In the description of many physical and technical problems, not only field quantities like velocity, pressure, temperature, and magnetic field are unknown, but also some geometric quantities like domain and boundary position must be computed. Such problems where the position of the domain boundary forms part of the solution are called free boundary value problems. We are interested in free boundary value problems with capillary surfaces. These surfaces are characterised by the adjacency of two fluids where at least one fluid is a liquid. The conditions on capillary surfaces which determine the surface position result from the balance between the normal stresses in the fluids and the surface tension on the surface. In the following we will consider two classes of free boundary value problems with capillary surfaces where the surface separates a liquid from the surrounding air or vacuum. Stationary problems form the first class. Here, we assume that the liquid is at rest and that the surface position doesn’t change in time. The second class consists of nonstationary problems which involve flow problems in the liquid and time-dependent surfaces. As an example of the first problem class we will investigate the equilibrium shape of a ferrofluid layer under the influence of an external magnetic field. Another example comes from the second class. We consider the oscillation of a drop whose motion is driven by surface tension only.
