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Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations. (English) Zbl 1134.35317

Summary: We consider complex-valued acoustic and elastic Helmholtz equations with the first order absorbing boundary condition in a star-shaped domain in \(\mathbb R^N\) for \(N \geq 2\). It is known that the elliptic regularity coefficients depend on the frequency \(\omega\), 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 \(\omega\) for large \(\omega\). 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 the author [Comput. Appl. Math. 14, No. 2, 141–156 (1995; Zbl 0842.35015) and with D. Sheen, Trans. Am. Math. Soc. 346, No. 2, 475–487 (1994; Zbl 0811.35021)], which were carried out based on layer potential representations of the solutions of the Helmholtz equations.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B65 Smoothness and regularity of solutions to PDEs
74J99 Waves in solid mechanics
76Q05 Hydro- and aero-acoustics
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References:

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