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| In [[mathematics]], there is in [[mathematical analysis]] a class of '''Sobolev inequalities''', relating norms including those of [[Sobolev space]]s. These are used to prove the '''Sobolev embedding theorem''', giving inclusions between certain [[Sobolev space]]s, and the [[Rellich–Kondrachov theorem]] showing that under slightly stronger conditions some Sobolev spaces are [[compactly embedded]] in others. They are named after [[Sergei Lvovich Sobolev]].
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| ==Sobolev embedding theorem==
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| Let ''W''<sup>''k'',''p''</sup>('''R'''<sup>''n''</sup>) denote the Sobolev space consisting of all real-valued functions on '''R'''<sup>''n''</sup> whose first ''k'' [[weak derivative]]s are functions in [[Lp space|''L''<sup>''p''</sup>]]. Here ''k'' is a non-negative integer and 1 ≤ ''p'' ≤ ∞. The first part of the Sobolev embedding theorem states that
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| if ''k'' > ''ℓ'' and 1 ≤ ''p'' < ''q'' ≤ ∞ are two extended real numbers such that ''(k-ℓ)p < ''n'' and :
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| :<math>\frac{1}{q} = \frac{1}{p}-\frac{k-\ell}{n},</math>
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| then
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| :<math>W^{k,p}(\mathbf{R}^n)\subseteq W^{\ell,q}(\mathbf{R}^n)</math>
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| and the embedding is continuous. In the special case of ''k'' = 1 and ''ℓ'' = 0, Sobolev embedding gives
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| :<math>W^{1,p}(\mathbf{R}^n) \subseteq L^{p^*}(\mathbf{R}^n)</math>
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| where ''p''<sup>∗</sup> is the '''Sobolev conjugate''' of ''p'', given by
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| :<math>\frac{1}{p^*} = \frac{1}{p} - \frac{1}{n}.</math>
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| This special case of the Sobolev embedding is a direct consequence of the [[#Gagliardo–Nirenberg–Sobolev inequality|Gagliardo–Nirenberg–Sobolev inequality]].
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| The second part of the Sobolev embedding theorem applies to embeddings in [[Hölder space]]s ''C''<sup>''r'',α</sup>('''R'''<sup>''n''</sup>). If (''k''−''r''−α)/''n'' = 1/''p'' with α ∈ (0,1), then one has the embedding
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| :<math>W^{k,p}(\mathbf{R}^n)\subset C^{r,\alpha}(\mathbf{R}^n).</math>
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| This part of the Sobolev embedding is a direct consequence of [[#Morrey's inequality|Morrey's inequality]]. Intuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity of the classical derivatives.
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| ;Generalizations
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| The Sobolev embedding theorem holds for Sobolev spaces ''W''<sup>''k'',''p''</sup>(''M'') on other suitable domains ''M''. In particular ({{harvnb|Aubin|1982|loc=Chapter 2}}; {{harvnb|Aubin|1976}}), both parts of the Sobolev embedding hold when
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| * ''M'' is a [[bounded set|bounded]] [[open set]] in '''R'''<sup>''n''</sup> with [[Lipschitz continuity|Lipschitz]] boundary (or whose boundary satisfies the cone condition; {{harvnb|Adams|1975|loc=Theorem 5.4}})
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| * ''M'' is a [[compact space|compact]] [[Riemannian manifold]]
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| * ''M'' is a compact Riemannian [[manifold with boundary]] with Lipschitz boundary
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| * ''M'' is a [[complete manifold|complete]] Riemannian manifold with [[injectivity radius]] δ > 0 and bounded [[sectional curvature]].
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| ;Kondrachov embedding theorem
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| {{main|Rellich–Kondrachov theorem}}
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| On a compact manifold with ''C''<sup>1</sup> boundary, the '''Kondrachov embedding theorem''' states that if ''k''> ''ℓ'' and ''k''−''n''/''p'' > ''ℓ''−''n''/''q'' then the Sobolev embedding
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| :<math>W^{k,p}(M)\subset W^{l,q}(M)</math>
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| is [[completely continuous]] (compact).
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| ==Gagliardo–Nirenberg–Sobolev inequality==
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| Assume that ''u'' is a continuously differentiable real-valued function on '''R'''<sup>''n''</sup> with [[compact support]]. Then for 1 ≤ ''p'' < ''n'' there is a constant ''C'' depending only on ''n'' and ''p'' such that
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| : <math>\|u\|_{L^{p^*}(\mathbf{R}^n)}\leq C \|Du\|_{L^{p}(\mathbf{R}^n)}</math>
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| where
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| :<math> p^*=\frac{pn}{n-p}>p</math>
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| is the [[Sobolev conjugate]] of ''p''.
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| The Gagliardo–Nirenberg–Sobolev inequality implies directly the Sobolev embedding
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| :<math>W^{1,p}(\mathbf{R}^n)\sub L^{p^*}(\mathbf{R}^n).</math>
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| The embeddings in other orders on '''R'''<sup>''n''</sup> are then obtained by suitable iteration.
