Source supports in electrostatics

Abstract

We investigate the inverse source problem of electrostatics in a bounded and convex domain with compactly supported source. We try to extract all information about the unknown source support from the given Cauchy data of the associated potential, adopting by this previous work of Kusiak and Sylvester to the case of electrostatics. We introduce, and for the unit disk we also compute numerically, what we call the discoidal source support, i.e., the smallest set made up by the intersection of disks within the domain, which carries a source compatible with the given data. (2000): 35R30, 65N21. AMS subject classification

Convex Source Support and Its Application to Electric Impedance Tomography

“…Since the Neumann boundary values of u 1 and u 2 can be made homogeneous by subtracting the same harmonic function, it follows easily that L ρ F 1 = L ρ F 2 =: g ρ . As in Example 3.3 of [10], we may argue that every source compatible with the data g ρ must contain both (−1/2, 0) and (1/2, 0) in its convex support, meaning that C ρ g ρ = Γ 1 . (Take note that Γ 2 could have been chosen as any smooth curve that is contained in the intersection of B ρ and the upper half-plane, does not intersect itself, and has the end points (−1/2, 0) and (1/2, 0).)…”

“…The corresponding theory was presented in [10] for electrostatics, and the notion of convex source support was introduced. The purpose of the present paper is to develop an efficient algorithm for computing the convex source support corresponding to the measurement u| ∂D = (v−v 0 )| ∂D in (1.2) and thus introduce a method for extracting information on the location of the support of the inhomogeneity σ − 1 from only one (or a few) measurement of EIT.…”

“…[10,19]). Furthermore, the intersection of the supports of the sources compatible with g is, in general, empty (cf.…”

“…Furthermore, the intersection of the supports of the sources compatible with g is, in general, empty (cf. [19]), and this may hold even if the holes in the supports are included before the intersection is taken [10]. However, intersecting the convex hulls of the supports of the compatible sources provides information on the original source as explained in the following (cf.…”

“…However, intersecting the convex hulls of the supports of the compatible sources provides information on the original source as explained in the following (cf. [10]). …”