We prove uniqueness and stability for the inverse boundary value problem of the two dimensional Schr "odinger equation. We do not assume the potentials to be continuous or even bounded. Instead, we assume that some of their positive fractional derivatives are in a specific Lorentz space. These spaces are a natural generalization to the usual fractional Sobolev spaces. The thesis consists of two parts. In the first part, we define the generalized fractional Sobolev spaces and prove some of their properties including embeddings and interpolation identities. In particular we sharpen the usual Sobolev embedding into the space of H "older-continuous functions, by showing that a particular kind of space embeds into the space of continuous functions without any modulus of continuity. The inverse problem is considered in the second part of the thesis. We prove a new Carleman estimate for @math . This estimate has a fast decay rate, which will allow us to consider potentials with very low regularity. After that we use Bukhgeim's oscillating exponential solutions, Alessandrini's identity and stationary phase to get information about the difference of the potentials from the difference of the Cauchy data. The stability estimate will be of logarithmic type, but works with potentials of low regularity.
One of the important early papers on inverse boundary value problems is by Calder 'on @cite_11 . He considered an isotropic body @math from which one would like to deduce the electrical conductivity @math by doing electrical measurements on the boundary. If we keep the voltage @math fixed as @math on the boundary, then the stationary state of @math can be modeled by the boundary value problem The weighted normal derivative @math is the current flux going out of @math . Calder 'on asked whether knowing the boundary measurements, or Dirichlet-Neumann map @math , is enough to determine the conductivity @math inside the whole domain @math . This is called the . inverse problem!Calder 'on He showed the injectivity of a linearized problem near @math .
@cite_11
2093747300 2009083582 2028156707 2117051440
Abstract We study the multi-channel Gelʼfand–Calderon inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation − Δ ψ + v ( x ) ψ = 0 , x ∈ D , where v is a smooth matrix-valued potential defined on a bounded planar domain D . We give an exact global reconstruction method for finding v from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide. It is proved that, in two dimensions, the Calderon inverse conductivity problem in Lipschitz domains is stable in the @math sense when the conductivities are uniformly bounded in any fractional Sobolev space @math @math . We prove for a two dimensional bounded domain that the Cauchy data for the Schrodinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results. It is proved that, in two dimensions, the Calderon inverse conductivity problem in Lipschitz domains is stable when the conductivities are Holder continuous with any exponent α> 0. © 2007 Elsevier Masson SAS. All rights reserved.