What methods are there to show symmetry properties of the minimizer of a problem $\inf_{u\in X}\mathcal{F}(u)$ in the calculus of variations? In general, the symmetry properties of $\mathcal{F}$ do not imply the symmetry of its minimizers. However, the minimizer may posses some weaker symmetry. 
More precisely,  I am interested in functions $u$ defined on the sphere $S^2$. Even if $\mathcal{F}(u(R\cdot))=\mathcal{F}(u)$ for all rotations $R$ this does not imply that the solution is rotationally invariant (and therefore constant). Is it possible to formulate conditions on $\mathcal{F}$ that guarantee that each minimizer is rotationally symmetric around an axis?