Limit behaviour of thin insulating layers around multiconnected domains

Abstract

Let $\Omega$ be a bounded domain of $R^n$, with boundary $\partial \Omega$. Let ${\Gamma }_0$ and $\Gamma$ be connected components of $\partial \Omega$. We assume that $\Omega$ is surrounded along ${\Gamma }_0$ and $\Gamma$ by thin insulating layers ${\Sigma }_0^{\epsilon }$ and ${\Sigma }^{\epsilon }$ of varying respective thicknesses $h_0^{\epsilon }(s)$ and $h^{\epsilon }(s)$, $s$ being the generic point of ${\Gamma }_0$ and $\Gamma$. We denote by ${\Gamma }_0^{\epsilon }$ and ${\Gamma }^{\epsilon }$ the parts of $\partial {\Sigma }_0^{\epsilon }$ and $\partial {\Sigma }^{\epsilon }$ which do not meet $\Omega$. We consider a class of quasilinear elliptic problems with different exponents ($p$ in $\Omega$, $q_0$ in ${\Sigma }_0^{\epsilon }$, $q$ in ${\Sigma }^{\epsilon }$) and with the following boundary conditions:

• on ${\Gamma }_0^{\epsilon }$, $u^{\epsilon }=0$,
• on ${\Gamma }^{\epsilon }$, the total flux is prescribed and $u^{\epsilon }$ is constant, but unprescribed,
• on ${\Gamma }_0$ and $\Gamma$, the natural transmission conditions.

The restricted equations in ${\Sigma }_0^{\epsilon }$ and ${\Sigma }^{\epsilon }$ have nonconstant coefficients, ${\mu }_0^{\epsilon }$ and ${\mu }^{\epsilon }$, in the form ${\mu }_0^{\epsilon }(x)={\mu }_0^{\epsilon }({\sigma }_0(x))$ (respectively ${\mu }^{\epsilon }(x)={\mu }^{\epsilon }(\sigma (x)))$, ${\sigma }_0$ and $\sigma$ being the respective projections on ${\Gamma }_0$ and $\Gamma$. We predict the asymptotic behaviour of this problem as $h_0^{\epsilon }$ and ${\mu }_0^{\epsilon }$ (respectively $h^{\epsilon }$ and ${\mu }^{\epsilon }$) tend to zero in a suitable sense, provided they are related in a convenient way.

Résumé

Soit $\Omega$ un domaine borné de $R^n$, de frontière $\partial \Omega$. Soient ${\Gamma }_0$ et $\Gamma$ des composantes connexes de $\partial \Omega$. Nous supposons que $\Omega$ est entouré le long de ${\Gamma }_0$ et $\Gamma$ par de fines couches isolantes ${\Sigma }_0^{\epsilon }$ et ${\Sigma }^{\epsilon }$ d'épaisseurs variables respectives $h_0^{\epsilon }(s)$ et $h^{\epsilon }(s)$, $s$ étant le point générique de ${\Gamma }_0$ ou $\Gamma$. Nous désignons par ${\Gamma }_0^{\epsilon }$ et ${\Gamma }^{\epsilon }$ les parties de $\partial {\Sigma }_0^{\epsilon }$ et $\partial {\Sigma }^{\epsilon }$ qui ne bordent pas $\Omega$. Nous considérons une classe de problèmes elliptiques quasilinéaires, avec des exposants de Sobolev différents ($p$ dans $\Omega$, $q_0$ dans ${\Sigma }_0^{\epsilon }$, $q$ dans ${\Sigma }^{\epsilon }$) et avec les conditions au bord suivantes :

• sur ${\Gamma }_0^{\epsilon }$, $u^{\epsilon }=0$,
• sur ${\Gamma }^{\epsilon }$, le flux total est prescrit et $u^{\epsilon }$ est constant, mais indéterminé,
• sur ${\Gamma }_0$ et $\Gamma$, les conditions de transmission naturelles.

Les équations restreintes à ${\Sigma }_0^{\epsilon }$ et ${\Sigma }^{\epsilon }$ ont des coefficients non-constants, ${\mu }_0^{\epsilon }$ et ${\mu }^{\epsilon }$, de la forme ${\mu }_0^{\epsilon }(x)={\mu }_0^{\epsilon }({\sigma }_0(x))$ (resp. ${\mu }^{\epsilon }(x)={\mu }^{\epsilon }(\sigma (x)))$, ${\sigma }_0$ et $\sigma$ étant les projections respectives sur ${\Gamma }_0$ et $\Gamma$. Nous prédisons le comportement asymptotique de ce problème, lorsque $h_0^{\epsilon }$ et ${\mu }_0^{\epsilon }$ (resp. $h^{\epsilon }$ et ${\mu }^{\epsilon }$) tendent vers zéro simultanément, tout en vérifiant une relation de corrélation convenable.