Inverse problem for the Yang-Mills equations
TL;DR: 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 the four dimensional 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 the four dimensional Minkowski space. 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.
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TL;DR: In this article, the authors considered the recovery of a potential associated with a semi-linear wave equation on the Dirichlet-to-Neumann map and showed that the potential can be recovered in a Holder stable way.
Abstract: We consider the recovery of a potential associated with a semi-linear wave equation on $\mathbb{R}^{n+1}$, $n\geq 1$ We show a Holder stability estimate for the recovery of an unknown potential $a$ of the wave equation $\square u +a u^m=0$ from its Dirichlet-to-Neumann map We show that an unknown potential $a(x,t)$, supported in $\Omega\times[t_1,t_2]$, of the wave equation $\square u +a u^m=0$ can be recovered in a Holder stable way from the map $u|_{\partial \Omega\times [0,T]}\mapsto \langle\psi,\partial_
u u|_{\partial \Omega\times [0,T]}\rangle_{L^2(\partial \Omega\times [0,T])}$ This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a measurement function $\psi$ We also prove similar stability result for the recovery of $a$ when there is noise added to the boundary data The method we use is constructive and it is based on the higher order linearization As a consequence, we also get a uniqueness result We also give a detailed presentation of the forward problem for the equation $\square u +a u^m=0$
19 citations
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TL;DR: In this article, the authors studied the inverse problem of determining unknown coefficients in various semi-linear and quasi-linear wave equations and proposed a method to solve inverse problems for non-linear equations using interaction of three waves.
Abstract: The paper studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations. We introduce a method to solve inverse problems for non-linear equations using interaction of three waves, that makes it possible to study the inverse problem in all dimensions $n+1\geq 3$. We consider the case when the set $\Omega_{\textrm{in}}$, where the sources are supported, and the set $\Omega_{\textrm{out}}$, where the observations are made, are separated. As model problems we study both a quasi-linear and also a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the first half we study multiple-fold linearization of the non-linear wave equation near real parts of Gaussian beams that results in a three-wave interaction. We show that the three-wave interaction can produce a three-to-one scattering data. In the second half of the paper, we study an abstract formulation of the three-to-one scattering relation showing that it recovers the topological, differential and conformal structures of the manifold in a causal diamond set that is the intersection of the future of the point $p_{in}\in \Omega_{\textrm{in}}$ and the past of the point $p_{out}\in \Omega_{\textrm{out}}$. The results do not require any assumptions on the conjugate or cut points.
7 citations
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TL;DR: In this paper, it was shown that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric and the coefficient up to natural obstructions.
Abstract: We consider the semilinear wave equation $\Box_g u+a u^4=0$, $a
eq 0$, on a Lorentzian manifold $(M,g)$ with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric $g$ and the coefficient $a$ up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.
6 citations
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 ∆ ) .
5 citations
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TL;DR: In this article, the conformal class of the Lorentzian metric $g$ can be recovered up to diffeomorphisms, from the knowledge of the Neumann-to-Dirichlet map.
Abstract: We study inverse problems for the nonlinear wave equation $\square_g u + w(x,u,
abla_g u) = 0$ in a Lorentzian manifold $(M,g)$ with boundary, where $
abla_g u$ denotes the gradient and $w(x,u, \xi)$ is smooth and quadratic in $\xi$. Under appropriate assumptions, we show that the conformal class of the Lorentzian metric $g$ can be recovered up to diffeomorphisms, from the knowledge of the Neumann-to-Dirichlet map. With some additional conditions, we can recover the metric itself up to diffeomorphisms. Moreover, we can recover the second and third quadratic forms in the Taylor expansion of $w(x,u, \xi)$ with respect to $u$ up to null forms.
4 citations
References
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01 Jan 1998
TL;DR: In this article, the analysis of linear partial differential operators i distribution theory and fourier rep are a good way to achieve details about operating certain products using instruction manuals, which are clearlybuilt to give step-by-step information about how you ought to go ahead in operating certain equipments.
Abstract: the analysis of linear partial differential operators i distribution theory and fourier rep are a good way to achieve details about operating certainproducts. Many products that you buy can be obtained using instruction manuals. These user guides are clearlybuilt to give step-by-step information about how you ought to go ahead in operating certain equipments. Ahandbook is really a user's guide to operating the equipments. Should you loose your best guide or even the productwould not provide an instructions, you can easily obtain one on the net. You can search for the manual of yourchoice online. Here, it is possible to work with google to browse through the available user guide and find the mainone you'll need. On the net, you'll be able to discover the manual that you might want with great ease andsimplicity
3,025 citations
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.
Abstract: Pseudo-differential operators have been developed as a tool for the study of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations, but their value is rather limited in genuinely non-elliptic problems. In this paper we shall therefore discuss some more general classes of operators which are adapted to such applications. For these operators we shall develop a calculus which is almost as smooth as that of pseudo-differential operators. It also seems that one gains some more insight into the theory of pseudo-differential operators by considering them from the point of view of the wider classes of operators to be discussed here so we shall take the opportunity to include a short exposition.
2,450 citations
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.
Abstract: In this paper, we show that 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. From a physical point of view, we show that an isotropic conductivity can be determined by steady state measurements at the boundary.
1,608 citations
Book•
01 Apr 1985
TL;DR: The Maximal Torus of a Compact Lie Group and Root Systems are discussed in detail in this paper, where they are used to represent elementary representation theory and representative functions, respectively.
Abstract: I Lie Groups and Lie Algebras.- II Elementary Representation Theory.- III Representative Functions.- IV The Maximal Torus of a Compact Lie Group.- V Root Systems.- VI Irreducible Characters and Weights.- Symbol Index.
1,018 citations