STABILITY FOR DETERMINATION OF RIEMANNIAN METRICS BY SPECTRAL DATA AND DIRICHLET-TO-NEUMANN MAP LIMITED ON ARBITRARY SUBBOUNDARY

摘要

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 g_1 and g_2 is small, then g_1 and g_2 are close in L~2(Ω) modulo a suitable diffeomorphism within a priori bounds of g_1 and g_2. Both stability estimates are of the same double logarithmic rate.