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Equivalence of inverse problems for 2D elasticity and for the thin plate with finite measurements and its applications

In this paper, we prove that the inverse problems for 2D elasticity and for the thin plate with boundary data (finite or full measurements) are equivalent. Having proved this equivalence, we can solve inverse problems for the plate equation with boundary data by solving the corresponding inverse problems for 2D elasticity, and vice versa. For example, we can derive bounds on the volume fraction of the two-phase thin plate from the knowledge of one pair of boundary measurements using the known result for 2D elasticity \cite{ml}. Similarly, we give another approach to the size estimate problem for the thin plate studied by Morassi, Rosset, and Vessella \cite{mrv07, mrv09}.