Bifurcations on hemispheres (Q1322677)

We consider in detail the steady-state bifurcations of reaction-diffusion equations defined on the hemisphere with Neumann boundary conditions along the equator. Such equations have a natural O(2)-symmetry but may be extended to the full sphere where the natural symmetry group is O(3). We also determine a large class of partial differential equations and domains where this kind of extension is possible for both Neumann and Dirichlet boundary conditions.