We study an inverse boundary problem for the diffusion equation in ℝ2. 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) - (cμa + iω0)
u = 0; ω0 ≠ 0, where D and
μa are the diffusion and absorption coefficients. The inverse problem is the reconstruction
of D and μa 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 ω0, 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.
Effects of the approximations of light propagation on quantitative photoacoustic tomography using two-dimensional photon diffusion equation and linearization
