Homogenization of a monotone problem in a domain with oscillating boundary

• Published in: ESAIM: Mathematical Modelling and Numerical Analysis Vol. 33; no. 5; pp. 1057 - 1070
• Main Authors: Blanchard, Dominique, Carbone, Luciano, Gaudiello, Antonio
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.09.1999
• Subjects: 35B25; 35J60; Exact sciences and technology; Homogenization; Mathematical analysis; Mathematics; nonlinear problem; oscillating boundary; Partial differential equations; Sciences and techniques of general use; Solution uniqueness; Neumann problem; Asymptotic behavior; Homogenization; Banach space; Poincaré Wirtinger inequality; Partial differential equation; Non linear equation; Lebesgue theorem; Lebesgue measure; Weak solution; Variational equation; Dirichlet problem
• ISSN: 0764-583X; 1290-3841
• DOI: 10.1051/m2an:1999134

We study the asymptotic behaviour of the following nonlinear problem: $$\{ \begin{array}{ll} -{\rm div}(a( Du_h))+ \vert u_h\vert^{p-2}u_h =f \quad\hbox{in }\Omega_h, a( Du_h)\cdot\nu = 0 \quad\hbox{on }\partial\Omega_h, \end{array} .$$ in a domain Ωh of $\mathbb{R}^n$ whose boundary ∂Ωh contains an oscillating part with respect to h when h tends to ∞. The oscillating boundary is defined by a set of cylinders with axis 0xn that are h-1-periodically distributed. We prove that the limit problem in the domain corresponding to the oscillating boundary identifies with a diffusion operator with respect to xn coupled with an algebraic problem for the limit fluxes. Nous étudions le comportement asymptotique du problème non linéaire monotone $$\{ \begin{array}{ll} -{\rm div}(a( Du_h))+ \vert u_h\vert^{p-2}u_h =f \quad\hbox{dans }\Omega_h, a( Du_h)\cdot\nu = 0 \quad\hbox{sur }\partial\Omega_h, \end{array} .$$ posé sur un ouvert Ωh de $\mathbb{R}^n$ dont une partie de la frontière oscille avec h lorsque h tend vers ∞. Cette partie oscillante est constituée d'un ensemble de cylindres d'axe Oxn distribués avec la période h-1. Nous démontrons que dans le domaine correspondant à la partie oscillante, le problème limite couple un problème de diffusion en xn et un problème algébrique pour les flux limites.