an:04078109 Zbl 0659.49009 Kohn, Robert V.; Vogelius, Michael Relaxation of a variational method for impedance computed tomography EN Commun. Pure Appl. Math. 40, No. 6, 745-777 (1987). 00166796 1987
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49J45 35J20 78A55 35R30 quasiconvexity; relaxed formulation; impedance computed tomography 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. G.Buttazzo