00:00:00 / 00:00:00

Apparaît dans la collection : Non standard diffusions in fluids, kinetic equations and probability / Diffusions non standards en mécanique des fluides, équations cinétiques et probabilités

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).

Informations sur la vidéo

Données de citation

  • DOI 10.24350/CIRM.V.19483203
  • Citer cette vidéo Serra, Joaquim (12/12/2018). Stable phase transitions: from nonlocal to local. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19483203
  • URL https://dx.doi.org/10.24350/CIRM.V.19483203

Bibliographie

  • Cabré, X., Cinti, E., & Serra, J. (2018). Flatness of stable nonlocal phase transitions in in $\mathbb {R}^ 3$, forthcoming preprint
  • 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
  • Cinti, E., Serra, J., & Valdinoci, E. Quantitative flatness results and BV-estimates for nonlocal minimal surfaces, to appear in Journal of Differential Geometry
  • Dipierro, S., Serra, J., & Valdinoci, E. Improvement of flatness for nonlocal phase transitions, to appear in American Journal of Mathematics
  • 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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