Given a general symmetric elliptic operator
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, -estimates and domination properties for the semigroup associated with this D-t-N operator. Given and of the previous type with bounded measurable coefficients and , we prove that if their partial D-t-N operators (with and replaced by and ) coincide for all , then the operators and , 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 Behrndt and Rohleder  for Lipschitz continuous coefficients. We provide a different proof, based on spectral theory, which works for bounded measurable coefficients and other boundary conditions.
Cite this article
El Maati Ouhabaz, A "milder" version of Calderón's inverse problem for anisotropic conductivities and partial data. J. Spectr. Theory 8 (2018), no. 2, pp. 435–457DOI 10.4171/JST/201