# Uniqueness of energy maximizers

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Posted online: 2018-12-18 10:25:24Z by Hayk Mikayelyan15

Cite as: P-181218.1

• Analysis of PDEs
• Classical Analysis and ODEs

### 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

Mathematische Annalen 276, 225-253, 1987fulltext