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

Kohn, Robert V.; Vogelius, Michael

Commun. Pure Appl. Math. 38, 643-667 (1985).

[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 \(\gamma\).

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

References:

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