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The norm of a function $f\in W^l_p(\Omega)$ is given by
 
The norm of a function $f\in W^l_p(\Omega)$ is given by
\begin{equation*}
+
\begin{equation}\label{eq:1}
 
   \lVert f\rVert_{W^l_p(\Omega)}=\sum_{\lvert k\rvert\leq l}
 
   \lVert f\rVert_{W^l_p(\Omega)}=\sum_{\lvert k\rvert\leq l}
 
   \lVert f^{(k)}\rVert_{L_p(\Omega)}.
 
   \lVert f^{(k)}\rVert_{L_p(\Omega)}.
\end{equation*}
+
\end{equation}
 
Here
 
Here
 
\begin{equation*}
 
\begin{equation*}
Line 39: Line 39:
 
<!-- END CODE TO BE REMOVED --->
 
<!-- 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 $p=\infty$, this norm is equal to the essential supremum:
 +
\begin{equation*}
 +
  \lVert \psi\rVert_{L_\infty(\Omega)}
 +
  =\operatorname*{ess sup}_{x\in\Omega}\lvert\psi(x)\rvert \qquad (p=\infty),
 +
\end{equation*}
 +
that is, to the greatest lower bound of the set of all $A$ for which $A<\lvert\psi(x)\rvert$ on a set of measure zero.
 +
 
 +
<!-- BEGIN 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:
  
 
<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/s08598015.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/s08598015.png" /></td> </tr></table>
  
that is, to the greatest lower bound of the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598016.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598017.png" /> on a set of measure zero.
+
that is, to the greatest lower bound of the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598016.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598017.png" /> on a set of measure zero. -->
 +
<!-- END CODE TO BE REMOVED --->
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598018.png" /> was defined and first applied in the theory of boundary value problems of mathematical physics by S.L. Sobolev (see [[#References|[1]]], [[#References|[2]]]).
+
The space $W^l_p(\Omega)$ was defined and first applied in the theory of boundary value problems of mathematical physics by S.L. Sobolev (see [[#References|[1]]], [[#References|[2]]]).
  
 
Since its definition involves generalized derivatives rather than ordinary ones, it is complete, that is, it is a [[Banach space|Banach space]].
 
Since its definition involves generalized derivatives rather than ordinary ones, it is complete, that is, it is a [[Banach space|Banach space]].
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598019.png" /> is considered in conjunction with the linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598020.png" /> consisting of functions having partial derivatives of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598021.png" /> that are uniformly continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598022.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598023.png" /> has advantages over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598024.png" />, although it is not closed in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598025.png" /> and is not a complete space. However, for a wide class of domains (those with a Lipschitz boundary, see below) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598026.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598029.png" />, that is, for such domains the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598030.png" /> acquires a new property in addition to completeness, in that every function belonging to it can be arbitrarily well approximated in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598031.png" /> by functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598032.png" />.
+
<!-- BEGIN CODE TO BE REMOVED --->
 +
<!-- The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598018.png" /> was defined and first applied in the theory of boundary value problems of mathematical physics by S.L. Sobolev (see [[#References|[1]]], [[#References|[2]]]).
 +
 
 +
Since its definition involves generalized derivatives rather than ordinary ones, it is complete, that is, it is a [[Banach space|Banach space]]. --->
 +
<!-- END CODE TO BE REMOVED --->
  
It is sometimes convenient to replace the expression (1) for the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598033.png" /> by the following:
+
$W^l_p(\Omega)$ is considered in conjunction with the linear subspace
 +
$W^l_{pc}(\Omega)$ consisting of functions having partial derivatives of order $l$ that are uniformly continuous on $\Omega$. $W^l_{pc}(\Omega)$ has advantages over $W^l_p(\Omega)$, although it is not closed in the metric of $W^l_p(\Omega)$ and is not a complete space. However, for a wide class of domains (those with a Lipschitz boundary, see below) the space $W^l_{pc}(\Omega)$ is dense in $W^l_p(\Omega)$ for all $p$, $1\leq p<\infty$, that is, for such domains the space $W^l_p(\Omega)$ acquires a new property in addition to completeness, in that every function belonging to it can be arbitrarily well approximated in the metric of $W^l_p(\Omega)$ by functions from $W^l_{pc}(\Omega)$.
 +
 
