Asymptotics for vectorial Allen–Cahn type problems
Fabrice Bethuel1
1 Laboratoire Jacques-Louis Lions Sorbonne Université 4 place Jussieu 75252 Paris Cedex 5 France
Journées équations aux dérivées partielles (2023), Exposé no. 1, 16 p.

These notes present some recent results concerning the convergence of solutions to the elliptic vectorial Allen–Cahn equation in dimension two as the parameter ε tends to zero, and its connections to minimal surface theory in the weak sense of stationary varifolds. We first describe the results obtained so far in the scalar theory, which can be considered as quite satisfactory, and provide some ideas about the proofs and their main steps. We then present some adaptations necessary to handle the vectorial case in dimension two.

Publié le :
DOI : 10.5802/jedp.672
Fabrice Bethuel 1

1 Laboratoire Jacques-Louis Lions Sorbonne Université 4 place Jussieu 75252 Paris Cedex 5 France 
@incollection{JEDP_2023____A1_0,
     author = {Fabrice Bethuel},
     title = {Asymptotics for vectorial {Allen{\textendash}Cahn} type problems},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     note = {talk:1},
     pages = {1--16},
     publisher = {R\'eseau th\'ematique AEDP du CNRS},
     year = {2023},
     doi = {10.5802/jedp.672},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.672/}
}
TY  - JOUR
AU  - Fabrice Bethuel
TI  - Asymptotics for vectorial Allen–Cahn type problems
JO  - Journées équations aux dérivées partielles
N1  - talk:1
PY  - 2023
SP  - 1
EP  - 16
PB  - Réseau thématique AEDP du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.672/
DO  - 10.5802/jedp.672
LA  - en
ID  - JEDP_2023____A1_0
ER  - 
%0 Journal Article
%A Fabrice Bethuel
%T Asymptotics for vectorial Allen–Cahn type problems
%J Journées équations aux dérivées partielles
%Z talk:1
%D 2023
%P 1-16
%I Réseau thématique AEDP du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.672/
%R 10.5802/jedp.672
%G en
%F JEDP_2023____A1_0
Fabrice Bethuel. Asymptotics for vectorial Allen–Cahn type problems. Journées équations aux dérivées partielles (2023), Exposé no. 1, 16 p. doi : 10.5802/jedp.672. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.672/

[1] Stanley Alama; Lia Bronsard; Changfeng Gui Stationary layered solutions in ℝ 2 for an Allen–Cahn system with multiple well potential, Calc. Var. Partial Differ. Equ., Volume 5 (1997) no. 4, pp. 359-390 | DOI | MR | Zbl

[2] Nicholas D. Alikakos; Santiago I. Betelú; Xinfu Chen Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energy densities, Eur. J. Appl. Math., Volume 17 (2006) no. 5, pp. 525-556 | DOI | MR | Zbl

[3] William. K. Allard; Frederick J. Almgren The structure of stationary one dimensional varifolds with positive density, Invent. Math., Volume 34 (1976), pp. 83-97 | DOI | MR | Zbl

[4] Sisto Baldo Minimal interface criterion for phase transitions in mixtures of Cahn- Hilliard fluids, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 7 (1990) no. 2, pp. 67-90 | DOI | Numdam | MR | Zbl

[5] Fabrice Bethuel Asymptotics for two-dimensional vectorial Allen–Cahn systems (2020) (to appear in Acta Math.) | arXiv

[6] Fabrice Bethuel; Ramon Oliver-Bonafoux Pseudo-profiles for vectorial Allen–Cahn systems (2023) (work in progress)

[7] Lia Bronsard; Changfeng Gui; Michelle Schatzman A three-layered minimizer in ℝ 2 for a variational problem with a symmetric three-well potential, Commun. Pure Appl. Math., Volume 49 (1996) no. 7, pp. 677-715 | DOI | Zbl

[8] Irene Fonseca; Luc Tartar The gradient theory of phase transitions for systems with two potential wells, Proc. R. Soc. Edinb., Sect. A, Math., Volume 111 (1989) no. 1-2, pp. 89-102 | DOI | MR | Zbl

[9] John E. Hutchinson; Yoshihiro Tonegawa Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory., Calc. Var. Partial Differ. Equ., Volume 10 (2000) no. 1, pp. 49-84 | DOI | MR | Zbl

[10] Tom Ilmanen Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature, J. Differ. Geom., Volume 38 (1993) no. 2, pp. 417-461 | DOI | MR | Zbl

[11] Luciano Modica A gradient bound and a Liouville theorem for nonlinear Poisson equations, Commun. Pure Appl. Math., Volume 38 (1985), pp. 679-684 | DOI | Zbl

[12] Luciano Modica; Stefano Mortola Un esempio di Γ - -convergenza, Boll. Unione Mat. Ital., V. Ser., B, Volume 14 (1977), pp. 285-299 | MR | Zbl

[13] Antonin Monteil; Filippo Santambrogio Metric methods for heteroclinic connections, Math. Methods Appl. Sci., Volume 41 (2018) no. 3, pp. 1019-1024 | DOI | MR | Zbl

[14] Ramon Oliver-Bonafoux Non-minimizing connecting orbits for multi-well systems, Calc. Var. Partial Differ. Equ., Volume 61 (2022) no. 2, p. 27 (Id/No 69) | DOI | MR | Zbl

[15] Frank Pacard; Juncheng Wei Stable solutions of the Allen–Cahn equation in dimension 8 and minimal cones, J. Funct. Anal., Volume 264 (2013) no. 5, pp. 1131-1167 | DOI | MR | Zbl

[16] David Preiss Geometry of measures in R n : Distribution, rectifiability, and densities, Ann. Math., Volume 125 (1987), pp. 537-643 | DOI | Zbl

[17] Peter Sternberg; William P. Zeimer Local minimisers of a three-phase partition problem with triple junctions, Proc. R. Soc. Edinb., Sect. A, Math., Volume 124 (1994) no. 6, pp. 1059-1073 | DOI | MR | Zbl

[18] Andres Zuniga; Peter Sternberg On the heteroclinic connection problem for multi-well gradient systems, J. Differ. Equations, Volume 261 (2016) no. 7, pp. 3987-4007 | DOI | MR | Zbl

Cité par Sources :