A ``milder'' version of Calderón's inverse problem for anisotropic conductivities and partial data (Q1650223)

Summary: Given a general symmetric elliptic operator \[ L_{a}:=\sum_{k,j=1}^d\partial_k(a_{kj}\partial_j)+\sum_{k=1}^da_k\partial_k-\partial_k(\overline{a}_k.)+a_0 \] we define the associated Dirichlet-to-Neumann (D-t-N) map with partial data, i.e., data supported in a part of the boundary. We prove positivity, \(L^p\)-estimates and domination properties for the semigroup associated with this D-t-N operator. Given \(L_a \) and \(L_b\) of the previous type with bounded measurable coefficients \(a=\{a_{kj},a_k,a_0\}\) and \(b=\{b_{kj},b_k,b_0\}\), we prove that if their partial D-t-N operators (with \(a_0\) and \(b_0\) replaced by \(a_0-\lambda\) and \(b_0-\lambda\)) coincide for all \(\lambda\), then the operators \(L_a\) and \(L_b\), endowed with Dirichlet, mixed or Robin boundary conditions are unitary equivalent. In the case of the Dirichlet boundary conditions, this result was proved recently by \textit{J. Behrndt} and \textit{J. Rohleder} [Commun. Partial Differ. Equations 37, No. 4--6, 1141--1159 (2012; Zbl 1244.35164)] for Lipschitz continuous coefficients. We provide a different proof, based on spectral theory, which works for bounded measurable coefficients and other boundary conditions.