Sharp constant in Nash's inequality

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Numbers

$n$

$C_n$

0.1709794973964449893115465941664153906511054322594160511769652413214171416237723070101656497494868510

0.08672129755403024746947152715029068355849151089236066483035079688376920106009605097329820073953371523

0.05851461595654745234047114587254863060444697262647863487520717336660872364438868614577431611981610002

0.04434359093260563490911633092226919944568847910081807633818687497923942789188916099979703149621853906

0.03579887825170516231749511702500371355342702416410858474738429597603790672550272328839842250545290143

0.03007432438364857468619460108891722581289225777861006732446436346506668353720877228129997757057342350

0.02596564945010615295143051211039333622612092352286612106323973562115097400240816618280084251965899728

0.02286976016687953878916178797743105923735706258528950866870397434204204790626987868333639687270431316

0.02045101956631760625589605622998766035243299126662558249060584006966922904008590057932286669853531233

10:

0.01850766968428833110710764055919497240011985893808366761297463892634740002523804319265218946208553665

11:

0.01691106855815396727228820294274904757814632986391713366097868707897295404805910105817498081790840461

12:

0.01557528251643495446545435544754290523241783937232789646085727886299189679076802823755603662856196386

13:

0.01444069720924529341504526936829808373222688730680415895098261219154652402975551862125233896601539192

14:

0.01346464286048123490579103155401902812453682529494379769927987946772958239181425492524784863655973496

15:

0.01261576328448276720127419711068555955198566360811975734430145398896583638826580433217105499633954934

16:

0.01187049249885674299136572146698636175151472436995720852822954393750050773102416156411021018433148047

17:

0.01121077218196128323189933349435366655677322880556701706797079439754061672376297047276370635402604119

18:

0.01062252812025179086489324858376886818878994246868655840528962987317878859142936386060137399059581274

19:

0.01009462647592431315502591550965877074040864478329997312554885596643563329009276047183324212173552059

20:

0.009618142246057788878819177797568012470946001432662355526993627992654326066285521532993136252223107198

Definition

Let $n$ be a positive integer. This table stores the least constant $C_n$ such that Nash's inequality [3] $\|u\|_2^{2+4/n}\leq C_n\|\nabla u\|_2^2\|u\|_1^{4/n}$ holds for every $u\in L^1(\mathbb{R}^n)\cap W^{1,2}(\mathbb{R}^n)$ [1].

Parameters

$n$

— dimension (positive integer)

Formulas

(1)

$C_n=\frac{2(1+n/2)^{1+2/n}}{n\lambda_n\kappa_n^{2/n}}$, where $\kappa_n=\pi^{n/2}/\Gamma(1+n/2)$ is the volume of the unit ball in $\mathbb{R}^n$, $j_{\nu,1}$ is the first positive zero of the Bessel function $J_\nu$, and $\lambda_n=j_{n/2,1}^2$ is the first positive radial Neumann eigenvalue of $-\Delta$ on the unit ball. [1], [2]

(2)

In dimension $1$, $j_{1/2,1}=\pi$, so $C_1=27/(16\pi^2)$.

Programs

(P1)

Python

from mpmath import mp, besseljzero, gamma, pi, mpf

mp.dps = 120
n = mpf(21)
lam = besseljzero(n / 2, 1) ** 2
kappa = pi ** (n / 2) / gamma(1 + n / 2)
2 * (1 + n / 2) ** (1 + 2 / n) / (n * lam * kappa ** (2 / n))

References

[1]

E. A. Carlen and M. Loss, Sharp constant in Nash's inequality, International Mathematics Research Notices 1993 (1993), no. 7, 213-215. (doi)

[2]

C. Haberl and F. E. Schuster, Asymmetric affine $L_p$ Sobolev inequalities, Journal of Functional Analysis 257 (2009), 641-658. (doi)

Links

[3]

Wikipedia: Sobolev inequality, Nash inequality section

Similar tables

Best Sobolev constant for $W^{1,p}(\mathbb{R}^n)$ — its $p=2$ Sobolev constant and interpolation between $L^1$ and $L^{2n/(n-2)}$ give a non-sharp constant in Nash's inequality for $n\geq3$

Sharp constants in the fractional Sobolev inequality — the fractional Sobolev member of the same sharp-inequality family

Sharp constants in the Hardy-Littlewood-Sobolev inequality — the Hardy-Littlewood-Sobolev member of the same sharp-inequality family

Sharp Gagliardo-Nirenberg constants of Del Pino and Dolbeault — another sharp Gagliardo-Nirenberg inequality on $\mathbb{R}^n$, in the Del Pino-Dolbeault range rather than the $L^1$ Nash form

Babenko-Beckner constants of the Hausdorff-Young inequality — another sharp upper-bound constant, for the Fourier transform rather than Nash interpolation

Sharp constants in Young's convolution inequality — another sharp upper-bound constant for an $L^p$ inequality on $\mathbb{R}^n$, for convolution rather than Nash interpolation

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every dimension $n\leq20$, matching the range in which the Bessel-zero table has the needed first-zero rows $j_{n/2,1}$)

How they were obtained:

The generator isolates $j_{n/2,1}$ by sign changes using Sage's arb-backed RealBallField. For integer orders it evaluates the Bessel function by a rigorously tailed alternating power series and recurrence; for half-integer orders it uses the spherical-Bessel recurrence from sine and cosine.

The one-dimensional value is checked against $27/(16\pi^2)$.