 +
<!-- BEGIN CODE TO BE REMOVED --->
 +
<!-- <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598019.png" /> is considered in conjunction with the linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598020.png" /> consisting of functions having partial derivatives of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598021.png" /> that are uniformly continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598022.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598023.png" /> has advantages over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598024.png" />, although it is not closed in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598025.png" /> and is not a complete space. However, for a wide class of domains (those with a Lipschitz boundary, see below) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598026.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598029.png" />, that is, for such domains the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598030.png" /> acquires a new property in addition to completeness, in that every function belonging to it can be arbitrarily well approximated in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598031.png" /> by functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598032.png" />. --->
 +
<!-- END CODE TO BE REMOVED --->
 +
 
 +
It is sometimes convenient to replace the expression \eqref{eq:1} for the norm of
 +
$f\in W^l_p(\Omega)$ by the following:
 +
\begin{equation}\label{eq:2}
 +
  \lVert f\rVert^\prime_{W^l_p(\Omega)}=\left( \int_\Omega
 +
  \sum_{\lvert k\rvert\leq l} \lvert f^{(k)}(x)\rvert^p \,dx \right)^{1/p}
 +
  \qquad (1\leq p<\infty).
 +
\end{equation}
 +
The norm \eqref{eq:2} is equivalent to the norm \eqref{eq:1}, i.e.
 +
$c_1 \lVert f\rVert \leq \lVert f\rVert^\prime \leq c_2 \lVert f\rVert$,
 +
where $c_1, c_2>0$ do not depend on $f$. When $p=2$, \eqref{eq:2} is a Hilbert norm, and this fact is widely used in applications.
 +
 
 +
<!-- BEGIN CODE TO BE REMOVED --->
 +
<!-- It is sometimes convenient to replace the expression (1) for the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598033.png" /> by the following:
  
 
<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/s08598034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</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/s08598034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>
Line 57: Line 89:
 
<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/s08598035.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/s08598035.png" /></td> </tr></table>
  
The norm (1prm) is equivalent to the norm (1) i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598037.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598038.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598039.png" />, (1prm) is a Hilbert norm, and this fact is widely used in applications.
+
The norm (1prm) is equivalent to the norm (1) i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598037.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598038.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598039.png" />, (1prm) is a Hilbert norm, and this fact is widely used in applications. --->
 +
<!-- END CODE TO BE REMOVED --->
 +
 
  
 
The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598040.png" /> of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598041.png" /> is said to be Lipschitz if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598042.png" /> there is a rectangular coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598043.png" /> with origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598044.png" /> so that the box
 
The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598040.png" /> of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598041.png" /> is said to be Lipschitz if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598042.png" /> there is a rectangular coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598043.png" /> with origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598044.png" /> so that the box

Revision as of 22:25, 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}\label{eq:1} \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 $p=\infty$, this norm is equal to the essential supremum: \begin{equation*} \lVert \psi\rVert_{L_\infty(\Omega)} =\operatorname*{ess sup}_{x\in\Omega}\lvert\psi(x)\rvert \qquad (p=\infty), \end{equation*} that is, to the greatest lower bound of the set of all $A$ for which $A<\lvert\psi(x)\rvert$ on a set of measure zero.


The space $W^l_p(\Omega)$ 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.


$W^l_p(\Omega)$ is considered in conjunction with the linear subspace $W^l_{pc}(\Omega)$ consisting of functions having partial derivatives of order $l$ that are uniformly continuous on $\Omega$. $W^l_{pc}(\Omega)$ has advantages over $W^l_p(\Omega)$, although it is not closed in the metric of $W^l_p(\Omega)$ and is not a complete space. However, for a wide class of domains (those with a Lipschitz boundary, see below) the space $W^l_{pc}(\Omega)$ is dense in $W^l_p(\Omega)$ for all $p$, $1\leq p<\infty$, that is, for such domains the space $W^l_p(\Omega)$ acquires a new property in addition to completeness, in that every function belonging to it can be arbitrarily well approximated in the metric of $W^l_p(\Omega)$ by functions from $W^l_{pc}(\Omega)$.


It is sometimes convenient to replace the expression \eqref{eq:1} for the norm of $f\in W^l_p(\Omega)$ by the following: \begin{equation}\label{eq:2} \lVert f\rVert^\prime_{W^l_p(\Omega)}=\left( \int_\Omega \sum_{\lvert k\rvert\leq l} \lvert f^{(k)}(x)\rvert^p \,dx \right)^{1/p} \qquad (1\leq p<\infty). \end{equation} The norm \eqref{eq:2} is equivalent to the norm \eqref{eq:1}, i.e. $c_1 \lVert f\rVert \leq \lVert f\rVert^\prime \leq c_2 \lVert f\rVert$, where $c_1, c_2>0$ do not depend on $f$. When $p=2$, \eqref{eq:2} 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://encyclopediaofmath.org/index.php?title=Sobolev_space&oldid=25915
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