Minimal surfaces and free boundaries: Recent developments

Abstract

Free boundaries occur in a lot of physical phenomena and are of major interest both mathematically and physically. The aim of this contribution is to describe new ideas and results developed in the last 20 years or so that deal with some nonlocal (sometimes called anomalous) free boundary problems. Actually, such free boundary problems have been known for several decades, one of the main instances being the thin obstacle problem, the so-called (scalar) Signorini free boundary problem. We will describe in this survey some new techniques that allow to deal with long-range interactions. We will not try to be exhaustive since the literature on this type of problem has been flourishing substantially, but rather we give an overview of the main current directions of research. In particular, we want to emphasize the link, very much well-known in the community, between minimal surfaces, their "approximation" by the Allen-Cahn equation and free boundary problems.