Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry

The method of moving planes is used to establish a weak set of conditions under which the nonlinear equation -Δu(x)=V(|x|)e^u(x), x∈ℝ² admits only rotationally symmetric solutions. Additional structural invariance properties of the equation then yield another set of conditions which are not originally covered by the moving plane technique. For instance, nonmonotonic V can be accommodated. Results for -Δu(y)=V(y)e^u(y)-c, with y∈S², are obtained as well. A nontrivial example of broken symmetry is also constructed. These equations arise in the context of extremization problems, but no extremization arguments are employed. This is of some interest in cases where the extremizing problem is neither manifestly convex nor monotone under symmetric decreasing rearrangements. The results answer in part some conjectures raised in the literature. Applications to logarithmically interacting particle systems and geometry are emphasized.