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