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_9: We consider the problem of determining the scalar coefficient γ in the elliptic equation div(γ grad u) = 0 in ω when, for every Dirichlet datum u = ∅ on ∂ω , the Neumann datum γ(∂ ∂ n)u = ∧.γ∅ is known. We prove a continuous dependence result
@cite_16: In this paper, we show that the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n ? 3. From a physical point of view, we show that an isotropic conductivity can be determined by steady state measurements at the boundary.
Sylvester and Uhlmann solved the problem in dimensions @math at least three for smooth conductivities bounded away from zero @cite_16 . They constructed solutions!complex geometric optics|textbf , that is, solutions of the form where the complex vectors @math satisfy where @math are perpendicular vectors satisfying @math . Using a well-known orthogonality relation for the potentials @math and @math , called the Alessandrini identity @cite_9 Alessandrini's identity , they got and after taking @math they saw that the Fourier transforms of @math and @math are the same, so the potentials are so too. Note that the only part that requires @math in this solution is the existence of the three vectors @math .