Inverse problem for the Yang-Mills equations

Abstract

We show that a connection can be recovered up to gauge from sourceto-solution type data associated with the Yang-Mills equations in Minkowski space R 1+3 . Our proof analyzes the principal symbols of waves generated by suitable nonlinear interactions and reduces the inversion to a broken non-abelian light ray transform. The principal symbol analysis of the interaction is based on a delicate calculation that involves the structure of the Lie algebra under consideration and the final result holds for any compact Lie group.

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

“…(Here and below we understand ϕ j (∂Ω) = ∅ if U j ∩ ∂Ω = ∅.) Since our equation ( 14) satisfies the compatibility conditions (13), one can verify by a direct calculation that (100) satisfies the compatibility conditions of [29,42] that were needed for unique solvability (98). In particular, at the intersection of {t = 0} and ∂ U j ∩ ϕ j (∂Ω) the compatibility conditions follow from the assumptions of the proposition we are proving.…”

“…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].…”

The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

“…Since the work [31], rapid progress has been made on the study of inverse problems for nonlinear equations. See [37,30,36,14,47,13,44,11,20,23,12,4,35,19,32] for results on hyperbolic equations and [3,10,8,34,33,21,29,27,2,28,26,9] for elliptic equations. For hyperbolic equations, the recovery of time-dependent coefficients is possible for some nonlinear equations, whereas the corresponding problems for linear equations are still largely open.…”

“…The approach was then further generalized to deal with other different types of nonlinear equations in [37,30,36,47,44,4]; recently [43] it was shown that recovery of a Riemannian metric is possible when one measures the solution of a forced semilinear wave equation only at a single point for some time. Distorted plane waves can also be used to recover the coefficients of (linear) lower order terms [11,12] and the nonlinear terms [37,13,23]. In the works [20,45,23,19], Gaussian beams, instead of distorted planes waves, are used to study inverse problems for nonlinear wave 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.…”