tailieunhanh - Đề tài " The Calder´on problem with partial data "

In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n ≥ 3, the knowledge of the Cauchy data for the Schr¨dinger equation measured on possibly very small subsets of the o boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem. This implies a similar result for the problem of Electrical Impedance Tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary. . | Annals of Mathematics The Calder ron problem with partial data By Carlos E. Kenig Johannes Sj ostrand and Gunther Uhlmann Annals of Mathematics 165 2007 567 591 The Calderon problem with partial data By Carlos E. Kenig Johannes Sjostrand and Gunther Uhlmann Abstract In this paper we improve an earlier result by Bukhgeim and Uhlmann 1 by showing that in dimension n 3 the knowledge of the Cauchy data for the Schrodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of 1 but use a richer set of solutions to the Dirichlet problem. This implies a similar result for the problem of Electrical Impedance Tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary. 1. Introduction The Electrical Impedance Tomography EIT inverse problem consists in determining the electrical conductivity of a body by making voltage and current measurements at the boundary of the body. Substantial progress has been made on this problem since Calderon s pioneer contribution 3 and is also known as Calderon s problem in the case where the measurements are made on the whole boundary. This problem can be reduced to studying the Dirichlet-to-Neumann DN map associated to the Schrodinger equation. A key ingredient in several of the results is the construction of complex geometrical optics for the Schrodinger equation see 14 for a survey . Approximate complex geometrical optics solutions for the Schrodinger equation concentrated near planes are constructed in 6 and concentrated near spheres in 8 . Much less is known if the DN map is only measured on part of the boundary. The only previous result that we are aware of without assuming any a priori condition on the potential besides being bounded is in 1 . It is shown there that if we measure the DN map restricted to roughly speaking slightly more than half of the boundary then one can determine uniquely .