On some inverse problems arising in elastography

This paper deals with some inverse problems for linear N-dimensional wave equations with origin in elastography where we try to identify a coefficient from some extra information on (a part of) the boundary. First, we assume that the total variation of the coefficient is a priori bounded. We reformulate the problem as the minimization of an appropriate function in an appropriate constraint set. We prove that this extremal problem possesses at least one solution, first in the one-dimensional case and then, with the help of some regularity results, in the general case, when N >= 2. In the final section, we consider a related (but different) one-dimensional problem, for which the total variation of the coefficient is not bounded a priori. Using some ideas from Pedregal (2005 ESAIM-COCV 15 357-81) and Maestre et al (2008 Interfaces Free Boundaries 10 87-117), we introduce an equivalent variational formulation. Then, we identify a relaxed problem whose solutions can be viewed as Young measures associated with minimizing sequences.