×

Determining the resistors in a network. (English) Zbl 0717.35092

A. P. Calderon, R. Kohn, M. Vogelius, J. Sylvester, G. Uhlmann and others have studied the so-called impedance tomography, i.e., the determination of the conductivity \(\gamma\) (x) satisfying div(\(\gamma\) (x)grad u)\(=0\) in a domain \(\Omega \subset {\mathbb{R}}^ n\) by means of measurements of the voltage u and its current \(\gamma (\partial u/\partial n)=\Lambda_{\gamma}u\) on \(\partial \Omega\), where u runs through a set of functions. In this paper a similar problem is studied relative to a rectangular network \(\Omega_ 0\subset {\mathbb{R}}^ 2\), where \(p=(i,j)\in \Omega_ 0\), i,j integers. This is a problem of discrete analysis. Following Kirchhoff’s laws a so-called \(\gamma\)-harmonic function f: \(\Omega\) \({}_ 0\to {\mathbb{R}}\) is introduced. A quadratic form \(Q_{\gamma}\) is defined. Similar to the paper of Calderon and others a uniqueness theorem is proved:
If \(\gamma_ 1\neq \gamma_ 2\) then \(Q_{\gamma_ 1}\neq Q_{\gamma_ 2}.\)
The paper contains further results for the problem under consideration.
Reviewer’s remark: The coefficients \(a_{\alpha}(x)\) of a partial differential equation are often densities which are to be determined by finitely many measurements on the boundary of a given domain. In general the values \(a_{\alpha}(x)\) cannot be uniquely determined by the boundary values. Therefore the differential equation has to be written in integral form (weak solutions) and certain average values (integrals) of \(a_{\alpha}(x)\) are to be introduced, i.e., the differential equation must be adapted to inverse theory. Then one can expect more useful results. This paper seems to be important in future.
Reviewer: G.Anger

MSC:

35R30 Inverse problems for PDEs
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
90B10 Deterministic network models in operations research
PDFBibTeX XMLCite
Full Text: DOI