Size estimates for the inverse boundary value problems of isotropic elasticity and complex conductivity in 3D

In the inverse boundary value problems of isotropic elasticity and complex conductivity, we derive estimates for the volume fraction of an inclusion whose physical parameters satisfy suitable gap conditions. For both the inclusion and the background medium we assume that the material coefficients are constant. In the elasticity case we require one measurement for the lower bound and another for the upper one. In the complex conductivity case we need three measurements for the lower bound and three for the upper. We accomplish this with the help of the 'translation method' which consists of perturbing the minimum principle associated with the equation by either a null-Lagrangian or a quasi-convex quadratic form.