For a absolutely continuous $u \colon \mathbb{R}^d \to \mathbb{R}$, we define $$ I(u) := \mathrm{tr}\bigg(\Big[\int_{\mathbb{R}^d} \nabla u(x) \otimes \nabla u(x) \, \mathrm{d}x\Big]^{-1}\bigg) $$ Let $\Omega$ be the unit ball (in Euclidean norm) in $\mathbb{R}^d$. Above, $\nabla u$ denotes the gradient of $u$, and $\mathrm{tr}(\cdot)$ the usual trace operator on $d \times d$ real matrices.
I am interested in computing the solution to the following variational problem over absolutely continuous $u$, $$ \sup\Big\{\,I(u) : \int_{\Omega} u^2 = 1,~~\, u\big|_{\partial \Omega} = 0\Big\} $$
In one-dimension, it is possible to give a characterization of this quantity; the optimal solution is then $u^2(x) = \cos^2(\pi x/2)$, restricted to $[-1, 1]$. This is shown by an argument based on the calculus of variations as done in a paper available online [1, Prop. 7].
I am not very familiar with calculus of variations techniques in higher dimensions, and am wondering if it is possible to derive an analagous characterization of the maximizer of $I(u)$.