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
| Bound | Reference | Comments |
|---|---|---|
| $\approx 2.15627$ | [BDZ2013] | Numerical evaluation; maximum achieved at $\delta^{\ast} \approx 3.92888$ (lies above the certified upper bound) |
| $2.1562256466^*$ | certificate | Cauchy–Schwarz dual bound with weights $\lvert k\rvert^2(1+\mu\lvert k\rvert^2)$; certified lattice sums with tail bounds; interval branch-and-bound over $\delta$ |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $\frac{\beta + \pi}{\pi} \approx 1.82283$ | [BDZ2013] | |
| $2.156224$ | [Pat2026] | Rigorous. Truncated conditional extremal of [BDZ2013] on $\mathbb{T}^2=[0,2\pi]^2$: $u(x)=\sum c_k e^{ik\cdot x}$ over $0<\lvert k\rvert^2\le 4000^2$ with $c_k = 1/(\lvert k\rvert^2(\lvert k\rvert^2+\mu))$, $\mu=41/5$. With $S_0=\sum c_k$, $S_1=\sum\lvert k\rvert^2c_k^2$, $S_2=\sum\lvert k\rvert^4c_k^2$: $\delta(u)=S_2/S_1\approx 3.93181$ and, using $\lVert u\rVert_\infty\ge u(0)=S_0$, $L\ge S_0^2/(\pi S_1)-\ln\delta-\ln(1+\ln\delta)=2.1562247\ldots$, evaluated in interval arithmetic. Increasing the cutoff approaches the numerical value $2.15627$. |
| $2.1562247706^*$ | certificate | Explicit trial function $u=\sum_{0<\lvert k\rvert\le 4000} e^{ik\cdot x}/(\lvert k\rvert^2(1+\mu\lvert k\rvert^2))$, $\mu=0.1221$; lattice sums enclosed with outward rounding; refines $2.156224$ from #226 |
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 $\lVert u\rVert_{\infty}^2 \le C\lVert \nabla u\rVert_{2}^2(\ln\delta + K)$ 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.
- [Pat2026] Patankar, V. Rigorous lower bounds for constants 16a, 20a, 20c and 70a, and an upper bound for 23b, with verification scripts. GitHub repository (2026): https://github.com/vidigoat/optimization-records/tree/main/tao.
Contribution notes
This entry was prepared with LLM assistance (Claude) for literature synthesis and formatting. The bound from [Pat2026] was found with search and proof assistance from Claude (Anthropic); its value is recomputed by the scripts linked in that reference.