Γ-convergence for nonlocal phase transitions
Savin, Ovidiu ; Valdinoci, Enrico
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 479-500 / Harvested from Numdam
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We discuss the Γ-convergence, under the appropriate scaling, of the energy functional ∥u∥ H s (Ω) 2 +∫ ΩW(u)dx, with s∈(0,1), where ∥u∥ H s (Ω) denotes the total contribution from Ω in the H s norm of u, and W is a double-well potential.When s∈[1/2,1), we show that the energy Γ-converges to the classical minimal surface functional – while, when s∈(0,1/2), it is easy to see that the functional Γ-converges to the nonlocal minimal surface functional.

@article{AIHPC_2012__29_4_479_0,
     author = {Savin, Ovidiu and Valdinoci, Enrico},
     title = {$\Gamma$-convergence for nonlocal phase transitions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {29},
     year = {2012},
     pages = {479-500},
     doi = {10.1016/j.anihpc.2012.01.006},
     mrnumber = {2948285},
     zbl = {1253.49008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2012__29_4_479_0}
}

Savin, Ovidiu; Valdinoci, Enrico. Γ-convergence for nonlocal phase transitions. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) pp. 479-500. doi : 10.1016/j.anihpc.2012.01.006. http://gdmltest.u-ga.fr/item/AIHPC_2012__29_4_479_0/

[1] G. Alberti, personal communication, 2010.

[2] G. Alberti, G. Bellettini, A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies, European J. Appl. Math. 9 no. 3 (1998), 261-284 | MR 1634336 | Zbl 0932.49018

[3] G. Alberti, G. Bellettini, M. Cassandro, E. Presutti, Surface tension in Ising systems with Kac potentials, J. Statist. Phys. 82 no. 3–4 (1996), 743-796 | MR 1372427 | Zbl 1042.82539

[4] G. Alberti, G. Bouchitté, P. Seppecher, Un résultat de perturbations singulières avec la norme H 1/2 , C. R. Acad. Sci. Paris Sér. I Math. 319 no. 4 (1994), 333-338 | MR 1289307 | Zbl 0845.49008

[5] G. Alberti, G. Bouchitté, P. Seppecher, Phase transition with the line-tension effect, Arch. Rational Mech. Anal. 144 no. 1 (1998), 1-46 | MR 1657316 | Zbl 0915.76093

[6] L. Ambrosio, G. de Philippis, L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, preprint, 2010. | MR 2765717 | Zbl 1207.49051

[7] A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications vol. 22 (2002) | MR 1968440 | Zbl 1198.49001

[8] L.A. Caffarelli, J.-M. Roquejoffre, O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 no. 9 (2010), 1111-1144 | MR 2675483 | Zbl 1248.53009

[9] L.A. Caffarelli, P.E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal. 195 no. 1 (2010), 1-23 | MR 2564467 | Zbl 1190.65132

[10] L.A. Caffarelli, E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations 41 no. 1–2 (2011), 203-240 | MR 2782803 | Zbl 05884582

[11] G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications vol. 8, Birkhäuser Boston, Inc., Boston, MA (1993) | MR 1201152

[12] E. De Giorgi, Sulla convergenza di alcune successioni dʼintegrali del tipo dellʼarea, Rend. Mat. (6) 8 (1975), 277-294 | MR 375037 | Zbl 0316.35036

[13] E. De Giorgi, T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 no. 6 (1975), 842-850 | MR 448194 | Zbl 0339.49005

[14] K.J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics vol. 85, Cambridge University Press, Cambridge (1986) | MR 867284 | Zbl 0599.28009

[15] M. Focardi, Aperiodic fractional obstacle problems, Adv. Math. 225 no. 6 (2010), 3502-3544 | MR 2729014 | Zbl 1213.49022

[16] A. Garroni, S. Müller, Γ-limit of a phase-field model of dislocations, SIAM J. Math. Anal. 36 no. 6 (2005), 1943-1964 | MR 2178227 | Zbl 1094.82008

[17] A. Garroni, S. Müller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal. 181 no. 3 (2006), 535-578 | MR 2231783 | Zbl 1158.74365

[18] A. Garroni, G. Palatucci, A singular perturbation result with a fractional norm, Variational Problems in Material Science, Progress in Nonlinear Differential Equations and Their Applications vol. 68, Birkhäuser, Basel (2006), 111-126 | MR 2223366 | Zbl 1107.82019

[19] E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics vol. 80, Birkhäuser Verlag, Basel (1984) | MR 775682 | Zbl 0545.49018

[20] M.D.M. González, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differential Equations 36 no. 2 (2009), 173-210 | MR 2546026 | Zbl 1175.49013

[21] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal. 98 no. 2 (1987), 123-142 | MR 866718 | Zbl 0616.76004

[22] G. Palatucci, O. Savin, E. Valdinoci, Local and global minimizers for a variational energy involving a fractional norm, preprint, 2010 (available from arxiv). | MR 3081641

[23] O. Savin, E. Valdinoci, Density estimates for a variational model driven by the Gagliardo norm, preprint, 2010. | MR 3133422

[24] L. Tartar, The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana vol. 7, Springer-Verlag, UMI, Berlin, Bologna (2009) | MR 2582099 | Zbl 1188.35004