Let
$$
\begin{cases}
-\Delta u_f=f & \text{in }D,\\
u_f=0 &\text{on }\partial D.
\end{cases}
$$
It is well-known that the maximization of the functional
$$
\Phi(f)=\int_D|\nabla u_f|^2dx,
$$
over the set
$$
\bar{\mathcal{R}}_\beta=\{f\,\,|\,\, 0\leq f\leq 1,\,\,\text{and}\,\, \int_Dfdx=\beta\}
$$
has a solution
$$
\hat{f}=\chi_{\{\hat{u}>\alpha\}}\in \mathcal{R}_\beta=\{\chi_E\,\,|\,\, |E|=\beta\},
$$
for some $\alpha>0$, and $\hat{u}=u_{\hat{f}}$.

Show that if $D$ is convex then the solution is unique.

It has been observed, at least numerically, that this is not true for non-convex domains.

Article
Rearrangements of functions, maximization of convex functionals, and vortex rings

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