Symmetry-breaking bifurcations for free boundary problems

Free boundary problems often possess solutions which are radially symmetric. In this paper we demonstrate how to establish symmetry-breaking bifurcation branches of solutions by reducing the bifurcation problem to one for which standard bifurcation theory can be applied. This reduction is performed by first introducing a suitable diffeomorphism which maps the near circular unknown domain onto a disc or a ball, and then verifying the assumptions of the Crandall-Rabinowitz theorem. We carry out the analysis in detail, for the case of one elliptic equation with a Neumann condition at the free boundary and with Dirichlet data given by the curvature of the free boundary. Other examples are briefly mentioned.