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.



Symmetry Properties of Minimizers - Calculus of Variations
Suppose that $\newcommand{\eF}{\mathscr{F}}$ $\newcommand{\bR}{\mathbb{R}}$ $\eF: C\to \bR$ is a convex functional defined on a closed convex subset $C$ of a say real Banach space $U$. (You can allow for more general topological spaces.) Suppose additionally that a compact Lie group $G$ acts on $C$. For any $u\in U$ we define the symmetrization


$$ [u]\_G:=\int\_G g\cdot u dg, $$


where $dg$ denotes the unique bi-invariant measure on $G$ of total volume $1$. (In other words, $dg$ is an invaraint probability measure on $G$.) Clearly $[u]\_G$ is a fixed point of the $G$ action and since $C$ is convex and $dg$ is a probability measure


$$ u\in C\Rightarrow \bar{u}\_G\in C. $$


Jensen's inequality implies


$$ \eF([u]\_G)\leq \int\_G \eF(g\cdot u) dg. $$


If $\eF$ happens to be $G$-invariant as well and $u\_0$ is a minimizer, then 


$$ \int\_G \eF(g\cdot u\_0) dg =\eF(u\_0)=\min\_C \eF. $$


In particular, this implies that


$$ \eF([u\_0]\_G)\leq \min\_C \eF. $$


so that $[u\_0]\_G$ is also a minimizer. To see how this work in practice, I refer to a [very old paper](http://www3.nd.edu/~lnicolae/opt-control.pdf) of mine where I used this symmetrization technique in an optimal control problem. The relevant part begins at page 19 Proposition 4.2. of [the paper.](http://www3.nd.edu/~lnicolae/opt-control.pdf)


The paper also presents applications to optimal control problems of symmetric rearrangement technique mentioned in Kelei Wang's answer. [The paper](http://www3.nd.edu/~lnicolae/opt-control.pdf) also has some references on various symmetrization techniques you might find useful.


Here is one instance when this works. Let $X$ be the Sobolev space $H^1(S^n)$ (functions with weak $L^2$ first order derivatives) where the sphere $S^n$ is equipped with the round metric. Let $f:\bR\to\bR$ be a convex function and consider the convex functional


$$ \eF: H^1(S^n)\to (-\infty,\infty],\;\;\eF(u)=\int\_{S^n} \Bigl(\; \frac{1}{2}|\nabla u|^2+ f(u)\;\Bigr) dV\_{S^n}, $$


where the gradient and the volume form are defined in terms of the round metric. Let $G$ be the group $SO(n+1)$ which acts transitively and isometrically on $S^n$. It induces an action by pullback on $H^1(S^n)$. The functional $\eF$ is invariant under this action. If it admits a minimizer, then it admits a symmetric minimizer, which has to be constant.


If instead you consider a functional


$$\eF\_1(u)=\int\_{S^n} \Bigl(\; \frac{1}{2}|\nabla u(x)|^2+ a(x)f(u(x))\;\Bigr) dV\_{S^n}(x), $$


where $a: S^n\to \bR$ is a nonnegative function which is invariant under the action of the closed subgroup $G\subset SO(n+1)$ and $\eF\_1$ has minimazers, then it will have $G$-invariant minimizers.


