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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" />).
 
  
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>
 
  
Here
 
  
<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>
+
{{MSC|46E35}}
 +
{{TEX|done}}
  
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
+
$\newcommand{\abs}[1]{\lvert #1\rvert}
 +
\newcommand{\norm}[1]{\lVert #1\rVert}
 +
\newcommand{\bfl}{\mathbf{l}}$
  
<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>
 
  
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:
+
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 derivative|generalized derivative]] up to and including order $l$ are integrable ($1\leq p\leq \infty$).
  
<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>
+
The norm of a function $f\in W^l_p(\Omega)$ is given by
 +
\begin{equation}\label{eq:1}
 +
  \norm{f}_{W^l_p(\Omega)}=\sum_{\abs{\alpha}\leq l}
 +
  \norm{D^{\alpha}f}_{L_p(\Omega)}.
 +
\end{equation}
 +
Here
 +
\begin{equation*}
 +
  D^{\alpha}f=\frac{\partial^{\lvert \alpha\rvert}f}{\partial x_1^{\alpha_1}\cdots
 +
      \partial x_n^{\alpha_n}},\qquad D^{0}f=f,
 +
\end{equation*}
 +
is the generalized partial derivative of $f$ of order
 +
$\abs{\alpha}=\sum_{j=1}^n \alpha_j$, and
 +
\begin{equation*}
 +
  \norm{\psi}_{L_p(\Omega)}
 +
  =\left( \int_\Omega \abs{\psi(x)}^p\,dx \right)^{1/p}
 +
  \qquad (1\leq p< \infty).
 +
\end{equation*}
  
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.
+
When $p=\infty$, this norm is equal to the essential supremum:
 +
\begin{equation*}
 +
  \norm{\psi}_{L_\infty(\Omega)}
 +
  =\operatorname*{ess sup}_{x\in\Omega}\abs{\psi(x)} \qquad (p=\infty),
 +
\end{equation*}
 +
that is, to the greatest lower bound of the set of all $A$ for which  
 +
$A<\abs{\psi(x)}$ on a set of measure zero.
  
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 {{Cite|So1}}, {{Cite|So2}}).
  
 
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" />.
 
 
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/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 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
 
 
<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/s08598045.png" /></td> </tr></table>
 
 
is such that the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598046.png" /> is described by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598047.png" />, with
 
 
<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/s08598048.png" /></td> </tr></table>
 
  
which satisfies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598049.png" /> (the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598050.png" /> onto the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598051.png" />) the Lipschitz condition
+
==Equivalent norm==
  
<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/s08598052.png" /></td> </tr></table>
+
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}
 +
  \norm{f}^\prime_{W^l_p(\Omega)}=\left( \int_\Omega
 +
  \sum_{\abs{\alpha}\leq l} \abs{D^{\alpha}f(x)}^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.
 +
$c_1 \norm{f}\leq\norm{f}^\prime\leq c_2\norm{f}$,
 +
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.
  
where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598053.png" /> does not depend on the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598054.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598055.png" />. All smooth and many piecewise-smooth boundaries are Lipschitz boundaries.
+
==Subspace $W^l_{pc}(\Omega)$==
  
For a domain with a Lipschitz boundary, (1) is equivalent to the following:
+
The 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 $\Omega$. The space $W^l_{pc}(\Omega)$ 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 $W^l_{pc}(\Omega)$.
  
<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/s08598056.png" /></td> </tr></table>
+
The boundary $\Gamma$ of a bounded domain $\Omega$ is said to be Lipschitz if for any $x^0\in\Gamma$ there is a rectangular coordinate system $\xi=(\xi_1,\ldots,\xi_n)$ with origin $x^0$ so that the box
 +
\begin{equation*}
 +
  \Delta=\{ \xi : \abs{\xi_j}<\delta,\ j=1,\ldots,n \}
 +
\end{equation*}
 +
is such that the intersection $\Gamma\cap\Delta$ is described by a function $\xi_n=\psi(\xi')$, with
 +
\begin{equation*}
 +
  \xi'=(\xi_1,\ldots,\xi_n)\in\Delta'=\{\abs{\xi_j}<\delta,\ j=1,\ldots,n-1\},
 +
\end{equation*}
 +
which satisfies on $\Delta'$ (the projection of $\Delta$ onto the plane $\xi_n=0$) the Lipschitz condition
 +
\begin{equation*}
 +
  \abs{\psi(\xi'_1)-\psi(\xi'_2)}\leq M \abs{\xi'_1-\xi'_2},\quad \xi'_1,\xi'_2\in\Delta',
 +
\end{equation*}
 +
where the constant $M$ does not depend on the points $\xi'_1,\xi'_2$, and $\abs{\xi}^2=\sum_{j=1}^{n-1}\xi_j^2$. All smooth and many piecewise-smooth boundaries are Lipschitz boundaries.
  
 +
For a domain with a Lipschitz boundary, \eqref{eq:1} is equivalent to the following:
 +
\begin{equation*}
 +
  \norm{f}_{W^l_p(\Omega)}=\norm{f}_{L_p(\Omega)}+\norm{f}'_{w^l_p(\Omega)},
 +
\end{equation*}
 
where
 
where
 +
\begin{equation*}
 +
  \norm{f}'_{w^l_p(\Omega)}=\sum_{\abs{\alpha}=l}\norm{D^{\alpha}f}_{L_p(\Omega)}.
 +
\end{equation*}
  
<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/s08598057.png" /></td> </tr></table>
+
==Anisotropic spaces==
 
 
One can consider more general anisotropic spaces (classes) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598059.png" /> is a positive vector (see [[Imbedding theorems|Imbedding theorems]]). For every such vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598060.png" /> one can define, effectively and to a known extent exhaustively, a class of domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598061.png" /> with the property that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598062.png" />, then any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598063.png" /> can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598064.png" /> within the same class. More precisely, it is possible to define a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598065.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598066.png" /> with the properties
 
 
 
<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/s08598067.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598068.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598069.png" /> (see [[#References|[3]]]).
 
