Inverse problems for real principal type operators

Abstract

We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determination methods for general operators, and prove global uniqueness results for determining coefficients in nonlinear real principal type equations. The article presents a unified approach for treating inverse boundary problems for transport and wave equations, and highlights the role of propagation of singularities in the solution of related inverse problems.

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

“…The separation matrix (53) corresponding to the functions v f k and points x k is invertible for ε ≤ ε 0 for ε 0 small enough. We set M := M ε 0 .…”

“…The research of inverse problems for non-linear equations is expanding fast. By using the higher-order linearization, inverse problems for nonlinear models have been studied for example in [3,11,12,13,16,17,21,20,32,36,37,38,41,45,46,53,60,62,63].…”

Inverse problem for the Yang-Mills equations

“…We mention that non-linear interactions have also been used to recover non-linear terms for scalar wave equations [33], scalar elliptic equations [13,31], and scalar real principal type equations [38]. In these four works, non-linear terms do not contain any derivatives, contrary to the Einstein and Yang-Mills equations.…”

Uniqueness and stability of an inverse problem for a semi-linear wave equation

“…The authors of [46] studied inverse problems for general semi-linear wave equations on Lorentzian manifolds, and in [45] they studied analogous problem for the Einstein-Maxwell equations. Recently, inverse problems for non-linear equations using the non-linearity as a tool, have been studied in [3,11,12,13,16,17,20,21,30,34,35,36,39,42,43,51,57,60,61]. The works mentioned above use the so-called higher order linearization method, which we will explain later.…”