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Start date: 2019 × End date: 2011 × DDC: 510 × Type: Report × Subject: uniqueness ×

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Now showing 1 - 10 of 16
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    Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Elschner, Johannes; Yamamoto, Masahiro
    We consider the inverse acoustic scattering problem of determining a sound-hard obstacle by far field measurements. It is proved that a polyhedral scatterer in $R^n, nge 2$, consisting of finitely many solid polyhedra, is uniquely determined by a single incoming plane wave.
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    Inverse scattering of electromagnetic waves by multilayered structures : uniqueness in TM mode
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, Guanghui
    Assume a time-harmonic electromagnetic wave is scattered by an infinitely long cylindrical conductor surrounded by an unknown piecewise homogenous medium remaining invariant along the cylinder axis. We prove that, in TM mode, the far field patterns for all observation directions at a fixed frequency uniquely determine the unknown surrounding medium as well as the shape of the cylindrical conductor. A similar uniqueness result is obtained for the scattering by multilayered penetrable periodic structures in a piecewise homogenous medium. The periodic interfaces and refractive indices can be uniquely identified from the near field data measured only above (or below) the structure for all quasi-periodic incident waves with a fixed phase-shift. The proofs are based on the singularity of the Green function to a two dimensional elliptic equation with piecewise constant leading coefficients
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    Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, Guanghui
    In this paper, we investigate the inverse problem of recovering a two-dimensional perfectly reflecting diffraction grating from the scattered waves measured above the structure. Inspired by a novel idea developed by Bao, Zhang and Zou [to appear in Trans. Amer. Math. Soc.], we present a complete characterization of the global uniqueness in determining polygonal periodic structures using a minimal number of incident plane waves. The idea in this paper combines the reflection principle for the Helmholtz equation and the dihedral group theory. We characterize all periodic polygonal structures that cannot be identified by one incident plane wave, including the resonance case where a Rayleigh frequency is allowed. Furthermore, we show that those unidentifiable gratings provide non-uniqueness examples for appropriately chosen wave number and incident angles. We also indicate and fix a gap in the proof of the main theorem of Elschner and Yamamoto [Z. Anal. Anwend., 26 (2007), 165-177], and generalize the uniqueness results of that paper.
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    Stationary solutions to an energy model for semiconductor devices where the equations are defined on different domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Glitzky, Annegret; Hünlich, Rolf
    We discuss a stationary energy model from semiconductor modelling. We accept the more realistic assumption that the continuity equations for electrons and holes have to be considered only in a subdomain $Omega_0$ of the domain of definition $Omega$ of the energy balance equation and of the Poisson equation. Here $Omega_0$ corresponds to the region of semiconducting material, $OmegasetminusOmega_0$ represents passive layers. Metals serving as contacts are modelled by Dirichlet boundary conditions. We prove a local existence and uniqueness result for the two-dimensional stationary energy model. For this purpose we derive a $W^1,p$-regularity result for solutions of systems of elliptic equations with different regions of definition and use the Implicit Function Theorem.
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    On a higher order convective Cahn-Hilliard type equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Korzec, Maciek Dominik; Rybka, Piotr
    A convective Cahn-Hilliard type equation of sixth order that describes the faceting of a growing surface is considered with periodic boundary conditions. By using a Galerkin approach the existence of weak solutions to this sixth order partial differential equation is established in $L^2(0,T; dot H^3_per)$. Furthermore stronger regularity results have been derived and these are used to prove uniqueness of the solutions. Additionally a numerical study shows that solutions behave similarly as for the better known convective Cahn-Hilliard equation. The transition from coarsening to roughening is analyzed, indicating that the characteristic length scale decreases logarithmically with increasing deposition rate
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    Inverse scattering of elastic waves by periodic structures : uniqueness under the third or fourth kind boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Elschner, Johannes; Hu, Guanghui
    The inverse scattering of a time-harmonic elastic wave by a two-dimensional periodic structure in R 2 is investigated. The grating profile is assumed to be a graph given by a piecewise linear function on which the third or fourth kind boundary conditions are satisfied. Via an equivalent variational formulation, existence of quasi-periodic solutions for general Lipschitz grating profiles is proved by applying the Fredholm alternative. However, uniqueness of solution to the direct problem does not hold in general. For the inverse problem, we determine and classify all the unidentifiable grating profiles corresponding to a given incident elastic field, relying on the reflection principle for the Navier equation and the rotational invariance of propagating directions of the total field. Moreover, global uniqueness for the inverse problem is established with a minimal number of incident pressure or shear waves, including the resonance case where a Rayleigh frequency is allowed. The gratings that are unidentifiable by one incident elastic wave provide non-uniqueness examples for appropriately chosen wave number and incident angles
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    An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Hu, Guanghui; Yang, Jiaqing; Zhang, Bo
    This paper is concerned with uniqueness for reconstructing a periodic inhomogeneous medium covered on a perfectly conducting plate. We deal with the problem in the frame of time-harmonic Maxwell systems without TE or TM polarization. An orthogonal relation for two refractive indices is obtained, and then inspired by Kirsch's idea, the refractive index can be identified by utilizing the eigenvalues and eigenfunctions of a quasi-periodic Sturm-Liouville eigenvalue problem.
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    Uniqueness in inverse elastic scattering with finitely many incident waves
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Elschner, Johannes; Yamamoto, Masahiro
    We consider the third and fourth exterior boundary value problems of linear isotropic elasticity and present uniqueness results for the corresponding inverse scattering problems with polyhedral-type obstacles and a finite number of incident plane elastic waves. Our approach is based on a reflection principle for the Navier equation.
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    Uniqueness in inverse scattering of elastic waves by three-dimensional polyhedral diffraction gratings
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Elschner, Johannes; Hu, Guanghui
    Literaturverz. S. 35 We consider the inverse elastic scattering problem of determining a three-dimensional diffraction grating profile from scattered waves measured above the structure. In general, a grating profile cannot be uniquely determined by a single incoming plane wave. We completely characterize and classify the bi-periodic polyhedral structures under the boundary conditions of the third and fourth kinds that cannot be uniquely recovered by only one incident plane wave. Thus we have global uniqueness for a polyhedral grating profile by one incident elastic plane wave if and only if the profile belongs to neither of the unidentifiable classes, which can be explicitly described depending on the incident field and the type of boundary conditions. Our approach is based on the reflection principle for the Navier equation and the reflectional and rotational invariance of the total field.
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    Inverse wave scattering by unbounded obstacles : uniqueness for the two-dimensional Helmholtz equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Hu, Guanghui
    In this paper we present some uniqueness results on inverse wave scattering by unbounded obstacles for the two-dimensional Helmholtz equation. We prove that an impenetrable one-dimensional rough surface can be uniquely determined by the values of the scattered field taken on a line segment above the surface that correspond to the incident waves generated by a countable number of point sources. For penetrable rough layers in a piecewise constant medium, the refractive indices together with the rough interfaces (on which the TM transmission conditions are imposed) can be uniquely identified using the same measurements and the same incident point source waves. Moreover, a Dirichlet polygonal rough surface can be uniquely determined by a single incident point source wave provided a certain condition is imposed on it.