IUMJ

Title: Symmetry-breaking bifurcations for free boundary problems

Authors: Andrei Borisovich and Avner Friedman

Issue: Volume 54 (2005), Issue 3, 927-947

Abstract:

Free boundary problems often possess solutions which are radially symmetric. In this paper we demonstrate how to establish sim\-me\-try-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.