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][1] 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.][2]

The paper   also presents   applications to optimal control problems of  symmetric rearrangement  technique     mentioned in Kelei Wang's answer.   [The paper][3] also has some  references on various symmetrization techniques you might find useful.


  [1]: http://www3.nd.edu/~lnicolae/opt-control.pdf
  [2]: http://www3.nd.edu/~lnicolae/opt-control.pdf
  [3]: http://www3.nd.edu/~lnicolae/opt-control.pdf