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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 \(\gamma\).
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
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[1] On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Rio de Janeiro, 1980, pp. 65–73.
[2] Cannon, Intl. J. Engrng. Sci. 1 pp 453– (1963)
[3] Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985. · Zbl 0695.35060
[4] Linear Partial Differential Operators, Grundlehren der Math. Wiss. 116, Springer, 1963.
[5] Isaacson, Comm. Pure Appl. Math. 36 pp 767– (1983)
[6] Kohn, Comm. Pure Appl. Math. 37 pp 289– (1984)
[7] and , Identification of an unknown conductivity by means of measurements at the boundary; Inverse Problems, ed., SIAM-AMS Proc. No. 14, 1984, pp. 113–123.
[8] Langer, Bull. AMS 39 pp 814– (1933)
[9] Lax, Comm. Pure Appl. Math. 9 pp 747– (1956)
[10] and , Non-Homogeneous Boundary Value Problems and Applications, I, Springer-Verlag, New York, 1972.
[11] Malgrange, Ann. Inst. Fourier Grenoble 6 pp 271– (1955-6) · Zbl 0071.09002
[12] Suzuki, Proc. Japan Acad. 56 pp 259– (1980)
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