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_28: 
@cite_18: 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.
@cite_25: Apparatus for use in shining shoes, including a pair of power driven shoe brushing means rotatable about a common horizontal axis, with each shoe brushing means operatively associated with a shoe supporting means. Each of the shoe supporting means is mounted for power rotation about a vertically disposed axis and includes releasable retaining means for engaging the interior of a supported shoe. The shoe supporting means are detailed in location relative to the brushing means so that all polished surfaces of a pair of supported shoes are contacted during rotation of the brushing means and shoe supporting means. A single motor is operatively associated with the shoe brushing means and the shoe supporting means for automatically effecting simultaneous rotation thereof in a shoe shining operation. The shoe shining apparatus includes polish dispensing means operatively associated with each of the brushing means, with the polish dispensing means including cover means supported between the polish dispensing means and the shoe brushing means and selectively movable between closed and opened conditions to permit polish to be applied to shoes during rotation of the brushes.
@cite_7: 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 .
Stability seemed to be proven first for the inverse problem of the conductivity equation. Liu @cite_25 showed it for potentials of the conductivity type. Barcel 'o, Faraco and Ru 'iz @cite_18 showed stability for H "older continuous conductivities. Clop, Faraco and Ru 'iz generalized it to @math , @math , in @cite_7 . For the Schr "odinger equation, there's the result of Novikov and Santacesaria for @math potentials in @cite_28 .