Stable recovery of Kelvin–Voigt viscoelasticity from a single magnetic resonance elastography measurement

We recover the shear modulus and shear viscosity in a scalar Kelvin–Voigt model from a single measured internal complex wave displacement amplitude in magnetic resonance elastography. We first justify the time-harmonic forward model by proving exponential convergence of the time-dependent solution after a smoothly initiated sinusoidal input. Subtracting the amplitude equations for two coefficient pairs gives a real first-order PDE system for the viscoelastic coefficient differences. We derive a measured nondegeneracy condition that makes this system uniformly elliptic and yields uniqueness, H1 Lipschitz stability, and a conditional Hölder estimate for L2 measurement errors in a subdomain of the region of interest. Full-domain and piecewise smooth results are also established. Finally, on a strictly convex planar domain, an affine complex boundary input generates a nondegenerate amplitude at all sufficiently low positive frequencies.