Recovering Lamé kernels in viscoelastic equations and ray transforms

The viscoelastic equations with the Lamé coefficients λ and μ, density ρ and viscoelastic coefficients <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>λ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {λ₁} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>μ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {μ₁} are considered. All these coefficients are functions of space variable <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝐱</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> </m:math> {x³} and they are known constants outside ball <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:mi>R</m:mi> </m:msub> </m:math> {BR} of radius R centered at the origin. It is supposed that λ, μ and ρ are given smooth functions in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> {R³} , while <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>λ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {λ₁} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>μ</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> {μ₁} are unknown in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:mi>R</m:mi> </m:msub> </m:math> {BR} . The problem of recovering the latter functions from a given information about solutions of some the Cauchy problems for viscoelastic equations is studied. At first, running plain waves going in the homogeneous media from infinity in direction <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ν</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>𝕊</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> {ν²} are considered. These waves are pure compressive or shear ones of the delta-image form. They are passed thorough ball <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:mi>R</m:mi> </m:msub> </m:math> {BR} and their amplitudes are measured on the boundary of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:mi>R</m:mi> </m:msub> </m:math> {BR} for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ν</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>𝕊</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> {ν²} . It is shown that this information allows to reduce the origin problem to two problems of the inversion of the ray transforms for a family of geodesic lines related to two Riemannian