A Class of Codimension-Two Free Boundary Problems

This review collates a wide variety of free boundary problems which are characterized by the uniform proximity of the free boundary to a prescribed surface. Such situations can often be approximated by mixed boundary value problems in which the boundary data switches across a “codimension-two” free boundary, namely, the edge of the region obtained by projecting the free boundary normally onto the prescribed surface. As in the parent problem, the codimension-two free boundary needs to be determined as well as the solution of the relevant field equations, but no systematic methodology has yet been proposed for nonlinear problems of this type. After presenting some examples to illustrate the surprising behavior that can sometimes occur, we discuss the relevance of traditional ideas from the theories of moving boundary problems, singular integral equations, variational inequalities, and stability. Finally, we point out the ways in which further refinement of these techniques is needed if a coherent theory is to emerge.