Uniqueness of the minimizer for a random non-local functional with double-well potential in $ d\leq 2$

Dirr, Nicolas; Orlandi, Enza

doi:10.1016/j.anihpc.2014.02.002

Uniqueness of the minimizer for a random non-local functional with double-well potential in

d \leq 2

Dirr, Nicolas ; Orlandi, Enza

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 3, pp. 593-622

Résumé

We consider a small random perturbation of the energy functional

{[u]}_{H^{s} (Λ, ℝ^{d})}^{2} + \int_{Λ} W (u (x)) d x

for

s \in (0, 1)

, where the non-local part

{[u]}_{H^{s} (Λ, ℝ^{d})}^{2}

denotes the total contribution from

Λ \subset ℝ^{d}

in the

H^{s} (ℝ^{d})

Gagliardo semi-norm of u and W is a double well potential. We show that there exists, as Λ invades

ℝ^{d}

, for almost all realizations of the random term a minimizer under compact perturbations, which is unique when

d = 2

,

s \in (\frac{1}{2}, 1)

and when

d = 1

,

s \in [\frac{1}{4}, 1)

. This uniqueness is a consequence of the randomness. When the random term is absent, there are two minimizers which are invariant under translations in space,

u = \pm 1

.

MR Zbl

DOI : 10.1016/j.anihpc.2014.02.002

Classification : 35R60, 80M35, 82D30, 74Q05
Keywords: Random functionals, Phase segregation in disordered materials, Fractional Laplacian

@article{AIHPC_2015__32_3_593_0,
     author = {Dirr, Nicolas and Orlandi, Enza},
     title = {Uniqueness of the minimizer for a random non-local functional with double-well potential in $ d\leq 2$
      },
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {593--622},
     year = {2015},
     publisher = {Elsevier},
     volume = {32},
     number = {3},
     doi = {10.1016/j.anihpc.2014.02.002},
     zbl = {1320.35355},
     mrnumber = {3353702},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2014.02.002/}
}

TY  - JOUR
AU  - Dirr, Nicolas
AU  - Orlandi, Enza
TI  - Uniqueness of the minimizer for a random non-local functional with double-well potential in $ d\leq 2$
      
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 593
EP  - 622
VL  - 32
IS  - 3
PB  - Elsevier
UR  - https://www.numdam.org/articles/10.1016/j.anihpc.2014.02.002/
DO  - 10.1016/j.anihpc.2014.02.002
LA  - en
ID  - AIHPC_2015__32_3_593_0
ER  -

%0 Journal Article
%A Dirr, Nicolas
%A Orlandi, Enza
%T Uniqueness of the minimizer for a random non-local functional with double-well potential in $ d\leq 2$
      
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 593-622
%V 32
%N 3
%I Elsevier
%U https://www.numdam.org/articles/10.1016/j.anihpc.2014.02.002/
%R 10.1016/j.anihpc.2014.02.002
%G en
%F AIHPC_2015__32_3_593_0

Dirr, Nicolas; Orlandi, Enza. Uniqueness of the minimizer for a random non-local functional with double-well potential in $ d\leq 2$. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 3, pp. 593-622. doi: 10.1016/j.anihpc.2014.02.002

Bibliographie
Cité par

[1] M. Aizenman, J. Wehr, Rounding effects on quenched randomness on first-order phase transitions, Commun. Math. Phys. 130 (1990), 489 -528 | MR | Zbl

[2] A. Bovier, Statistical Mechanics of Disordered Systems. A Mathematical Perspective, Cambridge University Press (2012) | MR | Zbl

[3] D. Brockmann, I.M. Sokolov, Lévy flights in external force fields: from models to equations, Chem. Phys. 284 (2002), 409 -421

[4] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521 -573 | MR | Zbl

[5] P.D. Ditlevsen, Anomalous jumping in a double-well potential, Phys. Rev. E 60 (1999), 172 -179

[6] N. Dirr, E. Orlandi, Sharp-interface limit of a Ginzburg–Landau functional with a random external field, SIAM J. Math. Anal. 41 (2009), 781 -824 | MR | Zbl

[7] N. Dirr, E. Orlandi, Unique minimizer for a random functional with double well potential in dimension 1 and 2, Commun. Math. Sci. 1 (2011), 331 -351 | Zbl | MR

[8] A. Garroni, S. Müller, A variational model for dislocation in the line tension limit, Arch. Ration. Mech. Anal. 81 (2006), 535 -578 | MR | Zbl

[9] A. Garroni, G. Palatucci, A singular perturbation result with a fractional norm, Variational Problem in Material Sciences, Prog. Nonlinear Differ. Equ. Appl. vol. 68 (2006), 111 -126 | MR | Zbl

[10] M.D.M. Gonzales, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differ. Equ. 36 (2009), 173 -210 | MR

[11] P. Hall, C.C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York (1980) | MR | Zbl

[12] G. Keller, Equilibrium States in Ergodic Theory, London Math. Soc. Stud. Texts vol. 42 (1998) | MR | Zbl

[13] R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamical approach, Phys. Rep. 339 (2000), 1 -77 | MR | Zbl

[14] G. Palatucci, O. Savin, E. Valdinoci, Local and global minimizers for a variational energy involving fractional norm, Ann. Mat. Pura Appl. 92 (2013), 673 -718 | MR | Zbl

[15] R. Cont, P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser. , Chapman & Hall, CRC, Boca Raton (2004) | MR | Zbl

[16] O. Savin, E. Valdinoci, Γ-convergence for nonlocal phase transitions, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 29 (2012), 479 -500 | MR | Zbl | Numdam

[17] O. Savin, E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, J. Math. Pures Appl. 9 (2014), 1 -26 | MR | Zbl

[18] R. Servadei, E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat. 58 (2014), 133 -154 | MR | Zbl

[19] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplacian operator, Commun. Pure Appl. Math. 60 (2007), 67 -112 | MR | Zbl

[20] B.J. West, P. Grigolini, R. Metzler, T. Nonnenmacher, Fractional diffusion and Lévy stable processes, Phys. Rev. E 55 (1997), 99 -106 | MR

Cité par Sources :