We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations.We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions.As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.

Abstract. We prove upper and lower estimates on the measure of an inclusion D in a conductor Ω in terms of one pair of current and potential boundary measurements. The growth rates of such estimates are essentially best possible.

We prove upper and lower bounds on the size of an unknown cavity, or of a perfectly conducting inclusion, in an electrical conductor in terms of boundary measurements of voltage and current. Such bounds, which might be used as a decision tool in quality testing of materials, are obtained by a nontrivial extension of previous results (Alessandrini G, Rosset E and Seo J K 2000 Proc. Am. Math. Soc. 128 53–64) regarding inclusions of finite, nonzero conductivity.

We prove a conditional stability estimate for the inverse problem of determining either cavities inside an elastic body or unknown boundary portions, from a single measurement of traction and displacement taken on the accessible part of the exterior boundary of .

We consider the problem of determining, within an elastic isotropic body Ω, the possible presence of an inclusion D made of different elastic material from boundary measurements of traction and displacement. We prove that the volume of D can be estimated, from above and below, by an easily expressed quantity related to work depending only on the boundary traction and displacement.