Γ-convergence of concentration problems
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, p. 151-179

In this paper, we use Γ-convergence techniques to study the following variational problem S ε F (Ω):=supε -2 * Ω F(u)dx: Ω |u| 2 dxε 2 ,u=0 on Ω, where 0F(t)|t| 2 * , with 2 * =2n n-2, and Ω is a bounded domain of n , n3. We obtain a Γ-convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem S ε F (Ω). Finally, a second order expansion in Γ-convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.

Classification:  35J60,  35C20,  49J45
@article{ASNSP_2003_5_2_1_151_0,
     author = {Amar, Micol and Garroni, Adriana},
     title = {$\Gamma $-convergence of concentration problems},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola normale superiore},
     volume = {Ser. 5, 2},
     number = {1},
     year = {2003},
     pages = {151-179},
     zbl = {1121.35048},
     mrnumber = {1990977},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_151_0}
}
Amar, Micol; Garroni, Adriana. $\Gamma $-convergence of concentration problems. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, pp. 151-179. http://www.numdam.org/item/ASNSP_2003_5_2_1_151_0/

[1] A. Bahri, “Critical points at infinity in some variational problems”, Vol. 182 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1989. | MR 1019828 | Zbl 0676.58021

[2] C. Bandle - M. Flucher, Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s equations ΔU=e U and ΔU=U (n+2)/(n-2) , Siam Rev., 38 (1996), 191-238. | MR 1391227 | Zbl 0857.35034

[3] F. Bethuel - H. Brézis - F. Hélein, Ginzburg-Landau vortices, In: “Progress in Nonlinear Differential Equations and their Applications”, 13, Birkhäuser Boston Inc., Boston, MA, 1994. | MR 1269538 | Zbl 0802.35142

[4] A. Braides, Γ-convergence for beginners, To appear, 2002. | MR 1968440 | Zbl pre01865939

[5] H. Brézis - L. Niremberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477. | MR 709644 | Zbl 0541.35029

[6] H. Brézis - L. A. Peletier, Asymptotic for elliptic equations involving critical growth, In: “Partial differential equations and the calculus of variation”, Vol. I, 1 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 1989, 149-192. | MR 1034005 | Zbl 0685.35013

[7] G. Dal Maso, “An introduction to Γ-convergence”, Birkhäuser, Boston, 1992. | MR 1201152 | Zbl 0816.49001

[8] G. Dal Maso - A. Malusa, Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains, Proc. Royal Soc. Edinburgh 125A (1995), 99-114. | MR 1318625 | Zbl 0828.35007

[9] E. De Giorgi - T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 842-850. | MR 448194 | Zbl 0339.49005

[10] E. De Giorgi - T. Franzoni, Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia 3 (1979), 63-101.

[11] M. Flucher, “Variational problems with concentration”, Birkhäuser, 1999. | MR 1711532 | Zbl 0940.35006

[12] M. Flucher - A. Garroni - S. Müller, Concentration of low energy extremals: identification of concentration points, Calc. Var. Partial Differential Equations 14 (2002), 483-516. | MR 1911826 | Zbl 1004.35040

[13] M. Flucher - S. Müller, Radial simmetry and decay rate of variational ground states in the zero mass case, Siam J. Math. Anal. 29 (1998), 712-719. | MR 1617704 | Zbl 0908.35005

[14] M. Flucher - S. Müller, Concentration of low energy extremals, Ann. Inst. H. Poincaré Anal. Non Linéaire (3) 10 (1999), 269-298. | Numdam | MR 1687286 | Zbl 0938.35042

[15] A. Garroni - S. Müller, Concentration phenomena for the volume functional in unbounded domains: identification of concentration points, to appear on J. Funct. Anal., 2002 | MR 1971258 | Zbl pre01915833

[16] Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 2 (1991), 159-174. | Numdam | MR 1096602 | Zbl 0729.35014

[17] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case., Rev. Mat. Iberoamericana 1 (1985), 145-201. | MR 834360 | Zbl 0704.49005

[18] O. Rey, Proof of two conjecture of H. Brezis and L.A. Peletier, Manuscripta Math. (1) 65 (1989), 19-37. | MR 1006624 | Zbl 0708.35032

[19] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 2 (1984), 479-495. | MR 788292 | Zbl 0576.53028