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The stability for the Cauchy problem for elliptic equations. (English) Zbl 1190.35228

The ill-posed Cauchy problem for elliptic equations which is pervasive in inverse boundary value problems modeled by elliptic equations is discussed in the present topical review. Essentially optimal stability results are provided in wide generality and under substantially minimal assumptions. In the special case of two variables, the Cauchy problem for Laplace’s equation is equivalent to the problem of continuation of a complex analytic function from values prescribed on an arc. The unique continuation property is also connected to the problem of the stability for the Cauchy problem. Usually proofs of the unique continuation property depend on two types of inequalities which can be applied - Carleman estimates and three-spheres inequalities. Various versions of the three-spheres inequality is formulated in the present review. The central theme of the review is to stress that three-spheres inequalities can be used as a universal building brick to derive optimal stability estimates.

MSC:

35R30 Inverse problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
35B35 Stability in context of PDEs
30E25 Boundary value problems in the complex plane
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