staple
Avec cedram.org
logo JEP
Table des matières de ce volume | Article précédent | Article suivant
Matteo Cozzi; Enrico Valdinoci
Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium
(Minimiseurs proches d’un plan pour une énergie non locale de type Ginzburg-Landau dans un milieu périodique)
Journal de l'École polytechnique — Mathématiques, 4 (2017), p. 337-388, doi: 10.5802/jep.45
Article PDF | TeX source
Class. Math.: 35R11, 35A15, 35B08, 82B26, 35B65
Mots clés: Énergies non locales, transitions de phase, minimiseurs de type plan, laplacien fractionnaire

Résumé - Abstract

Nous considérons une équation de transition de phase non locale dans un milieu périodique et nous construisons des solutions dont l’interface se trouve dans un domaine de direction prescrite et de largeur universelle. Les solutions construites jouissent aussi d’une propriété de minimalité locale par rapport à une certaine fonctionnelle d’énergie non locale.

Bibliographie

[AB06] F. Auer & V. Bangert, “Differentiability of the stable norm in codimension one”, Amer. J. Math. 128 (2006) no. 1, p. 215-238 Article
[BBM01] J. Bourgain, H. Brezis & P. Mironescu, Another look at Sobolev spaces, in J. L. Menaldi, E. Rofman, A. Sulem, éd., Optimal control and partial differential equations (Paris, 2000), IOS Press, 2001, p. 439–455
[BL17] L. Brasco & E. Lindgren, “Higher Sobolev regularity for the fractional $p$-Laplace equation in the superquadratic case”, Adv. Math. 304 (2017), p. 300-354 Article
[Bre11] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011
[BV08] I. Birindelli & E. Valdinoci, “The Ginzburg-Landau equation in the Heisenberg group”, Commun. Contemp. Math. 10 (2008) no. 5, p. 671-719 Article
[CC06] L. A. Caffarelli & A. Córdoba, “Phase transitions: uniform regularity of the intermediate layers”, J. reine angew. Math. 593 (2006), p. 209-235
[CC14] X. Cabré & E. Cinti, “Sharp energy estimates for nonlinear fractional diffusion equations”, Calc. Var. Partial Differential Equations 49 (2014) no. 1-2, p. 233-269 Article
[CC95] L. A. Caffarelli & A. Córdoba, “Uniform convergence of a singular perturbation problem”, Comm. Pure Appl. Math. 48 (1995) no. 1, p. 1-12 Article
[CdlL01] L. A. Caffarelli & R. de la Llave, “Planelike minimizers in periodic media”, Comm. Pure Appl. Math. 54 (2001) no. 12, p. 1403-1441 Article
[Coz16] M. Cozzi, “Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes”, arXiv:1609.09277, 2016
[Coz17] M. Cozzi, “Interior regularity of solutions of non-local equations in Sobolev and Nikol’skii spaces”, Ann. Mat. Pura Appl. (2017), online: doi:10.1007/s10231-016-0586-3 Article
[CS09] L. A. Caffarelli & L. Silvestre, “Regularity theory for fully nonlinear integro-differential equations”, Comm. Pure Appl. Math. 62 (2009) no. 5, p. 597-638 Article |  MR 2494809
[CS11] L. A. Caffarelli & L. Silvestre, “Regularity results for nonlocal equations by approximation”, Arch. Rational Mech. Anal. 200 (2011) no. 1, p. 59-88 Article
[CV17] M. Cozzi & E. Valdinoci, “Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media”, preprint, 2017
[DCKP14] A. Di Castro, T. Kuusi & G. Palatucci, “Nonlocal Harnack inequalities”, J. Funct. Anal. 267 (2014) no. 6, p. 1807-1836 Article
[DCKP16] A. Di Castro, T. Kuusi & G. Palatucci, “Local behavior of fractional $p$-minimizers”, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) no. 5, p. 1279-1299 Article
[DFV14] S. Dipierro, A. Figalli & E. Valdinoci, “Strongly nonlocal dislocation dynamics in crystals”, Comm. Partial Differential Equations 39 (2014) no. 12, p. 2351-2387 Article
