SMF

Inverse boundary value problems and applications

Inverse boundary value problems and applications

Gunther ULHMANN
  • Année : 1992
  • Tome : 207
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 35R30, 35P05, 81C10
  • Pages : 153-211
  • DOI : 10.24033/ast.149

In these notes we give an overview of inverse boundary value problems. In these problems one attemps to discover internal properties of a body by making measurements at the boundary. We concentrate mainly in the problem of determining the conductivity of a body by making measurements of voltage potentials and corresponding fluxes at the boundary. This problem is often referred to as Electric Impedance Tomography. We give applications to inverse scattering as well as inverse spectral problems. We consider first the isotropic case. In this case the conductivity does not depend on direction. We reduce the problem to an inverse boundary value problemfor the Schrödinger equation, at zero energy, for a compactly supported potential. More precisely the known information is encoded by the it Dirichlet to Neumann map. In this notes we describe how the construction of exponential growing solutions allows to prove that the potential is uniquely determined by knowledge of this map in dimension $n > 2$. We also discuss progress made in the two dimensional case. The method allow also to find reconstruction methods as well as to obtain estimates of the potential in terms of the given Dirichlet to Neumann map. The same techniques are used to prove similar results for the inverse scattering problem at a fixed energy. In the most general case in which the conductivity depends on direction, usually referred to as anisotropic case, there is a natural obstruction to uniqueness. We report on the progress made on this problem.

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