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A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859801.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859802.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859803.png" /> (usually open) such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859804.png" />-th power of the absolute value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859805.png" /> and of its generalized derivatives (cf. [[Generalized derivative|Generalized derivative]]) up to and including order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859806.png" /> are integrable (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859807.png" />).
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{{TEX|part}}
  
The norm of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859808.png" /> is given by
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A space $W^l_p(\Omega)$ of functions $f=f(x)=f(x_1,\ldots,x_n)$ on a set $\Omega\subset\RR^n$ (usually open) such that the $p$-th power of the absolute value of $f$ and of its generalized derivatives (cf. [[Generalized derivative|Generalized derivative]]) up to and including order $l$ are integrable ($1\leq p\leq \infty$).
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 +
<!-- BEGIN CODE TO BE REMOVED --->
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<!-- A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859801.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859802.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859803.png" /> (usually open) such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859804.png" />-th power of the absolute value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859805.png" /> and of its generalized derivatives (cf. [[Generalized derivative|Generalized derivative]]) up to and including order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859806.png" /> are integrable (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859807.png" />). --->
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<!-- END CODE TO BE REMOVED --->
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The norm of a function $f\in W^l_p(\Omega)$ is given by
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\begin{equation*}
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  \lVert f\rVert_{W^l_p(\Omega)}=\sum_{\lvert k\rvert\leq l}
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  \lVert f^{(k)}\rVert_{L_p(\Omega)}.
 +
\end{equation*}
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Here
 +
\begin{equation*}
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  f^{(k)}=\frac{\partial^{\lvert k\rvert}f}{\partial x_1^{k_1}\cdots
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      \partial x_n^{k_n}},\qquad f^{(0)}=f,
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\end{equation*}
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is the generalized partial derivative of $f$ of order
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$\lvert k\rvert=\sum_{j=1}^n k_j$, and
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\begin{equation*}
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  \lVert \psi\rVert_{L_p(\Omega)}
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  =\left( \int_\Omega \lvert\psi(x)\rvert^p\,dx \right)^{1/p}
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  \qquad (1\leq p\leq \infty).
 +
\end{equation*}
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 +
<!-- BEGIN CODE TO BE REMOVED --->
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<!-- The norm of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859808.png" /> is given by
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s0859809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598010.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598010.png" /></td> </tr></table>
  
is the generalized partial derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598011.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598012.png" />, and
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is the generalized partial derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598011.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598012.png" />, and  
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598013.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598013.png" /></td> </tr></table> --->
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<!-- END CODE TO BE REMOVED --->
  
 
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598014.png" />, this norm is equal to the essential supremum:
 
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598014.png" />, this norm is equal to the essential supremum:

Revision as of 23:38, 3 May 2012


A space $W^l_p(\Omega)$ of functions $f=f(x)=f(x_1,\ldots,x_n)$ on a set $\Omega\subset\RR^n$ (usually open) such that the $p$-th power of the absolute value of $f$ and of its generalized derivatives (cf. Generalized derivative) up to and including order $l$ are integrable ($1\leq p\leq \infty$).


The norm of a function $f\in W^l_p(\Omega)$ is given by \begin{equation*} \lVert f\rVert_{W^l_p(\Omega)}=\sum_{\lvert k\rvert\leq l} \lVert f^{(k)}\rVert_{L_p(\Omega)}. \end{equation*} Here \begin{equation*} f^{(k)}=\frac{\partial^{\lvert k\rvert}f}{\partial x_1^{k_1}\cdots \partial x_n^{k_n}},\qquad f^{(0)}=f, \end{equation*} is the generalized partial derivative of $f$ of order $\lvert k\rvert=\sum_{j=1}^n k_j$, and \begin{equation*} \lVert \psi\rVert_{L_p(\Omega)} =\left( \int_\Omega \lvert\psi(x)\rvert^p\,dx \right)^{1/p} \qquad (1\leq p\leq \infty). \end{equation*}


When , this norm is equal to the essential supremum:

that is, to the greatest lower bound of the set of all for which on a set of measure zero.

The space was defined and first applied in the theory of boundary value problems of mathematical physics by S.L. Sobolev (see [1], [2]).

Since its definition involves generalized derivatives rather than ordinary ones, it is complete, that is, it is a Banach space.

is considered in conjunction with the linear subspace consisting of functions having partial derivatives of order that are uniformly continuous on . has advantages over , although it is not closed in the metric of and is not a complete space. However, for a wide class of domains (those with a Lipschitz boundary, see below) the space is dense in for all , , that is, for such domains the space acquires a new property in addition to completeness, in that every function belonging to it can be arbitrarily well approximated in the metric of by functions from .

It is sometimes convenient to replace the expression (1) for the norm of by the following:

(1prm)

The norm (1prm) is equivalent to the norm (1) i.e. , where do not depend on . When , (1prm) is a Hilbert norm, and this fact is widely used in applications.

The boundary of a bounded domain is said to be Lipschitz if for any there is a rectangular coordinate system with origin so that the box

is such that the intersection is described by a function , with

which satisfies on (the projection of onto the plane ) the Lipschitz condition

where the constant does not depend on the points , and . All smooth and many piecewise-smooth boundaries are Lipschitz boundaries.

For a domain with a Lipschitz boundary, (1) is equivalent to the following:

where

One can consider more general anisotropic spaces (classes) , where is a positive vector (see Imbedding theorems). For every such vector one can define, effectively and to a known extent exhaustively, a class of domains with the property that if , then any function can be extended to within the same class. More precisely, it is possible to define a function on with the properties

where does not depend on (see [3]).

In virtue of this property, inequalities of the type found in imbedding theorems for functions automatically carry over to functions , .

For vectors , the domains have Lipschitz boundaries, and .

The investigation of the spaces (classes) () is based on special integral representations for functions belonging to these classes. The first such representation was obtained (see [1], [2]) for an isotropic space of a domain , star-shaped with respect to some sphere. For the further development of this method see, for example, [3].

The classes and can be generalized to the case of fractional , or vectors with fractional components .

The space can also be defined for negative integers . Its elements are usually generalized functions, that is, linear functionals on infinitely-differentiable functions with compact support in .

By definition, a generalized function belongs to the class () if

is finite, where the supremum is taken over all functions with norm at most one . The functions form the space adjoint to the Banach space .

References

[1] S.L. Sobolev, "On a theorem of functional analysis" Transl. Amer. Math. Soc. (2) , 34 (1963) pp. 39–68 Mat. Sb. , 4 (1938) pp. 471–497
[2] S.L. Sobolev, "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)
[3] O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , 1–2 , Wiley (1978) (Translated from Russian)
[4] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)


Comments

References

[a1] V.G. Maz'ja, "Sobolev spaces" , Springer (1985)
[a2] F. Trèves, "Basic linear partial differential equations" , Acad. Press (1975) pp. Sects. 24–26
[a3] R.A. Adams, "Sobolev spaces" , Acad. Press (1975)
How to Cite This Entry:
Sobolev space. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Sobolev_space&oldid=25912
This article was adapted from an original article by S.M. Nikol'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article