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ID: INPUT{id.yaml}

Title: >
  Best Sobolev constant for $W^{1,p}(\mathbb{R}^n)$

Definition: >
  Let $1 \leq p < n$ for some integer $n$, 
  and let $q = np(n-p)$. 
  The Sobolev inequality for $W^{1,p}(\mathbb{R}^n)$ states that 
  $S_{n,p} ||u||_{L^q} \leq ||\nabla u||_{L^p}$
  for any $u \in W^{1,p}(\mathbb{R}^n)$,
  where $S_{n,p}$ are the constants given in 
  CITE{formula-Snp} and CITE{formula-Sn1},
  which are best possible.
  This table lists these constants $S_{n,p}$ 
  for certain choices of $n$ and $p$.
  
  
Parameters:
  n:
    display: $n$
    type: Z
    constraints: $n > p$

p:
    display: $p$
    type: R
    constraints: $p \geq 1$
    
  q:
    display: $q$
    type: R
    constraints: $q = \frac{np}{n-p}$
   
Comments:
  
Formulas: 
  formula-Snp: >
    For $p > 1$, 
    $S_{n,p} = \pi^{1/2} n^{1/p} \big(\frac{n-p}{p-1}\big)^{1-1/p}
    \big(\frac{\Gamma(n/p)\Gamma(n+1-n/p)}{\Gamma(n)\Gamma(1+n/2)}\big)^{1/n}$.
    CITE{Aub}, CITE{Tal}
  formula-Sn1: >
    For $p = 1$,
    $S_{n,1} = \pi^{1/2} \frac{n}{\Gamma(1+n/2)^{1/n}}$.
    CITE{FedFle}, CITE{Maz}
    
Programs:

References:
  Aub:
    bib: >
      Aubin, T., "Problèmes isopérimétriques et espaces de Sobolev",
      J. Differ. Geom. 11 (1976), 573–598.
  FedFle:
    bib: >
      Federer, H., Fleming, W., "Normal and integral currents", 
      Ann. of Math. (2) 72 (1960), 458–520.
  Maz:
    bib: >
      Maz'ya, V.G., 
      "Classes of domains and imbedding theorems for function spaces",
      Soviet Math. Dokl. 1 (1960), 882–885.
  Tal:
    bib: >
      Talenti, G., "Best constant in Sobolev inequality", 
      Ann. Mat. Pura Appl. 110 (1976), 353–372.
Links:
  Wiki:
    title: "Wikipedia: Sobolev inequality"
    url: https://en.wikipedia.org/wiki/Sobolev_inequality

Similar tables:
  
Keywords:
- Sobolev inequality
  
Tags:
- functional analysis
- inequality

Data properties:
  type: R
  complete: no

Display properties:
  number-header: $S_{n,p}$

Numbers: INPUT{numbers.yaml}

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Best Sobolev constant for $W^{1,p}(\mathbb{R}^n)$

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Numbers

$n$

$p$

$q$

$S_{n,p}$

INPUT{numbers.yaml} (not shown in preview)

Definition

Let $1 \leq p < n$ for some integer $n$, and let $q = np(n-p)$. The Sobolev inequality for $W^{1,p}(\mathbb{R}^n)$ states that $S_{n,p} ||u||_{L^q} \leq ||\nabla u||_{L^p}$ for any $u \in W^{1,p}(\mathbb{R}^n)$, where $S_{n,p}$ are the constants given in (1) and (2), which are best possible. This table lists these constants $S_{n,p}$ for certain choices of $n$ and $p$.

Parameters

$n$

— integer ($n > p$)

$p$

— real number ($p \geq 1$)

$q$

— real number ($q = \frac{np}{n-p}$)

Formulas

(1)

For $p > 1$, $S_{n,p} = \pi^{1/2} n^{1/p} \big(\frac{n-p}{p-1}\big)^{1-1/p} \big(\frac{\Gamma(n/p)\Gamma(n+1-n/p)}{\Gamma(n)\Gamma(1+n/2)}\big)^{1/n}$. [1], [4]

(2)

For $p = 1$, $S_{n,1} = \pi^{1/2} \frac{n}{\Gamma(1+n/2)^{1/n}}$. [2], [3]

References

[1]

Aubin, T., "Problèmes isopérimétriques et espaces de Sobolev", J. Differ. Geom. 11 (1976), 573–598.

[2]

Federer, H., Fleming, W., "Normal and integral currents", Ann. of Math. (2) 72 (1960), 458–520.

[3]

Maz'ya, V.G., "Classes of domains and imbedding theorems for function spaces", Soviet Math. Dokl. 1 (1960), 882–885.

[4]

Talenti, G., "Best constant in Sobolev inequality", Ann. Mat. Pura Appl. 110 (1976), 353–372.

Links

[5]

Wikipedia: Sobolev inequality

Data properties

Entries are of type: real number

Table is complete: no