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

Picard, Colette

RAIRO, Modélisation Math. Anal. Numér. 21, 293-326 (1987).

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

MSC:

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)

Keywords:

gamma convergence; homogenization; domains with sieves; asymptotic behaviour

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