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.
Some more recent results in two dimensions have concerned partial data, stability and reducing smoothness requirements for the conductivities and potentials. Notable results of partial data include Imanuvilov, Uhlmann, Yamamoto @cite_33 and Guillarmou and Tzou @cite_5 . In the first paper the authors consider the Schr "odinger equation in a plane domain and in the second one on a Riemann surface with boundary. The results of both papers state that knowing the Cauchy data on any open subset on the boundary determines the potential uniquely if it is smooth enough.
@cite_5 @cite_33
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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. In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from (2+1)-dimensional Minkowski spacetime into the 2-sphere. Our results provide strong evidence for the conjecture that large-energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from 2 into S2. 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. We relax the regularity condition on potentials of Schr "odinger equations in the uniqueness results in EB and IY2 for the inverse boundary value problem of determining a potential by Dirichlet-to-Neumann map.