UNIQUENESS FOR AN ANISOTROPIC INVERSE BOUNDARY VALUE PROBLEM ARISING IN THE MECHANICS OF TENSION STRUCTURE

ABSTRACT. Given a linear second order elliptic equation in divergence form, we wish to dermine the matrix of the coefficients from the knowledge of the Dirichlet-Neumann mapping. It is well known that in general uniqueness doesn't hold if no additional assumptions on the coefficients are made. A relevant case is the isotropic one which has been widely studied. We prove here uniqueness in the case when the matrix of the coefficients, assumed to be nonisotropic, is divergence free. This case comes from the equilibrium problem for a membrane in tension structures. Consider an elastic or inextensible thin membrane stretched out a planar region, described by a simply connected bounded open set 2R⊂Ω, which is subject to a distributed plane traction state. Its transverse displacement)(xuu = is governed by the linearized equilibrium equation (1) Ω=σ − in)div ( fDu, where)(xff = represents the transverse load and the tensor {}2 1,) ( =σ=σ jiij x represents its plane traction state (see for instance (7) pag. 60 for a classical derivation). The mechanics of the problem requires that σ is a symmetric and positive definite matrix with bounded measurable entries that i