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

$$\Vert u{\Vert }_{L^{\infty }({\mathbb{T}}^2)}^2\le \frac{1}{4\pi }\Vert \nabla u{\Vert }_{L^2({\mathbb{T}}^2)}^2[\ln \delta (u)+\ln (1+\ln \delta (u))+L]$$

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

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

Equivalently, $L$ is defined via the constrained extremal problem $$L=\underset{\delta \ge 1}{\max }\left[4\pi \Theta (\delta )-\ln \delta -\ln (1+\ln \delta )\right]$$ where $\Theta (\delta ):=\sup \{|u(0){|}^2:\Vert \nabla u{\Vert }_2^2=1,\Vert \Delta u{\Vert }_2^2=\delta \}$.

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 $\Vert u{\Vert }_{\infty }^2\le C\Vert \nabla u{\Vert }_2^2(\ln \delta +K)$ 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.

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.