SIAM J. MATH. ANAL. c  2005 Society for Industrial and Applied Mathematics Vol. 37, No. 2, pp. 651–672 EXISTENCE, UNIQUENESS, AND REGULARITY RESULTS FOR PIEZOELECTRIC SYSTEMS ∗ D. MERCIER † AND S. NICAISE † Abstract. We investigate the time-harmonic piezoelectric system (a system coupling the elas- ticity system with the full Maxwell’s equations) in polyhedral domains of the space. Existence and uniqueness results of weak solutions are proved in different cases. We describe the corner and edge singularities of that system and deduce some regularity results. Key words. elasticity system, Maxwell’s system, singularities AMS subject classifications. 35J25, 35Q60 DOI. 10.1137/040617728 1. Introduction. Smart structures made of piezoelectric and/or piezomagnetic materials are gaining attention in applications since they are able to transform the energy from one type to another (magnetic, electric, and mechanical), allowing them to be used as sensors and/or actuators. Commonly used piezoelectric materials are ceramics and quartz. The mathematical model of this system starts to be well estab- lished [2, 8, 14, 24, 26] and corresponds to a coupling between the elasticity system and Maxwell’s equations (see below). A full mathematical analysis is not yet done, except in some particular cases [13, 19]. Namely, in these two works the electric field E is assumed to be curl free, i.e., E = ∇ϕ, where ϕ is an electric potential and a two- dimensional reduction is made. In [13], existence and uniqueness results in smooth domains are obtained using integral equations, while in [19] a variational formulation in polygonal domains is given and two-dimensional singularities are briefly described. On the other hand, there exists an extensive list of papers from mechanics liter- ature describing singularities of some particular piezoelectric materials with a plane crack [25, 27, 30] or along wedges [29]. But to our knowledge, an exact description of corner/edge singularities of the general piezoelectric system in three-dimensional polyhedral domains is not yet obtained. Such a description is very important since piezoelectric ceramics are very brittle, and therefore their fracture behavior must be understood. The knowledge of such singularities also has numerical implications, such as convergence speed. This paper has, therefore, the following goals: We present a general piezoelectric system, which includes standard models of ceramics like the PZT or the BaTiO 3 . We further develop some variational formulations which are the natural ones because they lead to solutions in the energy spaces (here called weak solutions). We prove existence and uniqueness results of weak solutions of the time-harmonic system in two different cases: the case when the magnetic permeability matrix is positive definite (BaTiO 3 ) and the case when the magnetic permeability matrix is zero (PZT ). In that second case, we even give two different formulations and show that generically they give rise to the same solutions. Moreover, we describe the corner and edge singularities of our ∗ Received by the editors October 27, 2004; accepted for publication (in revised form) February 22, 2005; published electronically November 9, 2005. http://www.siam.org/journals/sima/37-2/61772.html † Universit´ e de Valenciennes et du Hainaut Cambr´ esis, MACS, Institut des Sciences et Techniques de Valenciennes, F-59313 Valenciennes Cedex 9, France (Denis.Mercier@univ-valenciennes.fr, Serge. Nicaise@univ-valenciennes.fr). 651