Crack detection by the electric method: Uniqueness and~approximation

Let U be a bounded domain in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{R}^2 $$ \end{document} the boundary ∂ U of which is a closed nonself-intersecting curve. Let ω be a hole (a crack) in the interior of U the boundary of which is a closed nonself-intersecting curve. Let U and ω be star-shaped. It is shown that the problem of finding a harmonic function u and its domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Omega = U\backslash \bar \omega (\bar \omega \equiv \omega {\text{ }} \cup {\text{ }}\partial \omega ) $$ \end{document}, subject to given Cauchy conditions on an open subset of ∂ U and to the condition ∂ u/∂n=0 on ∂ω has at most one solution. This uniqueness result shows that an insulating crack (e.g. one made up of air) can be identified by the electric method. The problem of determining ∂ω is an ill-posed problem; it is here regularized using finite dimensional approximations.