Extendability of functions with partially vanishing trace

Let Ω ⊆ $R$$^d$ be open and D ⊆ ∂Ω be a closed part of its boundary. Under very mild assumptions on Ω, we construct a bounded Sobolev extension operator for the Sobolev space W$^{k,p}_D$ (Ω), 1 $\leq$ p $\leq$ ∞, which consists of all functions in W$^{k,p}$(Ω) that vanish in a suitable sense on D. In contrast to earlier work, this construction is global and does not use a localization argument, which allows to work with a boundary regularity that is sharp at the interface dividing D and ∂Ω \ D. Moreover, we provide homogeneous and local estimates for the extension operator. Also, we treat the case of Lipschitz function spaces with a vanishing trace condition on D.