En poursuivant votre navigation sur ce site, vous acceptez l'utilisation d'un simple cookie d'identification. Aucune autre exploitation n'est faite de ce cookie. OK

AUDIOVISUAL MATHEMATICS LIBRARY

AUDIOVISUAL MATHEMATICS LIBRARY

1

A problem of free boundary type for caloric measure

Sélection

Cochez un dossier pour ajouter l'enregistrement à vos favoris. Décochez-le pour l'en enlever.

4

gerer mes paniers

Signaler une erreur
Multi angle
Auteurs : Hofmann, Steven (Auteur de la Conférence)
CIRM (Editeur )

Loading the player...
Résumé : 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

Codes MSC :
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
    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
    arXiv category : Analysis of PDEs ; Classical Analysis and ODEs
    Domaine : Analysis and its Applications ; PDE
    Format : MP4 (.mp4) - HD
    Durée : 00:58:24
    Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2024-06-11_Hofman.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 Congrès : https://conferences.cirm-math.fr/2979.html

Données de citation

DOI : 10.24350/CIRM.V.20189403
Citer cette vidéo: 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

Voir aussi

Bibliographie

  • BORTZ, Simon, HOFMANN, Steven, MARTELL, José María, et al. Solvability of the $ L^ p $ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph. arXiv preprint arXiv:2306.17291, 2023. - https://doi.org/10.48550/arXiv.2306.17291

  • ATHANASOPOULOS, Ioannis, CAFFARELLI, Luis, et SALSA, Sandro. Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems. Annals of mathematics, 1996, vol. 143, no 3, p. 413-434. - https://doi-org.insmi.bib.cnrs.fr/10.2307/2118531

  • BORTZ, Simon, HOFFMAN, John, HOFMANN, Steve, et al. Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets. Anal. PDE, 2023, vol. 16, no 4, p. 1061-1088. - https://doi.org/10.2140/apde.2023.16.1061

  • BORTZ, Simon, HOFFMAN, John, HOFMANN, Steve, et al. Coronizations and big pieces in metric spaces. In : Annales de l'Institut Fourier. 2022. p. 2037-2078. - https://doi.org/10.5802/aif.3518

  • BORTZ, Simon, HOFFMAN, John, HOFMANN, Steve, et al. Corona decompositions for parabolic uniformly rectifiable sets. The Journal of Geometric Analysis, 2023, vol. 33, no 3, p. 96. - http://dx.doi.org/10.1007/s12220-022-01176-8

  • CAFFARELLI, Luis A. et SALSA, Sandro. A geometric approach to free boundary problems. American Mathematical Soc., 2005. - http://dx.doi.org/10.1090/gsm/068

  • DAHLBERG, Björn EJ. Estimates of harmonic measure. Archive for Rational Mechanics and Analysis, 1977, vol. 65, p. 275-288. - http://dx.doi.org/10.1007/BF00280445

  • DAVID, Guy et SEMMES, Stephen. Singular integrals and rectifiable sets in 𝑅ⁿ: Beyond Lipschitz graphs. Astérisque, 1991, no 193, p. 152. -

  • DAVID, Guy et SEMMES, Stephen. Analysis of and on uniformly rectifiable sets. American Mathematical Soc., 1993. -

  • DE SILVA, Daniela et SAVIN, Ovidiu. On the parabolic boundary Harnack principle. La Matematica, 2022, p. 1-18. - http://dx.doi.org/10.1007/s44007-021-00001-y

  • GARIEPY, LC Evans-RF et EVANS, Lawrence Craig. Measure theory and fine properties of functions, Revised edition. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 2015. - https://doi.org/10.1201/b18333

  • FABES, Eugene B., GAROFALO, Nicola, et SALSA, Sandro. Comparison theorems for temperatures in noncylindrical domains. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti, 1984, vol. 76, p. 1-12. - http://gdmltest.u-ga.fr/item/RLINA_1984_8_77_1-2_1_0/

  • FABES, Eugene B. et SAFONOV, Mikhail V. Behavior near the boundary of positive solutions of second order parabolic equations. Journal of Fourier Analysis and Applications, 1997, vol. 3 (special issue, proceedings of the conference dedicated to Professor Miguel de Guzmán), p. 871-882. - http://dx.doi.org/10.1007/BF02656492

