Brezis–Gallouet–Wainger remainder constant on the 2D torus

Description of constant

$C_{16} = L$ is the smallest constant for which the sharp Brezis–Gallouet inequality \(\|u\|_{L^\infty(\mathbb T^2)}^2 \le \frac{1}{4\pi}\,\|\nabla u\|_{L^2(\mathbb T^2)}^2 \Bigl[\ln\delta(u) + \ln\bigl(1+\ln\delta(u)\bigr) + L\Bigr]\) 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 $\Theta(\delta) := \sup\lbrace|u(0)|^2 : \lVert\nabla u\rVert_2^2 = 1,\, \lVert\Delta u\rVert_2^2 = \delta\rbrace$.

Known upper bounds

Known lower bounds

Here $\beta = \pi(2\gamma + 2\log 2 + 3\log\pi - 4\log\Gamma(1/4))$, where $\gamma$ is the Euler–Mascheroni constant.

Additional comments

• The leading coefficient $\frac{1}{4\pi}$ in front of the logarithmic terms is optimal, as is the doubly logarithmic correction; the remaining optimization is entirely in the additive constant $L$.
• The simpler “one-log” Brezis–Gallouet inequality $|u|{\infty}^2 \le C|\nabla u|{2}^2(\ln\delta + K)$ has infimum $C = \frac{1}{4\pi}$, but this infimum is not attained with any finite $K$—the log-log correction is necessary.
• The constant $L$ is expressed in terms of lattice sums over $\mathbb{Z}^2$ and does not have a known closed form.
• The maximum in the variational definition is unique and achieved at finite $\delta^{\ast}$; the corresponding conditional extremal $u_{\mu(\delta^{\ast})}(x)$ is an exact extremal function.
• Applications include sharp attractor dimension bounds for 2D Navier–Stokes equations on the torus.

References

• [BG1980] Brezis, H.; Gallouet, T. Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4 (1980), 677–681.
• [BDZ2013] Bartuccelli, M. V.; Deane, J. H. B.; Zelik, S. Asymptotic expansions and extremals for the critical Sobolev and Gagliardo–Nirenberg inequalities on a torus. Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 445–482. arXiv:1012.2061

For related results in Hölder space settings, see:

• [MSW2010] Morii, K.; Sato, T.; Wadade, H. Brézis–Gallouët–Wainger type inequality with a double logarithmic term in the Hölder space: Its sharp constants and extremal functions. Nonlinear Anal. 73 (2010), 1747–1766.

Contribution notes

This entry was prepared with LLM assistance (Claude) for literature synthesis and formatting.