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

Abstract: Given a general symmetric elliptic operator $$ L_{a} := \sum_{k,,j=1}^d \p_k (a_{kj} \p_j) + \sum_{k=1}^d a_k \p_k - \p_k(\overline{a_k} .) + a_0$$we define the associated Dirichlet-to-Neumann (D-t-N) operator 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 $a_0 -\la$ and $b_0 -\la$) coincide for all $\la$, 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 \cite{BR12} for Lipschitz continuous coefficients. We provide a different proof which works for bounded measurable coefficients and other boundary conditions.