  • FABES, Eugene, SAFONOV, Mikhail, et YUAN, Yu. Behavior near the boundary of positive solutions of second order parabolic equations. II. Transactions of the American Mathematical Society, 1999, vol. 351, no 12, p. 4947-4961. - http://dx.doi.org/10.1090/S0002-9947-99-02487-3

  • GENSCHAW, Alyssa et HOFMANN, Steve. A weak reverse Hölder inequality for caloric measure. The Journal of Geometric Analysis, 2020, vol. 30, no 2, p. 1530-1564. - http://dx.doi.org/10.1007/s12220-019-00212-4

  • HOFMANN, Steven et LEWIS, John L. L2 solvability and representation by caloric layer potentials in time-varying domains. Annals of mathematics, 1996, p. 349-420. - https://doi.org/10.2307/2118595

  • HOFMANN, Steve, LEWIS, John L., et NYSTRÖM, Kaj. Existence of big pieces of graphs for parabolic problems. Annales Fennici Mathematici, 2003, vol. 28, no 2, p. 355-384. - http://ttp://eudml.org/doc/123779

  • HOFMANN, Steve, LEWIS, John L., et NYSTRÖM, Kaj. Caloric measure in parabolic flat domains. 2004. - http://dx.doi.org/10.1215/S0012-7094-04-12222-5

  • HOFMANN, Steve. Parabolic singular integrals of Calderón-type, rough operators, and caloric layer potentials. Duke Math. J., 1997, vol. 90, no 1, p. 209-259. - http://dx.doi.org/10.1215/S0012-7094-97-09008-6

  • JERISON, David. Regularity of the Poisson kernel and free boundary problems. In : Colloquium Mathematicum. 1990. p. 547-568. - http://dx.doi.org/10.4064/cm-60-61-2-547-568

  • JERISON, David S. et KENIG, Carlos E. Boundary behavior of harmonic functions in non-tangentially accessible domains. Advances in Mathematics, 1982, vol. 46, no 1, p. 80-147. - https://doi.org/10.1016/0001-8708(82)90055-X

  • KINDERLEHRER, David et NIRENBERG, Louis. Regularity in free boundary problems. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1977, vol. 4, no 2, p. 373-391. - http://www.numdam.org/item/ASNSP_1977_4_4_2_373_0/

  • KAUFMAN, Robert et WU, Jang-Mei. Parabolic measure on domains of class Lip $1\over 2$. Compositio Mathematica, 1988, vol. 65, no 2, p. 201-207. - http://eudml.org/doc/89889

  • Lewis, John L., and Margaret Anne Marie Murray. The method of layer potentials for the heat equation in time-varying domains. Vol. 545. American Mathematical Soc., 1995. -

  • LEWIS, John L. et NYSTRÖM, Kaj. On a parabolic symmetry problem. ev. Mat. Iberoam. 23 (2007), no. 2, pp. 513–536 - https://doi.org/10.4171/rmi/504

  • LEWIS, John L. et SILVER, Judy. Parabolic measure and the Dirichlet problem for the heat equation in two dimensions. Indiana University mathematics journal, 1988, vol. 37, no 4, p. 801-839. - https://www.jstor.org/stable/24895357

  • NYSTRÖM, Kaj. The Dirichlet problem for second order parabolic operators. Indiana University Mathematics Journal, 1997, p. 183-245. - http://www.jstor.org/stable/24899425

  • NYSTRÖM, Kaj. On blow-ups and the classification of global solutions to parabolic free boundary problems. Indiana University mathematics journal, 2006, p. 1233-1290. - https://www.jstor.org/stable/24902414

  • NYSTRÖM, Kaj. On an inverse type problem for the heat equation in parabolic regular graph domains. Mathematische Zeitschrift, 2012, vol. 270, no 1-2, p. 197-222. - http://dx.doi.org/10.1007/s00209-010-0793-3

  • STEIN, Elias M. Singular integrals and differentiability properties of functions. Princeton university press, 1970. - https://doi.org/10.1515/9781400883882

  • STRICHARTZ, Robert S. Bounded mean oscillation and Sobolev spaces. Indiana University Mathematics Journal, 1980, vol. 29, no 4, p. 539-558. - https://www.jstor.org/stable/24892879


Imagette Video

Sélection

Cochez un dossier pour ajouter l'enregistrement à vos favoris. Décochez-le pour l'en enlever.

4

gerer mes paniers

Signaler une erreur