A "milder" version of Calderón's inverse problem for anisotropic conductivities and partial data

Abstract

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, $L^p$-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 Behrndt and Rohleder [6] for Lipschitz continuous coefficients. We provide a different proof, based on spectral theory, which works for bounded measurable coefficients and other boundary conditions.