This proposal concerns two inverse problems for hyperbolic equations. The first problem involves the wave operator P for the Laplace-Beltrami operator for a smooth, Riemannian metric on a bounded n-dimensional domain. The inverse problem is to describe material properties (represented by the coefficients of P) of an object (represented by the n-dimensional domain) given only measurements made at the surface of the object (modeled by a boundary operator, the Dirichlet-to-Neumann map). We consider the continuous dependence of the metric on the Dirichlet-to-Neumann map for P. For this inverse problem (and inverse problems in oil prospection and seismology, for example) it is known, though, that the material properties of the object are not uniquely determined by surface measurements, in general. In particular, a metric is not uniquely determined by the Dirichlet-to-Neumann map for P since the pullback of a metric by a diffeomorphism that fixes the boundary has the same associated Dirichlet-to-Neumann map as the original metric. Sylvester and Uhlmann have shown that unique determination does hold at the boundary, though,and in the case that surface measurements are available for all time, it follows from Belishev and Kurylev that uniqueness holds in the interior, up to the pull-back by a diffeomorphism that fixes the boundary. To study the dependence, then, of material properties of the object on surface measurements, for this inverse problem for which uniqueness does not hold, we propose showing that the metric depends continuously on the Dirichlet-to-Neumann map, up to the pull-back by a diffeomorphism that fixes the boundary. Having shown that continuous dependence holds, one can expect that material properties of the interior of the object (represented by the coefficients of the differential equation) can, in principle, be reconstructed arbitrarily accurately from measurements made only at the surface (that is, from the Dirichlet-to-Neumann map). In t he second problem we consider an inverse problem for the system of operators P for elastodynamics with residual stress. The linearly elastic, nonhomogeneous, isotropic object being studied is represented by a bounded, 3-dimensional region with smooth boundary. The behavior of the object is described in terms of solutions of the system of equations for elastodynamics. Coefficients of these equations represent material properties of the object, and surface measurements are modeled by the associated Dirichlet-to-Neumann map. It is assumed there are no body forces acting on the object after time zero but that events in the past have built up a residual stress in the object, which is modeled by a smooth, symmetric, second-rank tensor that is divergence-free and has zero traction on the boundary. The central question we pose here is to what extent the Dirichlet-to-Neumann map associated with P uniquely determines the density, coefficients of elasticity, and residual stress at the boundary.
