Stability estimate for the Helmholtz equation with rapidly jumping coefficients

The goal of this paper is to investigate the stability of the Helmholtz equation in the high- frequency regime with non-smooth and rapidly oscillating coefficients on bounded domains. Existence and uniqueness of the problem can be proved using the unique continuation principle in Fredholm's alternative. However, this approach does not give directly a coefficient-explicit energy estimate. We present a new theoretical approach for the one-dimensional problem and find that for a new class of coefficients, including coefficients with an arbitrary number of discontinuities, the stability constant (i.e., the norm of the solution operator) is bounded by a term independent of the number of jumps. We emphasize that no periodicity of the coefficients is required. By selecting the wave speed function in a certain \resonant" way, we construct a class of oscillatory configurations, such that the stability constant grows exponentially in the frequency. This shows that our estimates are sharp.Comment: a) Added references, b) rewritten the introduction with a summary of the results/techniques of the paper, c) Corrected typo

Stability estimate in an inverse problem for non-autonomous Schr\"odinger equations

We consider the inverse problem of determining the time dependent magnetic field of the Schr\"odinger equation in a bounded open subset of $R^n$, with $n \geq 1$, from a finite number of Neumann data, when the boundary measurement is taken on an appropriate open subset of the boundary. We prove the Lispchitz stability of the magnetic potential in the Coulomb gauge class by $n$ times changing initial value suitably

A global stability estimate for the Gel'fand-Calderon inverse problem in two dimensions

We prove a global logarithmic stability estimate for the Gel'fand-Calderon inverse problem on a two-dimensional domain