Local uniqueness for the inverse boundary problem for the two-dimensional diffusion equation

We study an inverse boundary problem for the diffusion equation in IR2. Our motivation is that this equation is an approximation of the linear transport equation and describes light propagation in highly scattering media. The diffusion equation in the frequency domain is the nonself-adjoint elliptic equation div(D grad u) - (c mu (alpha) + i omega (0)) u = 0, omega (0) double dagger 0, where D and mu (alpha) are the diffusion and absorption coefficients. The inverse problem is the reconstruction of D and mu (alpha) inside a bounded domain using only measurements at the boundary. In the two-dimensional case we prove that the Dirichlet-to-Neumann map, corresponding to any one positive frequency omega (0), determines uniquely both the diffusion and the absorption coefficients, provided they are sufficiently slowly-varying. In the null-background case we estimate analytically how large these coefficients can be to guarantee uniqueness of the reconstruction.