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H 1 Stable phase transitions: from nonlocal to local

Auteurs : Serra, Joaquim (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : The talk will review the motivations, state of the art, recent results, and open questions on four very related PDE models related to phase transitions: Allen-Cahn, Peierls-Nabarro, Minimal surfaces, and Nonlocal Minimal surfaces. We will focus on the study of stable solutions (critical points of the corresponding energy functionals with nonnegative second variation). We will discuss new nonlocal results on stable phase transitions, explaining why the stability assumption gives stronger information in presence of nonlocal interactions. We will also comment on the open problems and obstructions in trying to make the nonlocal estimates robust as the long-range (or nonlocal) interactions become short-range (or local).

    Keywords : phase transition; nonlocal minimal surfaces; stability; short-range interaction; long-range interaction

    Codes MSC :
    35B35 - Stability of solutions of PDE
    49Q05 - Minimal surfaces
    53A10 - Minimal surfaces, surfaces with prescribed mean curvature
    82B26 - Phase transitions (general)
    35R11 - Fractional partial differential equations

    Ressources complémentaires :
    https://www.cirm-math.fr/ProgWeebly/Renc1862/Serra.pdf

    Informations sur la rencontre

    Nom de la rencontre : Non standard diffusions in fluids, kinetic equations and probability / Diffusions non standards en mécanique des fluides, équations cinétiques et probabilités
    Organisateurs de la rencontre : Imbert, Cyril ; Mouhot, Clément ; Tristani, Isabelle
    Dates : 10/12/2018 - 14/12/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1862.html

    Citation Data

    DOI : 10.24350/CIRM.V.19483203
    Cite this video as: Serra, Joaquim (2018). Stable phase transitions: from nonlocal to local. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19483203
    URI : http://dx.doi.org/10.24350/CIRM.V.19483203


    Voir aussi

    Bibliographie

    1. Cabré, X., Cinti, E., & Serra, J. (2018). Flatness of stable nonlocal phase transitions in in $\mathbb {R}^ 3$, forthcoming preprint -

    2. Cabré, X., Cinti, E., & Serra, J. (2017). Stable $s$-minimal cones in $\mathbb {R}^ 3$ are flat for $s\sim 1$.〈arXiv:1710.08722〉 - https://arxiv.org/abs/1710.08722

    3. Cinti, E., Serra, J., & Valdinoci, E. Quantitative flatness results and BV-estimates for nonlocal minimal surfaces, to appear in Journal of Differential Geometry -

    4. Dipierro, S., Serra, J., & Valdinoci, E. Improvement of flatness for nonlocal phase transitions, to appear in American Journal of Mathematics -

    5. Figalli, A., & Serra, J. (2017). On stable solutions for boundary reactions: a De Giorgi type result in dimension 4+1.〈arXiv:1705.02781〉 - https://arxiv.org/abs/1705.02781

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