  We consider a PDE-constrained optimization problem governed by a free
boundary problem. The state system is based on coupling the Laplace equation in
the bulk with a Young-Laplace equation on the free boundary to account for
surface tension, as proposed by P.\ Saavedra and L.\ R.\ Scott
\cite{PSaavedra_RScott_1991}. This amounts to solving a second order system
both in the bulk and on the interface. Our analysis hinges on a convex control
constraint such that the state constraints are always satisfied. Using only
first order regularity we show that the control to state operator is twice
continuously Fr\'echet differentiable. We improve slightly the regularity of
the state variables and exploit it to show existence of a control together with
second order sufficient optimality conditions.
