Hydrodynamic limit for an open facilitated exclusion process with slow and fast boundaries

We study the symmetric facilitated exclusion process (FEP) on the finite one-dimensional lattice $\lbrace 1,\dots ,N-1\rbrace$ when put in contact with boundary reservoirs, whose action is subject to an additional kinetic constraint in order to enforce ergodicity, and whose speed is of order $N^{-θ}