Abstract:

We give most recent results on the increasing stability in the Cauchy problem for  Helmholtz type elliptic partial differential equations of second order without any convexity or non trapping assumptions. Proofs combine traditional energy estimates, Fourier analysis, and Carleman estimates. We give sharp increasing stability results for recovery of sources in the Helmholtz equation and stationary elasticity systems from the minimal boundary data on the interval [0,K] of the wave numbers. Proofs use analytic continuation in wave number, the Huygens' principle, and sharp bounds in the initial boundary value problems for the wave equation and dynamical elasticity system.