We prove that the standard second-kind integral equation formulation of the exterior Dirichlet problem for the Helmholtz equation is coercive (i.e., sign-definite) for all smooth convex domains when the wavenumber k is sufficiently large. (This integral equation involves the so-called combined potential, or combined field, operator.) This coercivity result yields k-explicit error estimates when the integral equation is solved using the Galerkin method, regardless of the particular approximation space used (and thus these error estimates apply to several hybrid numerical-asymptotic methods developed recently). Coercivity also gives k-explicit bounds on the number of GMRES iterations needed to achieve a prescribed accuracy when the integral equation is solved using the Galerkin method with standard piecewise-polynomial subspaces. The coercivity result is obtained by using identities for the Helmholtz equation originally introduced by Morawetz in her work on the local energy decay of solutions to the wave equation.

We study the spectral and polarization properties of acoustic waves propagating in nematic liquid-crystalline rubber materials. We apply the viscoelastic theory of nematic elastomers in the low-frequency (hydrodynamic) limit. Dynamic soft elasticity, exhibited by ideal nematic elastomers under certain geometries of shear at low frequencies, leads to anomalous anisotropy of energy transfer and attenuation of transverse waves. The results suggest an application of this class of materials as an acoustic polarizer medium.

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Properties of solutions of generic hyperbolic systems with multiple characteristics with diagonalizable principal part are investigated. Solutions are represented as a Picard series with terms in the form of iterated Fourier integral operators. It is shown that this series is an asymptotic expansion with respect to smoothness under quite general geometric conditions. Propagation of singularities and sharp regularity properties of solutions are obtained. Results are applied to establish regularity estimates for scalar weakly hyperbolic equations with involutive characteristics. They are also applied to derive the first and second terms of spectral asymptotics for the corresponding elliptic systems.