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A problem of free boundary type for caloric measure

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Authors : Hofmann, Steven (Author of the conference)
CIRM (Publisher )

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Abstract : For an open set $\Omega \subset \mathbb{R}^{d}$ with an Ahlfors regular boundary, solvability of the Dirichlet problem for Laplaces equation, with boundary data in $L^{p}$ for some $p<\infty$, is equivalent to quantitative, scale invariant absolute continuity (more precisely, the weak- $A_{\infty}$ property) of harmonic measure with respect to surface measure on $\partial \Omega$. A similar statement is true in the caloric setting. Thus, it is of interest to find geometric criteria which characterize the open sets for which such absolute continuity (hence also solvability) holds. Recently, this has been done in the harmonic case. In this talk, we shall discuss recent progress in the caloric setting, in which we show that quantitative absolute continuity of caloric measure, with respect to surface measure on the parabolic Ahlfors regular (lateral) boundary $\Sigma$, implies parabolic uniform rectifiability of $\Sigma$. We observe that this result may be viewed as the solution of a certain 1-phase free boundary problem. This is joint work with S. Bortz, J. M. Martell and K. Nyström.

Keywords : caloric measure; Dirichlet problem; free boundary; square function; Green function; level sets

MSC Codes :
35K05 - Heat equation
 35K20 - Boundary value problems for second-order, parabolic equations
 35R35 - Free boundary problems
 42B25 - Maximal functions, Littlewood-Paley theory
 42B37 - Harmonic analysis and PDE
    Information on the Video

    Film maker : Recanzone, Luca
    Language : English
    Available date : 10/07/2024
    Conference Date : 11/06/2024
    Subseries : Research talks
    arXiv category : Analysis of PDEs ; Classical Analysis and ODEs
    Mathematical Area(s) : Analysis and its Applications ; PDE
    Format : MP4 (.mp4) - HD
    Video Time : 00:58:24
    Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2024-06-11_Hofman.mp4
Information on the Event

Event Title : Harmonic analysis and partial differential equations / Analyse harmonique et équations aux dérivées partielles
Event Organizers : Bernicot, Frédéric ; Martell, José Maria ; Monniaux, Sylvie ; Portal, Pierre
Dates : 10/06/2024 - 14/06/2024
Event Year : 2024
Event URL : https://conferences.cirm-math.fr/2979.html

Citation Data

DOI : 10.24350/CIRM.V.20189403
Cite this video as: Hofmann, Steven (2024). A problem of free boundary type for caloric measure. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20189403
URI : http://dx.doi.org/10.24350/CIRM.V.20189403

See Also

Bibliography

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