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 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.

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

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 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.

The paper also presents applications to optimal control problems of symmetric rearrangement technique mentioned in Kelei Wang's answer. The paper 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.