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Monotone problems in cylinders with vanishing diameter. (Problèmes monotones dans des cylindres de faible diamètre.) (French. Abridged English version) Zbl 0808.35037

Summary: Let \(a: \mathbb{R}^ N\to \mathbb{R}^ N\) be a Lipschitz-continuous and strongly monotone function. Let \(\Omega= \omega\times (0,L)\) be the cylinder of height \(L\) and section \(\omega\). Let \(H_ D^ 1(\Omega)\) be the space of those functions of \(H^ 1(\Omega)\) which vanish on the sections \(x_ N= 0\) and \(x_ N=L\) of the cylinder. We consider the following nonlinear monotone boundary value problem: \[ u^ \varepsilon\in H_ D^ 1(\Omega), \quad \int_ \Omega a(D^ \varepsilon u^ \varepsilon) D^ \varepsilon v dx= \int_ \Omega h(\partial v/\partial x_ N), \quad \forall v\in H_ D^ 1(\Omega), \] where \(D^ \varepsilon v={}^ t((1/ \varepsilon) (\partial v/\partial x_ 1),\dots, (1/\varepsilon) (\partial v/\partial x_{N-1})\), \((\partial v/\partial x_ N))\) for any \(v: \Omega\to \mathbb{R}\).
We prove in the present note that \(u^ \varepsilon\) strongly converges in \(H_ D^ 1(\Omega)\) to the solution \(u\in H_ 0^ 1 (0,L)\) of a monotone problem defined on the segment \((0,L)\). We also prove that \(D^ \varepsilon u^ \varepsilon\) and \(a(D^ \varepsilon u^ \varepsilon)\) strongly converge in \((L^ 2 (\Omega))^ N\) to some limits which are computed explicitly. Finally we give an estimate in \(\varepsilon\) of the convergence rate of these quantities for some particular \(h\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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