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| ==Hardy–Littlewood–Sobolev lemma==
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| Sobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the Hardy–Littlewood–Sobolev [[fractional integration]] theorem. An equivalent statement is known as the '''Sobolev lemma''' in {{harv|Aubin|1982|loc=Chapter 2}}. A proof is in {{harv|Stein|loc=Chapter V, §1.3}}.
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| Let 0 < α <''n'' and 1 < ''p'' < ''q'' < ∞. Let ''I''<sub>α</sub> = (−Δ)<sup>−α/2</sup> be the [[Riesz potential]] on '''R'''<sup>''n''</sup>. Then, for ''q'' defined by
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| :<math>q = \frac{pn}{n-\alpha p}</math>
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| there exists a constant ''C'' depending only on ''p'' such that
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| :<math>\|I_\alpha f\|_q\le C\|f\|_p.</math>
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| If ''p'' = 1, then the weak-type estimate holds:
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| :<math>m\{x : |I_\alpha f(x)| > \lambda\} \le C\left(\frac{\|f\|_1}{\lambda}\right)^q</math>
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| where 1/''q'' = 1 − α/''n''.
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| The Hardy–Littlewood–Sobolev lemma implies the Sobolev embedding essentially by the relationship between the [[Riesz transform]]s and the Riesz potentials.
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| ==Morrey's inequality==
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| Assume ''n'' < ''p'' ≤ ∞. Then there exists a constant ''C'', depending only on ''p'' and ''n'', such that
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| :<math>\|u\|_{C^{0,\gamma}(\mathbf{R}^n)}\leq C \|u\|_{W^{1,p}(\mathbf{R}^n)}</math>
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| for all ''u'' ∈ C<sup>1</sup>('''R'''<sup>''n''</sup>) ∩ L<sup>''p''</sup>('''R'''<sup>''n''</sup>), where
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| :<math>\gamma=1-n/p.</math>
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| Thus if ''u'' ∈ ''W''<sup>1,''p''</sup>('''R'''<sup>''n''</sup>), then ''u'' is in fact [[Hölder continuous]] of exponent γ,
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| after possibly being redefined on a set of measure 0.
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| A similar result holds in a bounded domain ''U'' with ''C''<sup>1</sup> boundary. In this case,
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| :<math>\|u\|_{C^{0,\gamma}(U)}\leq C \|u\|_{W^{1,p}(U)}</math>
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| where the constant ''C'' depends now on ''n'', ''p'' and ''U''. This version of the inequality follows from the previous one by applying the norm-preserving extension of ''W''<sup>1,''p''</sup>(''U'') to ''W''<sup>1,''p''</sup>('''R'''<sup>''n''</sup>).
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| ==General Sobolev inequalities==
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| Let ''U'' be a bounded open subset of ''R''<sup>''n''</sup>, with a ''C''<sup>1</sup> boundary. (''U'' may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.) Assume ''u'' ∈ ''W''<sup>''k'',''p''</sup>(''U'').
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| (i) If
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| :<math>k<\frac{n}{p}</math>
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| then <math>u\in L^q(U)</math>, where
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| :<math>\frac{1}{q}=\frac{1}{p}-\frac{k}{n}.\ </math>
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| We have in addition the estimate
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| :<math>\|u\|_{L^q(U)}\leq C \|u\|_{W^{k,p}(U)}</math>,
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| the constant ''C'' depending only on ''k'', ''p'', ''n'', and ''U''.
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| (ii) If
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| :<math>k>\frac{n}{p}</math>
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| then ''u'' belongs to the [[Hölder space]] <math> C^{k-[n/p]-1,\gamma}(U)\,</math>, where
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| :<math>\gamma=\left[\frac{n}{p}\right]+1-\frac{n}{p}</math> if n/p is not an integer, or
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| :γ is any positive number < 1, if ''n''/''p'' is an integer
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| We have in addition the estimate
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| :<math>\|u\|_{C^{k-[n/p]-1,\gamma}(U)}\leq C \|u\|_{W^{k,p}(U)},</math>
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| the constant ''C'' depending only on ''k'', ''p'', ''n'', γ, and ''U''.
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| ==Case <math>p=n, k=1</math>==
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| If <math>u\in W^{1,n}(R^n)</math>, then <math>u</math> is a function of [[bounded mean oscillation]] and
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| :<math>\|u\|_{BMO}<C\|Du\|_{L^n(R^n)}</math>, for some constant ''C'' depending only on ''n''. | |
| This estimate is a corollary of the [[Poincaré inequality]].
