Stability for an inverse problem in potential theory

Abstract

Let D D be a subdomain of a bounded domain Ω \Omega in R n {\mathbb {R}^n} . The conductivity coefficient of D D is a positive constant k ≠ 1 k \ne 1 and the conductivity of Ω ∖ D \Omega \backslash D is equal to 1 1 . For a given current density g g on ∂ Ω \partial \Omega , we compute the resulting potential u u and denote by f f the value of u u on ∂ Ω \partial \Omega . The general inverse problem is to estimate the location of D D from the known measurements of the voltage f f . If D h {D_h} is a family of domains for which the Hausdorff distance d ( D , D h ) d(D,{D_h}) equal to O ( h ) O(h) ( h h small), then the corresponding measurements f h {f_h} are O ( h ) O(h) close to f f . This paper is concerned with proving the inverse, that is, d ( D , D h ) ≤ 1 c ‖ f h − f ‖ d(D,{D_h}) \leq \frac {1}{c}\left \| {f_h} - f\right \| , c > 0 c > 0 ; the domains D D and D h {D_h} are assumed to be piecewise smooth. If n ≥ 3 n \geq 3 , we assume in proving the above result, that D h ⊃ D {D_h} \supset D (or D h ⊂ D {D_h} \subset D ) for all small h h . For n = 2 n = 2 this monotonicity condition is dropped, provided g g is appropriately chosen. The above stability estimate provides quantitative information on the location of D h {D_h} by means of f h {f_h} .