Steady flows of compressible fluids in a rigid container with upper free boundary

We consider fluid in a smooth rigid container whose lateral boundary is a piece of vertical cylinder, bounded above by a free upper surface. As basic flow we consider the non homogeneous rest state in the presence of gravity, and of a surface tension. Under these assumptions, we study the existence of a steady free boundary Gamma and a steady motion in Omega of an isothermal viscous gas, resulting as perturbation to the rest state in correspondence of small non potential perturbations to the (large potential) gravitational force. We linearize the problem by prescribing the unknown domain Omega, then we make use of the iterative scheme introduced by Heywood and Padula. Our method is based on an iteration between the Neumann problem for a non homogeneous Stokes system for the velocity, the Neumann problem for an elliptic problem on Gamma for height, and a steady transport equation for the perturbation to the density. The difference of boundary condition between lateral boundary and free upper surfaces causes a singularity at the intersection (contact line). To avoid singularities on the contact line, we adopt weighted Sobolev spaces.