Analyse limite d’équations variationnelles dans un domaine contenant une grille. (Limit analysis of variational equations in a domain containing a sieve). (French) Zbl 0626.49008

Let \(\Omega\) be a bounded domain of \({\mathbb{R}}^ n(n\geq 3)\) and let \(\Sigma\) be an (n-1)-dimensional hyperplane in \({\mathbb{R}}^ n\) on which many small holes are periodically distributed. Every hole is obtained by rescaling a “unitary hole” \(T\subseteq {\mathbb{R}}^{n-1}\) with ratio \(r_{\epsilon}<\epsilon (diam T)^{-1}\) and we denote by \(T_{\epsilon}\) the union of all holes and by \(\Omega_ 1\), \(\Omega_ 2\) the two components of \(\Omega\)-\(\Sigma\). The problems considered are of Dirichlet and Neumann type \[ (D_{\epsilon})\quad -\Delta u=f\quad on\quad \Omega -T_{\epsilon};\quad u=0\quad on\quad \partial \Omega \cup T_{\epsilon} \]
\[ (N_{\epsilon})\quad -\Delta v+v=f\quad on\quad \Omega_{\epsilon}=\Omega_ 1\cup \Omega_ 2\cup T_{\epsilon};\quad \partial v/\partial \nu =0\quad on\quad \partial \Omega \cup (\Sigma -T_{\epsilon}) \] and the aim is to study the asymptotic behaviour (as \(\epsilon\to 0)\) of the solutions \(u_{\epsilon}\) of \((D_{\epsilon})\) and \(v_{\epsilon}\) of \((N_{\epsilon})\). The main tool is the \(\Gamma\)-convergence theory applied to the minimum problems \[ (M^ D_{\epsilon})\quad \min \{\int_{\Omega}(| Du|^ 2 2fu)dx:u\in H^ 1_ 0(\Omega),\quad u=0\quad on\quad T_{\epsilon}\} \]
\[ (M^ N_{\epsilon})\quad \min \{\int_{\Omega_{\epsilon}}(| Dv|^ 2+v^ 2-2fv)dx\quad: v\in H^ 1(\Omega_{\epsilon})\}. \] The author proves that the limit minimum problems are \[ (M^ D_ 0)\quad \min \{\int_{\Omega}(| Du|^ 2-2fu)dx\quad +\quad aC\int_{\Omega \cap \Sigma}u^ 2d\sigma \quad: u\in H^ 1_ 0(\Omega)\} \]
\[ (M^ N_ 0)\quad \min \{\int_{\Omega_ 1\cup \Omega_ 2}(| Dv|^ 2+v^ 2-2fv)dx\quad +\frac{aC}{4}\int_{\Omega \cap \Sigma}[v]^ 2d\sigma \quad: v\in H^ 1(\Omega_ 1\cup \Omega_ 2)\} \] where [v] is the jump of v along \(\Sigma\), C is the capacity of T, that is \(C=\inf \{\int_{{\mathbb{R}}^ n}| Dv|^ 2dx:\) \(v\in H^ 1({\mathbb{R}}^ n)\), \(v=1\) on \(T\}\), and \(a=\lim_{\epsilon \to 0}r_{\epsilon}^{n- 2}/\epsilon^{n-1}\). The paper contains also the study of some unilateral problems associated to \((D_{\epsilon})\) and \((N_{\epsilon})\), and in some particular cases the rate of convergence for \(u_{\epsilon}\) and \(v_{\epsilon}\) are determined.
Reviewer: G.Buttazzo


49J45 Methods involving semicontinuity and convergence; relaxation
35J25 Boundary value problems for second-order elliptic equations
49J40 Variational inequalities
35B40 Asymptotic behavior of solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
Full Text: DOI EuDML


[1] H. ATTOUCH, Variational convergence for fonctions and operators, Applicable Math. Series, Pitman, London (1984). Zbl0561.49012 · Zbl 0561.49012
[2] H. TTOUCH, A. DAMLAMIAN, F. MURAT, C. PICARD, The Neumann Sieve (à paraître).
[3] H. ATTOUCH, C. PICARD, Variational inequalities with varying obstacles, J. Functional Analasis, 50 (1983), 329-386. MR695419
[4] H. ATTOUCH, C. PICARD, Comportement limite de problèmes de transmission unilatéraux à travers des grilles de forme quelconque ; Rend. Sem. Math. Univers. Politecn. Torino (à paraître). Zbl0694.49006 · Zbl 0694.49006
[5] D. CIORANESCU, F. MURAT, Un terme étrange venu d’ailleurs, Collège de France Seminar, Research Notes in Mathematics, Pitman, London (1982) n^\circ 60, p. 98-138, n^\circ 70, p. 154-178. Zbl0496.35030 · Zbl 0496.35030
[6] G. DAL MASO, P. LONGO, Gamma-limits of obstacles, Ann. mat. Pura Appl., 128 (1981), 1-50. Zbl0467.49004 MR640775 · Zbl 0467.49004
[7] A. DAMLAMIAN, Le problème de la passoire de Neumann, Rend. Sem. Mat. Univers. Politecn. Torino, 43 (1985), 427-454. Zbl0629.35031 MR884870 · Zbl 0629.35031
[8] E. DE GIORGI, G. DAL MASO, P. LONGO, \(\Gamma\)-limiti di obstacoli, Rend. Acad. Naz. Lincei, 68 (1980), 481-487, Zbl0512.49011 MR639976 · Zbl 0512.49011
[9] E. DE GIORGI, T. FRANZONI, Su un tipo di convergenza variazionale, Rend, Acad. Naz. Lincei, 58 (1975), 842-850. Zbl0339.49005 MR448194 · Zbl 0339.49005
[10] J. L. LIONS, Some methods in the mathematical Analysis of Systems and their control, Science Press, Beijing, China (1981). Zbl0542.93034 MR664760 · Zbl 0542.93034
[11] V. A. MARCHENKO, E. A. HROUSLOV, Problèmes aux limites dans les domaine à frontière finement granulée, Kiev, 1974.
[12] F. MURAT, The Neumann sieve, Non-linear variational problems, ed. by A. Marino, Research Notes in Mathematics, Pitman, London, 127 (1985), 24-32, Zbl0586.35037 MR807534 · Zbl 0586.35037
[13] G. NGUETSENG, Problèmes d’écrans perforés pour l’équation de Laplace, M2AN, 19 (1985), 33-63. Zbl0572.31001 MR813688 · Zbl 0572.31001
[14] C. PICARD, Comportement limite d’un problème de Neumann dans un domaine conteant une grille, Public. U.E.R. de Math, de Lille (1983), vol. 5, fasc. 2.
[15] C. PICARD, Analyse limite d’équations variationnelles dans un domaine contenant une grille, in Thèse d’État, Université de Paris-Sud (Orsay), (1984).
[16] E. SANCHEZ-PALENCIA, Boundary value problems in domains containing perforated walls, Collège de France Seminar, Research Note in Math., Pitman, London, n^\circ 70 (1982), 309-325. Zbl0505.35020 MR670282 · Zbl 0505.35020
[17] E. SANCHEZ-PALENCIA, Problèmes mathématiques liés à l’écoulement d’un fluide visqueux à travers une grille, E. De Giorgi Colloquium, Krée ed., Research Notes in Math., Pitman, London, n^\circ 125 (1985). Zbl0602.76036 MR909713 · Zbl 0602.76036
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