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Inverse electrostatic and elasticity problems for checkered distributions

We study the inverse electrostatic and elasticity problems associated with Poisson and Navier equations. These problems arise in a number of applications, such as diagnostic of electronic devices and analysis of residual stresses in materials. In microelectronics, piecewise constant distributions of electric charge having a checkered structure (i.e., that are constant on rectangular blocks) are of particular importance. We prove that the inverse electrostatic problem has a unique solution for such distributions. We also show that the inverse elasticity problem has a unique solution for checkered distributions of body forces. General necessary and sufficient conditions for the uniqueness of solutions of both inverse problems are discussed as well.