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Uniqueness in the two-dimensional inverse conductivity problems of determining convex polygonal supports: case of variable conductivity. (English) Zbl 1081.35142

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.

MSC:

35R30 Inverse problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
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