tailieunhanh - Đề tài " Calder´on’s inverse conductivity problem in the plane "

We show that the Dirichlet to Neumann map for the equation ∇·σ∇u = 0 in a two-dimensional domain uniquely determines the bounded measurable conductivity σ. This gives a positive answer to a question of A. P. Calder´n o from 1980. Earlier the result has been shown only for conductivities that are sufficiently smooth. In higher dimensions the problem remains open. Contents Introduction and outline of the method | Annals of Mathematics Calder On s inverse conductivity problem in the plane By Kari Astala and Lassi P aiv arinta Annals of Mathematics 163 2006 265 299 Calderon s inverse conductivity problem in the plane By Kari Astala and Lassi Paivarinta Abstract We show that the Dirichlet to Neumann map for the equation V ơVu 0 in a two-dimensional domain uniquely determines the bounded measurable conductivity Ơ. This gives a positive answer to a question of A. P. Calderón from 1980. Earlier the result has been shown only for conductivities that are sufficiently smooth. In higher dimensions the problem remains open. Contents 1. Introduction and outline of the method 2. The Beltrami equation and the Hilbert transform 3. Beltrami operators 4. Complex geometric optics solutions 5. ỡpequations 6. From Aơ to T 7. Subexponential growth 8. The transport matrix 1. Introduction and outline of the method Suppose that Q c Rra is a bounded domain with connected complement and Ơ Q 0 to is measurable and bounded away from zero and infinity. Given the boundary values ộ G H1 2 dQ let u G H1 Q be the unique solution to V ơVu 0 in Q u ỚQ ộ G H 1 2 ỠQ . This so-called conductivity equation describes the behavior of the electric potential in a conductive body. The research of both authors is supported by the Academy of Finland. 266 KARI ASTALA AND LASSI PAIVARINTA In 1980 A. P. Calderon 11 posed the problem whether one can recover the conductivity Ơ from the boundary measurements . from the Dirichlet to Neumann map A . J . . 9u I d Here V is the unit outer normal to the boundary and the derivative ơdu dv exists as an element of H-1 2 ỠQ defined by ơ ỉ ơVu Ụýdm dV Ja where f e H 1 Q and dm denotes the Lebesgue measure. The aim of this paper is to give a positive answer to Calderon s question in dimension two. More precisely we prove Theorem 1. Let Q c R2 be a bounded simply connected domain and ơi e L x- Q i 1 2. Suppose that there is a constant c 0 such that c-1 ơi c. If AƠ1 Aơ2 .