The inverse conductivity problem in two dimensions
The author considers the problem of recovering the coefficient \(\gamma\) entering the Dirichlet problem \[ div(\gamma \nabla u)=0\text{ in } \Omega,\quad u=g\text{ on } \partial \Omega \] from the Dirichlet-to- Neumann map \(g\mapsto \partial u/\partial N\) on \(\partial \Omega\). Here \(\Omega\) is a bounded domain in \({\mathbb{R}}^ 2\) with smooth boundary and \(\gamma \in L_{\infty}(\Omega)\), \(\gamma >0\). When \(\Delta \gamma \in L_{\infty}(\Omega)\) the global uniqueness result was obtained by \textit{J. Sylvester} and \textit{G. Uhlmann} [Ann. Math., II. Ser. 125, 153-169 (1987; Zbl 0625.35078)]; see also their review paper ``The Dirichlet-to- Neumann map and applications'' [Inverse problems in partial differential equations, Proc. Conf., Pt. Summer Res. Conf. Math. Sci., Arcata/CA (US) 189, 101-139 (1990)]. In the plane case uniqueness is known only if \(\gamma\) is close to a constant. In this paper by using an expansion of almost exponential solutions to the equation and some orthogonality relations the author proves global uniqueness and stability results when \(\Delta\) log \(\gamma\) \(=0\) or \(\Delta \gamma^{\beta}=0\) for some \(\beta\).
scientific article; zbMATH DE number 4179970
scientific article