On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data

Journal de Mathématiques Pures et Appliquées

2018

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Abstract. We give a new stability estimate for the problem of determining the time-dependent zero order coefficient in a parabolic equation from a partial parabolic Dirichlet-to-Neumann map. The novelty of our result is that, contrary to the previous works, we do not need any measurement on the final time. We also show how this result can be used to establish a stability estimate for the problem of determining the nonlinear term in a semilinear parabolic equation from the corresponding "linearized" Dirichlet-to-Neumann map. The key ingredient in our analysis is a parabolic version of an elliptic Carleman inequality due to Bukhgeim and Uhlmann [6]. This parabolic Carleman inequality enters in an essential way in the construction of CGO solutions that vanish at a part of the lateral boundary.

Section: 3)mentioning

confidence: 95%

Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term

Choulli

Journal de Mathématiques Pures et Appliquées

2018

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“…First, Theorem 1.1 is stated in a general unbounded domain subject only to condition (1.1). This makes an important difference with other related results which, to our best knowledge, have all been stated in specific unbounded domains like a slab, the half space or a cylindrical domain (see [33,37,14,15]). In particular, Theorem 1.1 holds true with domains having different types of geometrical deformations like bends or twisting, which are frequently used in problems of transmission for improving the propagation.…”

Section: Comments About Our Resultsmentioning

confidence: 82%

“…Here the goal of the inverse problem can be described as the unique recovery of an electromagnetic impurity perturbing the guided propagation (see [10,25]). Let us also mention that in this paper we consider general closed waveguides, only subjected to condition (1.1), that have not necessary a cylindrical shape comparing to other related works like [14,15,30]. This means that we can consider our inverse problem in closed waveguides with different types of geometrical deformations, including bends and twisting, which can be used in several context for improving the propagation of signals (see for instance [46]).…”

Section: Physical Motivationsmentioning

confidence: 99%

“…We refer also to [24,35,36,44,52] for other related inverse problems stated in a slab. In [14,15], the authors considered the stable recovery of coefficients periodic along the axis of an infinite cylindrical domain. More recently, [30] considered, for what seems to be the first time, the recovery of non-compactly supported and non-periodic electric potentials appearing in an infinite cylindrical domain.…”

Section: Physical Motivationsmentioning

confidence: 99%

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Determination of non-compactly supported electromagnetic potentials in an unbounded closed waveguide

2019

Rev. Mat. Iberoam.

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We study the inverse problem of determining a magnetic Schrödinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with general bounded and non-compactly supported electromagnetic potentials. By assuming that the electromagnetic potentials are known on the neighborhood of the boundary outside a compact set, we even prove the unique determination of the magnetic field and the electric potential from measurements restricted to a bounded subset of the infinite boundary. Finally, in the case of a waveguide taking the form of an infinite cylindrical domain, we prove the recovery of the magnetic field and the electric potential from partial data corresponding to restriction of Neumann boundary measurements to slightly more than half of the boundary. We establish all these results by mean of a new class of complex geometric optics solutions and of Carleman estimates suitably designed for our problem stated in an unbounded domain and with bounded electromagnetic potentials.

“…In [30], periodic potentials are stably retrieved from the asymptotics of the boundary spectral data of the Dirichlet Laplacian. Finally, we refer to [19,20], for the analysis of the Calderón problem in a periodic waveguide.…”

mentioning

confidence: 99%

An Inverse Problem for the Magnetic Schrödinger Equation in Infinite Cylindrical Domains

Bellassoued

Publ. Res. Inst. Math. Sci.

Soccorsi

2018

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We study the inverse problem of determining the magnetic field and the electric potential entering the Schrödinger equation in an infinite 3D cylindrical domain, by Dirichlet-to-Neumann map. The cylindrical domain we consider is a closed waveguide in the sense that the cross section is a bounded domain of the plane. We prove that the knowledge of the Dirichlet-to-Neumann map determines uniquely, and even Hölderstably, the magnetic field induced by the magnetic potential and the electric potential. Moreover, if the maximal strength of both the magnetic field and the electric potential, is attained in a fixed bounded subset of the domain, we extend the above results by taking finitely extended boundary observations of the solution, only.Keywords: Inverse problem, magnetic Schrödinger equation, Dirichlet-to-Neumann map, infinite cylindrical domain.1.3. State of the art. Inverse coefficients problems for partial differential equations such as the Schrödinger equation, are the source of challenging mathematical problems, and have attracted many attention over the last decades. For instance, using the Bukhgeim-Klibanov method (see [14,35,36]), [3] claims Lipschitz stable determination of the time-independent electric potential perturbing the dynamic (i.e. non stationary) Schrödinger equation, from a single boundary measurement of the solution. In this case, the observation is performed on a sub boundary fulfilling the geometric optics condition for the observability, derived by Bardos, Lebeau and Rauch in [2]. This geometrical condition was removed by [8] for potentials which are a priori known in a neighborhood of the boundary, at the expense of weaker stability. In the same spirit, [22] Lipschitz stably determines by means of the Bughkgeim-Klibanov technique, the magnetic potential in the Coulomb gauge class, from a finite number of boundary measurements of the solution. Uniqueness results in inverse problems for the DN map related to the magnetic Schrödinger equation are also available in [24], but they are based on a different approach involving geometric optics (GO) solutions. The stable recovery of the magnetic field by the DN map of the dynamic magnetic Schrödinger equation is established in [9] by combining the approach used for determining the potential in hyperbolic equations (see [5,7,11,28,44,47,49]) with the one employed for the idetification of the magnetic field in elliptic equations (see [23,45,50]). Notice that in the one-dimensional case, [1] proved by means of the boundary control method introduced by [4], that the DN map uniquely determines the time-independent electric potential of the Schrödinger equation. In [10] the time-independent electric potential is stably determined by the DN map associated with the dynamic magnetic Schrödinger equation on a Riemannian manifold. This result was recently extended by [6] to simultaneous determination of both the magnetic field and the electric potential. As for inverse coefficients problems of the Schrödinger equation with either Neumann, spectral, or scatte...