  The aim of this paper is to prove the existence of ${\mathcal R}$-bounded
solution operator families for a resolvent problem on the upper half-space
arising from a compressible fluid model of Korteweg type with free boundary
condition. Such a compressible fluid model was derived by Dunn and Serrin
(1985) and studied by Kotschote (2008) as a boundary value problem with
non-slip boundary condition.
