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 possess 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?
Edit: If there is more than one minimizer $$\mathcal{F}(u(R\cdot))=\mathcal{F}(u) \text{ for all rotations } R$$ then this does not imply that all minimizers are symmetric with respect to any rotation. Instead, for each minimizer $u(\cdot)$ its rotated version is $u(R\cdot)$ is a minimizer, too. What tools are available to show that all minimizers possess a weaker symmetry e.g. symmetric with respect to one axis. The functional $\mathcal{F}$ of interest is a the sum of a convex and a non-convex bilinear part.