## Free boundaries subject to topological constraints. (arXiv:1902.00158v1 [math.AP])

We discuss the extent to which solutions to one-phase free boundary problems
can be characterized according to their topological complexity. Our questions
are motivated by fundamental work of Luis Caffarelli on free boundaries and by
striking results of T. Colding and W. Minicozzi concerning finitely connected,
embedded, minimal surfaces. We review our earlier work on the simplest case,
one-phase free boundaries in the plane in which the positive phase is simply
connected. We also prove a new, purely topological, effective removable
singularities theorem for free boundaries. At the same time, we formulate some
open problems concerning the multiply connected case and make connections with
the theory of minimal surfaces and semilinear variational problems.查看全文