Artificial conditions for the linear elasticity equations

34 pagesInternational audienc

Doctor of Philosophy

dissertationWe study three problems in this dissertation. In the rst problem, we derive bounds on the volume occupied by an inclusion in a body through the use of a single measurement of the complex voltage and current ux around the boundary of the body. We assume that the conductivities of the inclusion and the body are complex. In the second problem, we derive a formula that gives the exact volume fraction occupied by a linearly elastic inclusion in a linearly elastic body when both the inclusion and the body have the same shear modulus. The formula for the volume of the inclusion is based on an appropriate measurement of the displacement and traction around the boundary of the body, tailored to force the body to behave as if it were embedded in an innite medium. In the third problem, we prove that the power dissipated in a nonsymmetric slab superlens blows up in the limit as the dissipation parameters in the lens and the surrounding medium go to zero when certain charge density distributions are placed within a critical distance of the slab. The critical distance that leads to this blow-up of the power dissipation depends nontrivially on the relative amount of dissipation in the slab and surrounding medium. This behavior of the power dissipation, in combination with the fact that the potential remains bounded far away from the slab as the dissipation parameters go to zero, leads to cloaking by anomalous localized resonance