Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium
Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 337-388.

We consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality property with respect to a suitable non-local energy functional.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.45
Classification: 35R11, 35A15, 35B08, 82B26, 35B65
Keywords: Non-local energies, phase transitions, plane-like minimizers, fractional Laplacian
Matteo Cozzi 1; Enrico Valdinoci 2

1 BGSMath Barcelona Graduate School of Mathematics and Departament de Matemàtiques, Universitat Politècnica de Catalunya Diagonal 647, E-08028 Barcelona (Spain)
2 Weierstraß Institut für Angewandte Analysis und Stochastik Mohrenstraße 39, D-10117 Berlin (Germany) and Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano (Italy) and School of Mathematics and Statistics, University of Melbourne Grattan Street, Parkville, VIC-3010 Melbourne (Australia)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JEP_2017__4__337_0,
     author = {Matteo Cozzi and Enrico Valdinoci},
     title = {Plane-like minimizers for {a~non-local~Ginzburg-Landau-type} energy in~a~periodic~medium},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {337--388},
     publisher = {\'Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.45},
     zbl = {06754330},
     mrnumber = {3623357},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.45/}
}
TY  - JOUR
AU  - Matteo Cozzi
AU  - Enrico Valdinoci
TI  - Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2017
SP  - 337
EP  - 388
VL  - 4
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.45/
DO  - 10.5802/jep.45
LA  - en
ID  - JEP_2017__4__337_0
ER  - 
%0 Journal Article
%A Matteo Cozzi
%A Enrico Valdinoci
%T Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium
%J Journal de l’École polytechnique — Mathématiques
%D 2017
%P 337-388
%V 4
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.45/
%R 10.5802/jep.45
%G en
%F JEP_2017__4__337_0
Matteo Cozzi; Enrico Valdinoci. Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium. Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 337-388. doi : 10.5802/jep.45. https://jep.centre-mersenne.org/articles/10.5802/jep.45/

[AB06] F. Auer & V. Bangert - “Differentiability of the stable norm in codimension one”, Amer. J. Math. 128 (2006) no. 1, p. 215-238 | DOI | MR | Zbl

[BBM01] J. Bourgain, H. Brezis & P. Mironescu - “Another look at Sobolev spaces”, in Optimal control and partial differential equations (Paris, 2000) (J. L. Menaldi, E. Rofman & A. Sulem, eds.), IOS Press, Amsterdam, 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 | DOI | MR | Zbl

[Bre11] H. Brezis - Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011 | Zbl

[BV08] I. Birindelli & E. Valdinoci - “The Ginzburg-Landau equation in the Heisenberg group”, Commun. Contemp. Math. 10 (2008) no. 5, p. 671-719 | DOI | MR | Zbl

[CC95] L. A. Caffarelli & A. Córdoba - “Uniform convergence of a singular perturbation problem”, Comm. Pure Appl. Math. 48 (1995) no. 1, p. 1-12 | DOI | MR | Zbl

[CC06] L. A. Caffarelli & A. Córdoba - “Phase transitions: uniform regularity of the intermediate layers”, J. reine angew. Math. 593 (2006), p. 209-235 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[CdlL01] L. A. Caffarelli & R. de la Llave - “Planelike minimizers in periodic media”, Comm. Pure Appl. Math. 54 (2001) no. 12, p. 1403-1441 | DOI | MR | Zbl

[Coz16] M. Cozzi - “Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes” (2016), arXiv:1609.09277

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[CS11] L. A. Caffarelli & L. Silvestre - “Regularity results for nonlocal equations by approximation”, Arch. Rational Mech. Anal. 200 (2011) no. 1, p. 59-88 | DOI | MR | Zbl

[CV17] M. Cozzi & E. Valdinoci - “Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media” (2017), preprint | Zbl

[DCKP14] A. Di Castro, T. Kuusi & G. Palatucci - “Nonlocal Harnack inequalities”, J. Funct. Anal. 267 (2014) no. 6, p. 1807-1836 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[DFV14] S. Dipierro, A. Figalli & E. Valdinoci - “Strongly nonlocal dislocation dynamics in crystals”, Comm. Partial Differential Equations 39 (2014) no. 12, p. 2351-2387 | DOI | MR | Zbl

[DK15] B. Dyda & M. Kassmann - “Regularity estimates for elliptic nonlocal operators” (2015), arXiv:1509.08320v2

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[Dáv13] G. Dávila - “Plane-like minimizers for an area-Dirichlet integral”, Arch. Rational Mech. Anal. 207 (2013) no. 3, p. 753-774 | DOI | MR | Zbl

[Fri12] A. Friedman - “PDE problems arising in mathematical biology”, Netw. Heterog. Media 7 (2012) no. 4, p. 691-703 | DOI | MR | Zbl

[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), vol. 11, Edizioni della Normale, Pisa, 2012 | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[Kas11] M. Kassmann - “Harnack inequalities and Hölder regularity estimates for nonlocal operators revisited” (2011), available at http://www.math.uni-bielefeld.de/sfb701/files/preprints/sfb11015.pdf

[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 | DOI | MR | Zbl

[Nab97] F. R. N. Nabarro - “Fifty-year study of the Peierls-Nabarro stress”, Mater. Sci. Eng. A 234 (1997), p. 67-76 | DOI

[NV07] M. Novaga & E. Valdinoci - “The geometry of mesoscopic phase transition interfaces”, Discrete Contin. Dynam. Systems 19 (2007) no. 4, p. 777-798 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[PV05] A. Petrosyan & E. Valdinoci - “Geometric properties of Bernoulli-type minimizers”, Interfaces Free Bound. 7 (2005) no. 1, p. 55-77 | DOI | MR | Zbl

[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 | DOI | Zbl

[SV12] O. Savin & E. Valdinoci - “Γ-convergence for nonlocal phase transitions”, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012) no. 4, p. 479-500 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[SV14] R. Servadei & E. Valdinoci - “Weak and viscosity solutions of the fractional Laplace equation”, Publ. Mat. 58 (2014) no. 1, p. 133-154 | DOI | MR | Zbl

[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 | MR | Zbl

Cited by Sources: