Reconstruction in the Calderon Problem with Partial Data

Details Reconstruction in the Calderon Problem with Partial Data Reconstruction in the Calderon Problem with Partial Data

2009-04-30

arxiv.org/abs/0904.4794v2

Mathematics

Analysis of PDEs

Final version, 17 pages, to appear in Comm in PDE

Scientific paper

We consider the problem of recovering the coefficient \sigma(x) of the elliptic equation \grad \cdot(\sigma \grad u)=0 in a body from measurements of the Cauchy data on possibly very small subsets of its surface. We give a constructive proof of a uniqueness result by Kenig, Sj\"ostrand, and Uhlmann. We construct a uniquely specified family of solutions such that their traces on the boundary can be calculated by solving an integral equation which involves only the given partial Cauchy data. The construction entails a new family of Green's functions for the Laplacian, and corresponding single layer potentials, which may be of independent interest.

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