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| ==Nash inequality==
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| The Nash inequality, introduced by {{harvs|first=John|last=Nash|authorlink=John Forbes Nash, Jr.|year=1958|txt}}, states that there exists a constant ''C'' > 0, such that for all ''u'' ∈ L<sup>1</sup>('''R'''<sup>''n''</sup>) ∩ W<sup>1,2</sup>('''R'''<sup>''n''</sup>),
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| :<math>\|u\|_{L^2(\mathbf{R}^n)}^{1+2/n}\leq C\|u\|_{L^1(\mathbf{R}^n)}^{2/n} \| Du\|_{L^2(\mathbf{R}^n)}.</math> | |
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| The inequality follows from basic properties of the [[Fourier transform]]. Indeed, integrating over the complement of the ball of radius ρ,
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| {{NumBlk|:|<math>\int_{|x|\ge\rho} |\hat{u}(x)|^2\,dx \le \int_{|x|\ge\rho} \frac{x^2}{\rho^2}|\hat{u}(x)|^2\,dx\le \rho^{-2}\int_{\mathbf{R}^n}|D u|^2\,dx</math>|{{EquationRef|1}}}}
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| by [[Parseval's theorem]]. On the other hand, one has
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| :<math>|\hat{u}| \le \|u\|_{L^1}</math>
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| which, when integrated over the ball of radius ρ gives
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| {{NumBlk|:|<math>\int_{|x|\le\rho} |\hat{u}(x)|^2\,dx \le \rho^n\omega_n \|u\|_{L^1}^2</math>|{{EquationRef|2}}}}
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| where ω<sub>''n''</sub> is the volume of the [[n sphere|''n''-ball]]. Choosing ρ to minimize the sum of ({{EquationNote|1}}) and ({{EquationNote|2}}) and again applying Parseval's theorem <math>\scriptstyle{\|\hat{u}\|_{L^2} = \|u\|_{L^2}}</math> gives the inequality.
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| In the special case of ''n'' =1, the Nash inequality can be extended{{fact|date=July 2012}} to the ''L''<sup>''p''</sup> case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality {{harv|Brezis|1999}}. In fact, if ''I'' is a bounded interval, then for all 1 ≤ ''r'' < ∞ and all 1 ≤ ''q'' ≤ ''p'' < ∞ the following inequality holds
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| :<math>\| u\|_{L^p(I)}\le C\| u\|^{1-a}_{L^q(I)} \|u\|^a_{W^{1,r}(I)}</math>
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| where ''a'' is defined by
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| :<math>a\left(\frac{1}{q}-\frac{1}{r}+1\right)=\frac{1}{q}-\frac{1}{p}.</math>
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| ==References==
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| *{{citation|mr=0450957|last= Adams|first= Robert A. |title=Sobolev spaces|series=Pure and Applied Mathematics, |volume= 65.|publisher= Academic Press |publication-place= New York-London|year= 1975|pages= xviii+268|isbn=978-0-12-044150-1|unused_data=isbn status=May be invalid - please double check}}.
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| *{{Citation | last1=Aubin | first1=Thierry | title=Espaces de Sobolev sur les variétés riemanniennes | mr=0488125 | year=1976 | journal=Bulletin des Sciences Mathématiques. 2e Série | issn=0007-4497 | volume=100 | issue=2 | pages=149–173}}
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| *{{Citation | last1=Aubin | first1=Thierry | title=Nonlinear analysis on manifolds. Monge-Ampère equations | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-0-387-90704-8 | mr=681859 | year=1982 | volume=252}}.
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| * {{citation|first=Haïm|last=Brezis|authorlink=Haïm Brezis|title=Analyse fonctionnelle : théorie et applications|publisher=[[Masson]]|location=Paris|year=1983|isbn=0-8218-0772-2}}
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| *{{citation|first=Lawrence|last=Evans|authorlink=Lawrence C. Evans| title=Partial Differential Equations | publisher=American Mathematical Society, Providence | year=1998 | isbn=0-8218-0772-2}}
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| * {{citation|first=Maz'ja|last=Vladimir G.|authorlink= Vladimir Maz'ja|title=Sobolev spaces|series=Springer Series in Soviet Mathematics|publisher=Springer-Verlag|publication-place=Berlin|year=1985}}, Translated from the Russian by T. O. Shaposhnikova.
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| *{{citation|last=Nash|first=J.|authorlink=John Forbes Nash, Jr.|title=Continuity of solutions of parabolic and elliptic equations|journal=Amer. J. Math.|volume=80|year=1958|pages=931–954|doi=10.2307/2372841|jstor=2372841|issue=4|publisher=American Journal of Mathematics, Vol. 80, No. 4}}.
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| *{{springer|id=i/i050230|title=Imbedding theorems|first=S.M.|last= Nikol'skii}}
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| * {{citation|first=Elias|last=Stein|authorlink=Elias Stein|title=Singular integrals and differentiability properties of functions|publisher=[[Princeton University Press]]|location=Princeton, NJ|year=1970|isbn=0-691-08079-8}}
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| [[Category:Inequalities]]
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| [[Category:Sobolev spaces]]
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| [[Category:Compactness theorems]]
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