  This paper describes a technique to determine the linear well-posedness of a
general class of vector elliptic problems that include a steady interface, to
be determined as part of the problem, that separates two subdomains. The
interface satisfies mixed Dirichlet and Neumann conditions. We consider ``2+2''
models, meaning two independent variables respectively on each subdomain. The
governing equations are taken to be vector Laplacian, to be able to make
analytic progress. The interface conditions can be classified into four large
categories, and we concentrate on the one with most physical interest. The
well-posedness criteria in this case are particularly clear. In many physical
cases, the movement of the interface in time-dependent situations can be
reduced to a normal motion proportional to the residual in one of the steady
state interface conditions (the elliptic interior problems and the other
interface conditions are satisfied at each time). If only the steady state is
of interest, one can consider using other residuals for the normal velocity.
Our analysis can be extended to give insight into choosing residual velocities
that have superior numerical properties. Hence, in the second part, we discuss
an iterative method to solve free boundary problems. The advantages of the
correctly chosen, non-physical residual velocities are demonstrated in a
numerical example, based on a simplified model of two-phase flow with phase
change in porous media.
