A remark on precomposition on $\sH^{1/2}(S^1)$ and $\eps$-identifiability of disks in tomography

Details A remark on precomposition on $\sH^{1/2}(S^1)$ and $\eps$-identifiability of disks in tomography A remark on precomposition on $\sH^{1/2}(S^1)$ and $\eps$-identifiability of disks in tomography

2006-07-07

arxiv.org/abs/math/0607205v1

Mathematics

Optimization and Control

Scientific paper

We consider the inverse conductivity problem with one measurement for the equation $div((\sigma\_1+(\sigma\_2-\sigma\_1)\chi\_D)\nabla{u})=0$ determining the unknown inclusion $D$ included in $\Omega$. We suppose that $\Omega$ is the unit disk of $\mathbb{R}^2$. With the tools of the conformal mappings, of elementary Fourier analysis and also the action of some quasi-conformal mapping on the Sobolev space $\sH^{1/2}(S^1)$, we show how to approximate the Dirichlet-to-Neumann map when the original inclusion $D$ is a $\epsilon-$ approximation of a disk. This enables us to give some uniqueness and stability results.

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