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**Determining conductivity by boundary measurements. II: Interior results.**
*(English)*
Zbl 0595.35092

[For Part I, see ibid. 37, 289-297 (1984; Zbl 0576.35116).]

Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ n\) \(n\geq 2\), with boundary \(\Gamma\). For any \(\gamma \in L^{\infty}(\Omega)\) \(0<\gamma_ 0\leq \gamma (x)\), let \(L_{\gamma}\) denote the operator \(L_{\gamma}(u)=\nabla \cdot (\gamma \nabla u)\) acting on \(H^ 1(\Omega)\). For \(\phi \in H^{1/2}(\Gamma)\) \(Q_{\gamma}(\phi)\) denotes the energy of the solution of associated boundary value problem \[ (1)\quad L_{\gamma}(u)=0,\quad u|_{\Gamma}=\phi \] in other words \[ \quad (2)\quad Q_{\gamma}(\phi)=\int_{\Omega}\gamma | \nabla u|^ 2dx. \] For sufficiently smooth \(\Gamma\) the Green’s formula gives \[ (3)\quad Q_{\gamma}(\phi)=\int_{\Gamma}u\gamma (\partial u/\partial \nu)ds, \] where ds is the surface element, \(\partial u/\partial \nu\) is the normal derivate. The aim is to determine \(\gamma\) given knowledge of the quadratic form \(Q_{\gamma}\). The \(\gamma\) is identifiable by boundary measurements if the map \(\gamma \to Q_{\gamma}\) is injective.

In Part I (loc. cit.) the authors have proved that, \(Q_{\gamma}\) determines \(\gamma\) and all its derivatives at the boundary provided \(\gamma\) is smooth near the boundary. The above mentioned paper concerned to real-analytic conductivity \(\gamma\).

In the present paper, the authors extend their analysis for piecewise real-analytic conductivities. It is proved that similar results hold for piecewise real analytic conductivities in two dimensional case as for real analytic conductivities. In a special case of a layered structure the authors show that a three times continuously differentiable conductivity is identifiable by boundary measurements. The paper gives a convergence reconstruction algorithm to obtain the real analytic conductivity \(\gamma\).

Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ n\) \(n\geq 2\), with boundary \(\Gamma\). For any \(\gamma \in L^{\infty}(\Omega)\) \(0<\gamma_ 0\leq \gamma (x)\), let \(L_{\gamma}\) denote the operator \(L_{\gamma}(u)=\nabla \cdot (\gamma \nabla u)\) acting on \(H^ 1(\Omega)\). For \(\phi \in H^{1/2}(\Gamma)\) \(Q_{\gamma}(\phi)\) denotes the energy of the solution of associated boundary value problem \[ (1)\quad L_{\gamma}(u)=0,\quad u|_{\Gamma}=\phi \] in other words \[ \quad (2)\quad Q_{\gamma}(\phi)=\int_{\Omega}\gamma | \nabla u|^ 2dx. \] For sufficiently smooth \(\Gamma\) the Green’s formula gives \[ (3)\quad Q_{\gamma}(\phi)=\int_{\Gamma}u\gamma (\partial u/\partial \nu)ds, \] where ds is the surface element, \(\partial u/\partial \nu\) is the normal derivate. The aim is to determine \(\gamma\) given knowledge of the quadratic form \(Q_{\gamma}\). The \(\gamma\) is identifiable by boundary measurements if the map \(\gamma \to Q_{\gamma}\) is injective.

In Part I (loc. cit.) the authors have proved that, \(Q_{\gamma}\) determines \(\gamma\) and all its derivatives at the boundary provided \(\gamma\) is smooth near the boundary. The above mentioned paper concerned to real-analytic conductivity \(\gamma\).

In the present paper, the authors extend their analysis for piecewise real-analytic conductivities. It is proved that similar results hold for piecewise real analytic conductivities in two dimensional case as for real analytic conductivities. In a special case of a layered structure the authors show that a three times continuously differentiable conductivity is identifiable by boundary measurements. The paper gives a convergence reconstruction algorithm to obtain the real analytic conductivity \(\gamma\).

Reviewer: I.Ecsedi

### MSC:

35Q99 | Partial differential equations of mathematical physics and other areas of application |

35R30 | Inverse problems for PDEs |

35A99 | General topics in partial differential equations |

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65N99 | Numerical methods for partial differential equations, boundary value problems |

### Keywords:

Green’s formula; quadratic form; piecewise real-analytic conductivities; layered structure; reconstruction algorithm### Citations:

Zbl 0576.35116
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\textit{R. V. Kohn} and \textit{M. Vogelius}, Commun. Pure Appl. Math. 38, 643--667 (1985; Zbl 0595.35092)

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### References:

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