Double logarithmic stability in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map

We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schrödinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a $\log\log$ stability estimate for the $L^2$-norm (resp. the $H^{-1}$-norm) of $H^t$, for $t>0$, and bounded (resp. $L^2$) potentials.