[DK15] B. Dyda & M. Kassmann, “Regularity estimates for elliptic nonlocal operators”, arXiv:1509.08320v2, 2015
[dlLV07] R. de la Llave & E. Valdinoci, “Multiplicity results for interfaces of Ginzburg-Landau-Allen-Cahn equations in periodic media”, Adv. Math. 215 (2007) no. 1, p. 379-426 Article
[DNPV12] E. Di Nezza, G. Palatucci & E. Valdinoci, “Hitchhiker’s guide to the fractional Sobolev spaces”, Bull. Sci. Math. 136 (2012) no. 5, p. 521-573 Article
[DPV15] S. Dipierro, G. Palatucci & E. Valdinoci, “Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting”, Comm. Math. Phys. 333 (2015) no. 2, p. 1061-1105 Article |  MR 3296170
[Dáv13] G. Dávila, “Plane-like minimizers for an area-Dirichlet integral”, Arch. Rational Mech. Anal. 207 (2013) no. 3, p. 753-774 Article
[Fri12] A. Friedman, “PDE problems arising in mathematical biology”, Netw. Heterog. Media 7 (2012) no. 4, p. 691-703 Article
[GM12] M. Giaquinta & L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) 11, Edizioni della Normale, Pisa, 2012
[Hed32] G. A. Hedlund, “Geodesics on a two-dimensional Riemannian manifold with periodic coefficients”, Ann. of Math. (2) 33 (1932) no. 4, p. 719-739 Article
[Kas09] M. Kassmann, “A priori estimates for integro-differential operators with measurable kernels”, Calc. Var. Partial Differential Equations 34 (2009) no. 1, p. 1-21 Article
[Kas11] M. Kassmann, “Harnack inequalities and Hölder regularity estimates for nonlocal operators revisited”, available at http://www.math.uni-bielefeld.de/sfb701/files/preprints/sfb11015.pdf, 2011
[Mat90] J. N. Mather, “Differentiability of the minimal average action as a function of the rotation number”, Bol. Soc. Brasil. Mat. (N.S.) 21 (1990) no. 1, p. 59-70 Article
[Nab97] F. R. N. Nabarro, “Fifty-year study of the Peierls-Nabarro stress”, Mater. Sci. Eng. A 234 (1997), p. 67-76 Article
[NV07] M. Novaga & E. Valdinoci, “The geometry of mesoscopic phase transition interfaces”, Discrete Contin. Dynam. Systems 19 (2007) no. 4, p. 777-798 Article
[Pon04] A. C. Ponce, “An estimate in the spirit of Poincaré’s inequality”, J. Eur. Math. Soc. (JEMS) 6 (2004) no. 1, p. 1-15 Article
[PSV13] G. Palatucci, O. Savin & E. Valdinoci, “Local and global minimizers for a variational energy involving a fractional norm”, Ann. Mat. Pura Appl. (4) 192 (2013) no. 4, p. 673-718 Article |  MR 3081641
[PV05] A. Petrosyan & E. Valdinoci, “Geometric properties of Bernoulli-type minimizers”, Interfaces Free Bound. 7 (2005) no. 1, p. 55-77 Article
[Sil06] L. Silvestre, “Hölder estimates for solutions of integro-differential equations like the fractional Laplace”, Indiana Univ. Math. J. 55 (2006) no. 3, p. 1155-1174 Article
[SV12] O. Savin & E. Valdinoci, “$\Gamma $-convergence for nonlocal phase transitions”, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012) no. 4, p. 479-500 Article
[SV13] R. Servadei & E. Valdinoci, “Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators”, Rev. Mat. Iberoamericana 29 (2013) no. 3, p. 1091-1126 Article
[SV14] O. Savin & E. Valdinoci, “Density estimates for a variational model driven by the Gagliardo norm”, J. Math. Pures Appl. (9) 101 (2014) no. 1, p. 1-26 Article
[SV14] R. Servadei & E. Valdinoci, “Weak and viscosity solutions of the fractional Laplace equation”, Publ. Mat. 58 (2014) no. 1, p. 133-154 Article
[Val04] E. Valdinoci, “Plane-like minimizers in periodic media: jet flows and Ginzburg-Landau-type functionals”, J. reine angew. Math. 574 (2004), p. 147-185