Uniqueness of energy maximizers

Problem's Description

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.

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

   • G. R. Burton

   Mathematische Annalen 276, 225-253, 1987fulltext