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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

Citations:

Zbl 0561.00014