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.
There are also some results for the inverse boundary value problem of the Schr "odinger equation whose potential is not assumed to be of the conductivity type. Jerison and Kenig proved, according to @cite_8 , that if @math with @math , @math , then the Dirichlet-Neumann map @math determines the potential @math uniquely. The case @math was open until the paper of Bukhgeim. In @cite_36 , he introduced new kinds of solutions to the Schr "odinger equation, which allow the use of stationary phase. This led to an elegant solution of this long standing open problem. There is a point in the argument that requires differentiability of the potentials. Imanuvilov and Yamamoto published the paper @cite_38 in arXiv after the writing of this thesis. They seem to have fixed that problem and hence proven uniqueness for @math , @math . non-smooth potential L@ @math potential|see non-smooth potential
@cite_36 @cite_38 @cite_8
1551397227 2093747300 2963034365 2028156707
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. 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. On a fixed smooth compact Riemann surface with boundary (M0, g), we show that for the Schrodinger operator ∆ + V with potential V ∈ C(M0) for some α > 0, the Dirichlet-to-Neumann map N|Γ measured on an open set Γ ⊂ ∂M0 determines uniquely the potential V . We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends. 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.