Symmetry and monotonicity results for solutions of semilinear PDEs in sector-like domains

In this paper we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates (r, theta, z), we investigate the shape of possibly sign-changing solutions whose derivative in theta vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of theta or strictly monotone with respect to theta. In the special case of a planar domain, the result holds in a circular sector as well as in an annular one, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case where its opening is larger than pi.