SHARP REGULARITY COEFFICIENT ESTIMATES FOR COMPLEX-VALUED ACOUSTIC AND ELASTIC HELMHOLTZ EQUATIONS

Abstract

We consider complex-valued acoustic and elastic Helmholtz equations with the first order absorbing boundary condition in a star-shaped domain in ℜN for N ≥ 2. It is known that the elliptic regularity coefficients depend on the frequency ω, and have singularities for both zero and infinite frequency. In this paper, we obtain sharp estimates for the coefficients with respect to large frequencies. It is proved that the elliptic regularity coefficients are bounded by first or second order polynomials in ω for large ω. The crux of our analysis is to establish and make use of Rellich identities for the solutions to the acoustic and elastic Helmholtz equations. Our results improve the earlier estimates of Refs. 10 and 11, which were carried out based on layer potential representations of the solutions of the Helmholtz equations.