A Note on Bifurcation for Harmonic Maps on Annular Domains

In this paper we consider harmonic maps u(r, theta) from an annular domain Omega(rho) = B-1\(B) over bar (rho) to S (2) with the boundary conditions: u(rho, theta) = (cos theta, sin theta, 0) and u(1, theta) = (cos (theta +theta(0)), sin (theta +theta(0)), 0), where theta(0) is an element of [0, pi[ is a fixed angle. This problem arises from the theory of liquid crystals. We prove, with elementary time map arguments, a bifurcation result, namely the existence of a not trivial (that is not planar) harmonic map of minimum energy u(theta 0), for suitable combination of value of rho and theta(0). This result improves the one in Greco (Proc Am Math Soc 129(4):1199-1206, 2000). In the case theta(0) = pi, so that u(1, theta) = (-cos theta, -sin theta, 0), no bifurcation occurs, since the minimum of the energy is not trivial, and we study the behavior of the harmonic maps u(theta 0) as theta(0) -> pi.