Stability for determination of riemannian metrics by spectral data and dirichlet-to-neumann map limited on arbitrary subboundary

In this paper, we establish conditional stability estimates for two inverse problems of determining metrics in two dimensional Laplace-Beltrami operators. As data, in the first inverse problem we adopt spectral data on an arbitrarily fixed subboundary, while in the second, we choose the Dirichlet-to-Neumann map limited on an arbitrarily fixed subboundary. The conditional stability estimates for the two inverse problems are stated as follows. If the difference between spectral data or Dirichlet-to-Neumann maps related to two metrics g1 and g2 is small, then g1 and g2 are close in L2(Ω) modulo a suitable diffeomorphism within a priori bounds of g1 and g2. Both stability estimates are of the same double logarithmic rate. © 2019 American Institute of Mathematical Sciences.