Title: On a free boundary problem and minimal surfaces,
Abstract: From minimal surfaces such as Simons' cone and catenoids, using refined
Lyapunov-Schmidt reduction method, we construct new solutions for a free
boundary problem whose free boundary has two components. In dimension $8$,
using variational arguments, we also obtain solutions which are global
minimizers of the corresponding energy functional. This shows that Savin's
theorem is optimal.