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On some inverse problems arising in elastography. (English) Zbl 1253.35218

Summary: 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 \(N \geq 2\). 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 P. Pedregal [ESAIM, Control Optim. Calc. Var. 11, 357–381 (2005; Zbl 1089.49022)] and F. Maestre, A. Münch and P. Pedregal [Interfaces Free Bound. 10, No. 1, 87–117 (2008; Zbl 1231.49002)], we introduce an equivalent variational formulation. Then, we identify a relaxed problem whose solutions can be viewed as Young measures associated with minimizing sequences.

MSC:

35R30 Inverse problems for PDEs
35L10 Second-order hyperbolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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