Singularities of Bernoulli Free Boundaries

In this article we show that, for a large class of Bernoulli problems, a free boundary which is symmetric with respect to a vertical line through an isolated singular point must necessarily have a corner at that point, and we give a formula for the contained angle. The assumptions used admit the possibility of other singular points, even uncountably many, on the free boundary. This result is an extension of the first Stokes conjecture in the theory of hydrodynamic waves. We also show that, even in the presence of singularities, a geometrically simple Bernoulli free boundary is necessarily symmetric.