## The Neumann sieve.(English)Zbl 0586.35037

Nonlinear variational problems, Int. Workshop, Elba/Italy 1983, Res. Notes Math. 127, 24-32 (1985).
[For the entire collection see Zbl 0561.00014.]
This paper presents results obtained in a joint work with H. Attouch, A. Damlamian and C. Picard. Let $$\Omega$$ be an open bounded set of $${\mathbb{R}}^ N$$ divided into two parts $$\Omega_ a=\Omega \cap \{x_ N>0\}$$ and $$\Omega_ b=\Omega \cap \{x_ N<0\}$$ by $$\Gamma =\Omega \cap \{x_ N=0\}.$$ The sieve $$\Gamma^{\epsilon}$$ is constructed by perforating holes of size $$r^{\epsilon}$$ and of shape $${\mathcal H}$$ at each vertex of a lattice of $$\{x_ N=0\}$$ of size $$2\epsilon$$. Let $$\Omega^{\epsilon}=\Omega \setminus \Gamma^{\epsilon}$$ and consider for any given f in $$L^ 2(\Omega)$$ the ”Neumann Sieve’s problem”: $- \Delta u^{\epsilon}+u^{\epsilon}=f\quad in\quad \Omega^{\epsilon};\quad \partial u^{\epsilon}/\partial n=0\quad on\quad \Gamma^{\epsilon};\quad u^{\epsilon}=0\quad on\quad \partial \Omega.$ If the size of the holes is carefully chosen $$(r^{\epsilon}\sim C_ 0\epsilon^{(N-1)/(N-2)}$$ if $$N\geq 3)$$, $$u^{\epsilon}$$ converges weakly in $$H^ 1(\Omega^{\epsilon})$$ to some function u. This function, defined through its restriction $$u_ a$$ and $$u_ b$$ to $$\Omega_ a$$ and $$\Omega_ b$$, is not continuous across $$\Gamma$$. It is the solution of the problem: $-\Delta u_ a+u_ a=f\quad in\quad \Omega_ a;\quad \partial u_ a/\partial n_ a+\mu (u_ a-u_ b)=0\quad on\quad \Gamma;$
$-\Delta u_ b+u_ b=f\quad in\quad \Omega_ b;\quad \partial u_ b/\partial n_ b+\mu (u_ b- u_ a)=0\quad on\quad \Gamma;\quad u=0\quad on\quad \partial \Omega,$ where $$n_ a$$ $$(=-e_ N)$$ and $$n_ b=(+e_ N)$$ are the outer normals to $$\Omega_ a$$ and $$\Omega_ b$$ on $$\Gamma$$, and where $$\mu$$ is a real positive constant related to the capacity in $${\mathbb{R}}^ N$$ of the shape $${\mathcal H}$$ of the holes: $$\mu =(C_ 0^{N- 2}/2^{N+1})Cap_{{\mathbb{R}}^ N}(\bar {\mathcal H})$$ if $$N\geq 3$$.

### MSC:

 35J25 Boundary value problems for second-order elliptic equations 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 58J32 Boundary value problems on manifolds 46E99 Linear function spaces and their duals

Zbl 0561.00014