Global uniqueness for a two-dimensional inverse boundary value problem


We show that the coefficient $\gamma(x)$ of the elliptic equation $\nabla \cdot (\gamma\nabla u) = 0$ in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary, and gave a reconstruction procedure. For the equation $\Sigma \partial_i(\gamma^{ij}\partial_j u) = 0$, two matrix-valued functions $\gamma_1$ and $\gamma_2$ yield the same Dirichlet-to-Neumann map if and only if there is a diffeomorphism which fixes the boundary and transforms $\gamma_1$ into $\gamma_2$.



Adrian I. Nachman