### Abstract or Additional Information

We consider a Stokes free boundary problem with surface tension effects in variational form. This model is an extension of the coupled system proposed by P. Saavedra and L. R. Scott, where they consider a Laplace equation in the bulk with Young-Laplace equation on the free boundary to account for surface tension. The two main difficulties for the Stokes free boundary problem are: the vector curvature on the interface, and the existence of solution to Stokes equations with Navier-slip boundary conditions for $W^{2-1/p}_p$ domains. We will demonstrate the existence of solution to Stokes equations using a perturbation argument for the bended half space followed by a standard localization technique. The $W^{2-1/p}_p$ regularity of the interface allows us to write the variational form for the entirefree boundary problem, we will conclude with the well-posedness of this system.