Doubling inequalities for anisotropic plate equations and applications to size estimates of inclusions

We prove upper and lower estimates of the area of an unknown elastic inclusion in a thin plate by one boundary measurement. The plate is made of non-homogeneous linearly elastic material belonging to a general class of anisotropy and the domain of the inclusion is a measurable subset of the plate. The size estimates are expressed in terms of the work exerted by a couple field applied at the boundary and of the induced transversal displacement and its normal derivative taken at the boundary of the plate. Main new mathematical tool is a doubling inequality for solutions to fourth order elliptic equations whose principal part $P(x,D)$ is the product of two second order elliptic operators $P_1(x,D), P_2(x,D)$ such that $P_1(0,D)=P_2(0,D)$. The proof of the doubling inequality is based on Carleman method, a sharp three spheres inequality and a bootstrapping argument.