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.
@cite_11: This paper is a reprint of the original work by A. P. Calderon published by the Brazilian Mathematical Society (SBM) in ATAS of SBM (Rio de Janeiro), pp. 65-73, 1980. The original paper had no abstract, so this reprint to be truthful to the original work is published with no abstract.
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 .