Inverse Problem for the Yang–Mills Equations
TLDR
In this article, it was shown that a connection can be recovered up to gauge from source-to-solution type data associated with the Yang-Mills equations in Minkowski space.Abstract:
We show that a connection can be recovered up to gauge from source-to-solution type data associated with the Yang–Mills equations in Minkowski space $${\mathbb {R}}^{1+3}$$
. 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.read more
Citations
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Book ChapterDOI
Fourier Integral Operators
Yu. V. Egorov,Mikhail Shubin +1 more
TL;DR: In this paper, the Fourier integral operators (FIFO) were examined for hyperbolic types of elliptic differential equations, and a wider class of operators, the so-called FIFO-integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).
Book ChapterDOI
Representations of Compact Lie Groups
TL;DR: The theory of compact Lie algebras was introduced in this paper, where it was shown that a compact Lie group can be identified with the set of left-invariant vector fields on the group, or with the sets of appropriate differential operators of order one.
Journal ArticleDOI
Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation
TL;DR: In this article , the authors considered the recovery of a potential associated with a semi-linear wave equation on R n + 1 , n ≥ 1 , and showed that an unknown potential a (x, t ) of the wave equation □ u + a u m = 0 can be recovered in a Hölder stable way from the map u | ∂ Ω × [ 0, T ] ↦ 〈 ψ , ∂ ǫ u , ∆ , ∀ ǒ u |∆ ǔ u |, ∀ L 2 ( ∆ L 2 ∆ ) .
Journal ArticleDOI
Inverse problems for nonlinear hyperbolic equations with disjoint sources and receivers
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Determining Riemannian Manifolds From Nonlinear Wave Observations at a Single Point
TL;DR: In this article, it was shown that on an a-prior unknown Riemannian manifold, measuring the source-to-solution map for the semilinear wave equation at a single point determines the topological, differential, and geometric structure.
References
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Journal ArticleDOI
Fourier Integral Operators. I
TL;DR: In this paper, a more general class of pseudo-differential operators for non-elliptic problems is discussed. But their value is rather limited in genuinely nonelliptical problems.
Journal ArticleDOI
A global uniqueness theorem for an inverse boundary value problem
John Sylvester,Gunther Uhlmann +1 more
TL;DR: In this paper, the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n? 3.
Book ChapterDOI
Quasi-linear equations of evolution, with applications to partial differential equations
Journal ArticleDOI
Reconstructions from boundary measurements
TL;DR: In this paper, the authors define la forme quadratique Qγ sur H 1/2 (∂Ω) par Qγ(f)=∫ Ω γ(x)|⊇u (x)| 2 dx ou u∈H 1 (Ω), est la solution unique a Lγu=0 dans Ω, u| ∂ Ω =f.