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Determining anisotropic real-analytic conductivities by boundary measurements. (English) Zbl 0702.35036

From the introduction: “If an electrical potential is applied to the surface of a solid body, the current flux across the surface depends on the conductivity in the interior of the body. We want to consider the inverse problem of determining the conductivity by these boundary measurements. More precisely, if \(\Omega\) is a smoothly bounded domain in \({\mathbb{R}}^ n\), an (anisotropic) conductivity is a smooth, symmetric, positive definite matrix-valued function \(\gamma =(\gamma^{ij})\) on \({\bar \Omega}\). If a potential \(f\in C^{\infty}(\partial \Omega)\) is applied to the boundary, then the potential u in \(\Omega\) solves the Dirichlet problem \[ L_{\gamma}u=0\text{ in } \Omega,\quad u|_{\partial \Omega}=f, \] where \(L_{\gamma}u=\sum^{n}_{i,j=1}\frac{\partial}{\partial x^ i}(\gamma^{ij}\frac{\partial u}{\partial x^ j}).\)
The current flux at the boundary is then given by the (n-1)-form \[ \Lambda_{\gamma}f=(\gamma du)\rfloor (dx^ 1\wedge...\wedge dx^ n)|_{\partial \Omega}, \] where \(\gamma du=\sum^{n}_{i,j=1}\gamma^{ij}(\partial u/\partial x^ i)(\partial /\partial x^ j)\). The map \(\Lambda_{\gamma}\) from functions on \(\partial \Omega\) to (n-1)-forms on \(\partial \Omega\) is called the Dirichlet-to-Neumann map or voltage-to-current map associated with \(\gamma\). The inverse problem we want to address is to recover \(\gamma\) from \(\Lambda_{\gamma}\).”

MSC:

35B60 Continuation and prolongation of solutions to PDEs
35R30 Inverse problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
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