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on Brezis–Gallouet–Wainger remainder constant on the 2D torus ($C_{16}$)

The statement

$C_{16} = L$ is the smallest constant for which the sharp Brezis–Gallouet inequality

\|u\|_{L^\infty(\mathbb T^2)}^2 \le \frac{1}{4\pi}\,\|\nabla u\|_{L^2(\mathbb T^2)}^2 \Bigl[\ln\delta(u) + \ln\bigl(1+\ln\delta(u)\bigr) + L\Bigr]

holds for all zero-mean functions $u \in H^2(\mathbb T^2)$ with sufficiently large frequency ratio

\delta(u) := \frac{\|\Delta u\|_{L^2(\mathbb T^2)}^2}{\|\nabla u\|_{L^2(\mathbb T^2)}^2}.

Equivalently, $L$ is defined via the constrained extremal problem

L = \max_{\delta \geq 1} \left[ 4\pi\Theta(\delta) - \ln\delta - \ln(1+\ln\delta) \right]

where $\Theta(\delta) := \sup\lbrace|u(0)|^2 : \lVert\nabla u\rVert_2^2 = 1,\, \lVert\Delta u\rVert_2^2 = \delta\rbrace$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.