INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS Inverse Problems 18 (2002) 1333–1353 PII: S0266-5611(02)35325-5 Detecting cavities by electrostatic boundary measurements Giovanni Alessandrini 1 , Antonino Morassi 2 and Edi Rosset 1 1 Dipartimento di Scienze Matematiche, Universit` a degli Studi di Trieste, Italy 2 Dipartimento di Ingegneria Civile, Universit` a degli Studi di Udine, Italy E-mail: alessang@univ.trieste.it, antonino.morassi@dic.uniud.it and rossedi@univ.trieste.it Received 21 March 2002, in final form 26 June 2002 Published 19 August 2002 Online at stacks.iop.org/IP/18/1333 Abstract 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. 1. Introduction Suppose that  is an electrically conducting body having, for simplicity of discussion, constant conductivity σ ≡ 1 and suppose that it might contain an unknown cavity D. The electrostatic potential u i in , corresponding to an assigned (nontrivial) current density ϕ on ∂ is given as the solution of the Neumann problem u i = 0 in \ D, ∂ u i ∂ν = ϕ on ∂, ∂ u i ∂ν = 0 on ∂ D. (1.1) The inverse problem of detecting D from boundary measurements consists of finding D when the boundary values of the voltage u | ∂ are given. It is well known by now that, if \ D is connected and D is a priori assumed to be an open set whose boundary satisfies some smoothness condition, then it is uniquely determined by the boundary measurement u | ∂ , [ARon01, ABRV01, CHY01]. However, there are examples that show that the stability of this inverse problem is no better than logarithmic (see [ARon01, section 4]), and this fact poses serious limitations to the possibility of finding efficient procedures of reconstruction. 0266-5611/02/051333+21$30.00 © 2002 IOP Publishing Ltd Printed in the UK 1333