Stability estimate for the Helmholtz equation with rapidly jumping coefficients

TL;DR

This paper investigates the stability of the high-frequency Helmholtz equation with non-smooth, rapidly oscillating coefficients, providing new bounds on the solution operator that are independent of the number of discontinuities.

Contribution

The paper introduces a new theoretical approach for 1D Helmholtz problems with discontinuous coefficients, establishing stability bounds independent of jump count, and constructs oscillatory configurations demonstrating sharpness.

Findings

Stability constant bounded independently of the number of jumps for certain coefficients.

Existence and uniqueness proved via unique continuation principle.

Constructed oscillatory configurations with exponential growth in stability constant.

Abstract

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.