## 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

### 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)
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### References:

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