Inverse problems and Carleman estimates.

*(English)*Zbl 0755.35151The author describes the current state of the method of Carleman type estimates suggested by A. L. Bukhgejm and himself [Dokl. Akad. Nauk SSSR 260, 269-272 (1981; Zbl 0497.35082)]. This method is based on powerful estimates of solutions of PDE in weighted Sobolev spaces which are a classical tool in the uniqueness of continuation study as well as in the uniqueness of propagation of singularities. He formulates general Carleman estimates and gives examples for elliptic, parabolic and hyperbolic second order equations, then he exposes a general scheme of the method (dividing by some function and differentiating to eliminate unknown right side terms) and gives application to the mentioned equations.

This method is applicable to non-overdetermined inverse problems. A certain disadvantage is a requirement of nonzero initial data in hyperbolic problems. It looks that this method proved to be quite useful and it is still promising.

This method is applicable to non-overdetermined inverse problems. A certain disadvantage is a requirement of nonzero initial data in hyperbolic problems. It looks that this method proved to be quite useful and it is still promising.

Reviewer: V.Isakov (Wichita)

##### MSC:

35R30 | Inverse problems for PDEs |

35B45 | A priori estimates in context of PDEs |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |