Relaxation of a variational method for impedance computed tomography. (English) Zbl 0659.49009

Kohn, Robert V.; Vogelius, Michael

Commun. Pure Appl. Math. 40, No. 6, 745-777 (1987).

The problem studied in the paper is the following: find the electrical conductivity of a body by means of voltage and current flux measurements at the boundary. Mathematically, the problem can be studied by means of an elliptic partial differential equation \[ (1)\quad div(\gamma (x)Du)=0\quad in\quad \Omega \subset {\mathbb{R}}^ n\quad (n\geq 2) \] where u is the voltage, \(\gamma\) the unknown conductivity, and \(\gamma\) Du the current flux. The boundary measurements are given by the map \(\Lambda_{\gamma}: H^{1/2}(\partial \Omega)\to H^{-1/2}(\partial \Omega)\) which maps a function \(\phi \in H^{1/2}(\partial \Omega)\) into \(\gamma Du_{\phi}\nu \in H^{-1/2}(\partial \Omega)\), where \(\mu_{\phi}\) is the solution of (1) with \(u=\phi\) on \(\partial \Omega.\)
The problem is then reduced to the minimization for a functional of the form (2) \(\int_{\Omega}g(DU)dx\) \((U=U_ 0\) on \(\partial \Omega)\) where U is a vector-valued function and g is Borel measurable. By using the fact that the relaxed problem associated to (2) is \(\int_{\Omega}Qg(DU)dx\), where Qg denotes the quasiconvex envelope of g, the authors deduce a relaxed formulation for the impedance computed tomography problem, which seems to be more efficient than the original one from the point of view of practical calculations.

Reviewer: G.Buttazzo

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
35J20	Variational methods for second-order elliptic equations
78A55	Technical applications of optics and electromagnetic theory
35R30	Inverse problems for PDEs

Keywords:

quasiconvexity; relaxed formulation; impedance computed tomography

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