We consider a linear coupled system of quasi-electrostatic equations which govern the evolution of a 3-D layered piezoelectric body. Assuming that a dissipative effect is effective at the boundary, we study the uniform stabilization problem. We prove that this is indeed the case, provided some geometric conditions on the region and the interfaces hold. We also assume a monotonicity condition on the coefficients. As an application, we deduce exact controllability of the system with boundary control...

Ce papier porte sur l’étude mathématique d’une équation du type de Grad-Mercier qui décrit, dans certaines circonstances, l’équilibre d’un plasma confiné [H. Grad, P.N. Hu et D.C. Stevens, Proc. Nat. Acad. Sci. USA, 72,n${}^{\circ }$10 (1975), 3789–3793, C. Mercier, Publication of Euratom, CEA, Luxembourg (1974), C. Mercier, Communications personnelles à R. Temam et aux auteurs]. Il s’agit de trouver une fonction “régulière” $u$ solution du système$$\left\{\begin{array}{cc}-\Delta u+\lambda g\left[\delta \right(u\left)\right]=0\hfill & \text{dans}\Omega ,\hfill \\ u=\text{constante}\text{(inconnue)}\>0\hfill & \text{sur}\partial \Omega ,\hfill \\ {\int }_{\partial \Omega }\frac{\partial u}{\partial n}=I,\hfill \end{array}\right.$$où $\Omega$ est un ouvert borné régulier de ${\mathbf{R}}^n$, et$$\delta \left(u\right)\left(x\right)=\mathrm{mes}\{y\in \Omega \mid u(x)\<u(y)\<0\}.$$L’opérateur non linéaire...