Optimal stability estimate in the inverse boundary value problem for periodic potentials with partial data

We consider the inverse boundary value problem for operators of the form −△+q in an infinite domain Ω=R×ω⊂R1+n, n≥3, with a periodic potential q. For Dirichlet-to-Neumann data localized on a portion of the boundary of the form Γ1=R×γ1, with γ1 being the complement either of a flat or spherical portion of ∂ω, we prove that a log-type stability estimate holds.