Symmetry and monotonicity of solutions to some variational problems in cylinders and annuli

Abstract

We prove symmetry and monotonicity properties for local minimizers and stationary solutions of some variational problems related to semilinear elliptic equations in a cylinder $(-a,a)imes omega$, where $omega $ is a bounded smooth domain in $mathbb{R}^{N-1}$. The admissible functions satisfy periodic boundary conditions on ${pm a} imes omega $, and some other conditions. We show also symmetry properties for related problems in annular domains. Our proofs are based on rearrangement arguments and on the Moving Plane Method.