$\Gamma$-convergence of concentration problems

Amar, Micol; Garroni, Adriana

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 151-179.

In this paper, we use $\Gamma$-convergence techniques to study the following variational problem

$$S_{\epsilon }^F\left(\Omega \right):=sup\left\{{\epsilon }^{-2^{*}}{\int }_{\Omega }F\left(u\right)dx:{\int }_{\Omega }{\left|\nabla u\right|}^{2}dx\le {\epsilon }^2,u=0\mathrm{on}\partial \Omega \right\},$$

where $0\le F\left(t\right)\le {\left|t\right|}^{2^{*}}$, with $2^{*}=\frac{2n}{n-2}$, and $\Omega$ is a bounded domain of ${ℝ}^n$, $n\ge 3$. We obtain a $\Gamma$-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 $S_{\epsilon }^F\left(\Omega \right)$. Finally, a second order expansion in $\Gamma$-convergence permits to identify the concentration points of the maximizing sequences, also in some non-homogeneous case.

[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 | Zbl

[2] C. Bandle - M. Flucher, Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s equations $\Delta U={\mathrm{e}}^U$ and $\Delta U=U^{(n+2)/(n-2)}$, Siam Rev., 38 (1996), 191-238. | MR | Zbl

[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 | Zbl

[4] A. Braides, $\Gamma$-convergence for beginners, To appear, 2002. | MR

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

[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 | Zbl

[7] G. Dal Maso, “An introduction to $\Gamma$-convergence”, Birkhäuser, Boston, 1992. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

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

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

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