibliographie

1. Azzam, J., Mourgoglou, M., & Tolsa, X. (2018). A geometric characterization of the weak-$A_\infty$ condition for harmonic measure. - https://arxiv.org/abs/1803.07975

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