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Singular solutions of elliptic equations and the determination of conductivity by boundary measurements. (English) Zbl 0778.35109
The author considers the inverse problem of determining the positive coefficient \(a=a(x)\) of the elliptic equation (i) \(\text{div}(a\nabla u)=0\) in \(\Omega\subseteq\mathbb{R}^ n\), from the knowledge of the Dirichlet to Neumann operator \(\Lambda_ a:u|_{\partial\Omega}\mapsto(\partial u/\partial\nu)|_{\partial\Omega}\) (here \(\nu\) is the exterior normal to \(\partial\Omega)\). This paper is the latest advance in a series of results which date back to the work of Kohn and Vogelius, Sylvester and Uhlmann, and the author himself. It is chiefly concerned with the problems of uniqueness and stability. A first result shows that, if \(a\) and \(\partial\Omega\) are Lipschitz continuous, then \(\Lambda_ a\) determines \(a\) on \(\partial\Omega\) among all piecewise analytic perturbations of \(a\). Later on, uniqueness is also achieved in the Sobolev class \(W^{2,\infty}(\Omega)\).
The main stability estimate can be stated as follows. If \(a,b\in\text{Lip}(\overline\Omega)\) and \(b-a\) has bounded \(C^{k+\alpha}\) norm in a neighborhood of \(\partial\Omega\), then \(\| D^ \eta(a-b)\|_{L^ \infty(\partial\Omega)}\leq C\|\Lambda_ a-\Lambda_ b\|_ *^{\delta_ k}\), where \(\|\cdot\|_ *\) denotes the norm for linear operators from \(H^{1/2}(\partial\Omega)\) to its dual and \(\delta_ k=\prod^ k_{j=0}\alpha/(\alpha+j)\). In the class \(W^{2,\infty}(\Omega)\), a global stability estimate of logarithmic type is also proved. The method employed by the author relies on the construction of solutions of (i) which have an isolated singularity with prescribed asymptotic behaviour.
Finally, the above results are extended to the case of the anisotropic equation \(\text{div}(A(a(x))\nabla u)=0\), where \(t\mapsto A(t)\) is a one- parameter family of symmetric, positive-definite matrices with the property that \(dA/dt\) is also positive-definite.

MSC:
35R30 Inverse problems for PDEs
35J15 Second-order elliptic equations
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