Kim, Sungwhan; Yamamoto, Masahiro Uniqueness in the two-dimensional inverse conductivity problems of determining convex polygonal supports: case of variable conductivity. (English) Zbl 1081.35142 Inverse Probl. 20, No. 2, 495-506 (2004). Summary: In a bounded domain \(\Omega\subset\mathbb{R}^2\), we consider \[ \text{div} ((\ell(x)+m(x)\chi_D(x))\nabla u(x))+k(x)u(x)=0, \] where \(\ell,m\in C^2(\overline\Omega)\) satisfy \(\ell,\ell+m>0\) on \(\Omega\), the subdomain \(D\) is the union of a finite number of polygons, \(\chi_D\) denotes the characteristic function of \(D\) and \(k\in L^\infty(\Omega)\), \(k\geq 0\) on \(\overline\Omega\). We discuss an inverse problem of determining \(D\) by the boundary data of \(u\). Our main results are stated as follows. (i) Case \(k\equiv 0\): a suitably given single Dirichlet datum of \(u\) on \(\partial\Omega\) yields the uniqueness of the convex hull of \(D\). (ii) Case \(0<k(x)<\lambda_0\), where \(\lambda_0\) is the product of the minimum of \(\ell\) and the first eigenvalue of \(-\Delta\) with the zero Dirichlet boundary condition. By changing Dirichlet inputs to \(u\) on \(\partial\Omega\) twice and suitably, the resulting two Neumann data guarantee the uniqueness of the convex hull. Cited in 2 Documents MSC: 35R30 Inverse problems for PDEs 35J25 Boundary value problems for second-order elliptic equations PDFBibTeX XMLCite \textit{S. Kim} and \textit{M. Yamamoto}, Inverse Probl. 20, No. 2, 495--506 (2004; Zbl 1081.35142) Full Text: DOI