diff --git a/constants/16a.md b/constants/16a.md index 785a049..c423cdf 100644 --- a/constants/16a.md +++ b/constants/16a.md @@ -3,11 +3,14 @@ ## Description of constant $C_{16} = L$ is the smallest constant for which the sharp Brezis–Gallouet inequality + $$ -\|u\|_{L^\infty(\mathbb T^2)}^2 \le \frac{1}{4\pi}\,\|\nabla u\|_{L^2(\mathbb T^2)}^2 +\|u\|\_{L^\infty(\mathbb T^2)}^2 \le \frac{1}{4\pi}\,\|\nabla u\|_{L^2(\mathbb T^2)}^2 \Bigl[\ln\delta(u) + \ln\bigl(1+\ln\delta(u)\bigr) + L\Bigr] $$ + holds for all zero-mean functions $u \in H^2(\mathbb T^2)$ with sufficiently large frequency ratio + $$ \delta(u) := \frac{\|\Delta u\|_{L^2(\mathbb T^2)}^2}{\|\nabla u\|_{L^2(\mathbb T^2)}^2}. $$