 
 
In virtue of this property, inequalities of the type found in imbedding theorems for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598070.png" /> automatically carry over to functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598072.png" />.
 
 
 
For vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598073.png" />, the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598074.png" /> have Lipschitz boundaries, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598075.png" />.
 
 
 
The investigation of the spaces (classes) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598076.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598077.png" />) is based on special integral representations for functions belonging to these classes. The first such representation was obtained (see [[#References|[1]]], [[#References|[2]]]) for an isotropic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598078.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598079.png" />, star-shaped with respect to some sphere. For the further development of this method see, for example, [[#References|[3]]].
 
 
 
The classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598081.png" /> can be generalized to the case of fractional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598082.png" />, or vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598083.png" /> with fractional components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598084.png" />.
 
 
 
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598085.png" /> can also be defined for negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598086.png" />. Its elements are usually generalized functions, that is, linear functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598087.png" /> on infinitely-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598088.png" /> with compact support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598089.png" />.
 
  
By definition, a [[Generalized function|generalized function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598090.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598091.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598092.png" />) if
+
One can consider more general anisotropic spaces (classes) $W^\bfl_p(\Omega)$, where $\bfl=(l_1,\ldots,l_n)$ is a positive vector (see [[Imbedding theorems|Imbedding theorems]]). For every such vector $\bfl$ one can define, effectively and to a known extent exhaustively, a class of domains $\mathfrak{M}^{(\bfl)}$ with the property that if $\Omega\in\mathfrak{M}^{(\bfl)}$, then any function $f\in W^\bfl_p(\Omega)$ can be extended to $\R^n$ within the same class. More precisely, it is possible to define a function $\bar{f}$ on $\R^n$ with the properties
 +
\begin{equation*}
 +
  \bar{f}(x)=f(x),\quad x\in\Omega,
 +
  \quad \norm{\bar{f}}_{W^\bfl_p(\R^n)}\leq c \norm{f}_{W^\bfl_p(\R^n)},
 +
\end{equation*}
 +
where $c$ does not depend on $f$ (see {{Cite|BeIlNi}}).
  
<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/s08598093.png" /></td> </tr></table>
+
In virtue of this property, inequalities of the type found in imbedding theorems for functions $f\in W^\bfl_p(\R^n)$ automatically carry over to functions
 +
$f\in W^\bfl_p(\Omega)$, $\Omega\in\mathfrak{M}^{(\bfl)}$.
  
is finite, where the supremum is taken over all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598094.png" /> with norm at most one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598095.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598096.png" /> form the space adjoint to the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085980/s08598097.png" />.
+
For vectors $\bfl=(l,\ldots,l)$, the domains $\Omega\in\mathfrak{M}^{(\bfl)}$ have Lipschitz boundaries, and $W^\bfl_p(\Omega)=W^l_p(\Omega)$.
  
====References====
+
The investigation of the spaces (classes) $W^\bfl_p(\Omega)$ ($\Omega\in\mathfrak{M}^{(\bfl)}$) is based on special integral representations for functions belonging to these classes. The first such representation was obtained (see {{Cite|So1}}, {{Cite|So2}}) for an isotropic space $W^\bfl_p(\Omega)$ of a domain $\Omega$, star-shaped with respect to some sphere. For the further development of this method see, for example, {{Cite|BeIlNi}}.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,  "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,   "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR></table>
 
  
 +
The classes $W^\bfl_p$ and $W^l_p$ can be generalized to the case of fractional $l$, or vectors $\bfl=(l_1,\ldots,l_n)$ with fractional components $l_j$.
  
 +
==Adjoint space==
  
====Comments====
+
The space $W^l_p(\Omega)$ can also be defined for negative integers $l$. Its elements are usually generalized functions, that is, linear functionals $(f,\phi)$ on infinitely-differentiable functions $\phi$ with compact support in $\Omega$.
  
 +
By definition, a [[Generalized function|generalized function]] $f$ belongs to the class $W^{-l}_p(\Omega)$ ($l=1,2,\ldots$) if
 +
\begin{equation*}
 +
  \norm{f}_{W^{-l}_p(\Omega)}=\sup(f,\phi)
 +
\end{equation*}
 +
is finite, where the supremum is taken over all functions $\phi\in W^l_q(\Omega)$ with norm at most one ($1/p+1/q=1$). The functions $f\in W^{-l}_p(\Omega)$ form the  [[ Adjoint_space | adjoint space]] to the Banach space $W^l_q(\Omega)$.
  
====References====
+
==References==
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V.G. Maz'ja,  "Sobolev spaces" , Springer  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> F. Trèves,  "Basic linear partial differential equations" , Acad. Press (1975)  pp. Sects. 24–26</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R.A. Adams,  "Sobolev spaces" , Acad. Press  (1975)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ad}}||valign="top"|  R.A. Adams,  "Sobolev spaces" , Acad. Press  (1975)  {{MR|0450957}}  {{ZBL|0314.46030}}
 +
|-
 +
|valign="top"|{{Ref|BeIlNi}}||valign="top"|  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)  {{MR|0519341}} {{MR|0521808}}  {{ZBL|0392.46022}}
 +
|-
 +
|valign="top"|{{Ref|Ma}}||valign="top"| V.G. Maz'ja,  "Sobolev spaces" , Springer  (1985)   {{ZBL|0692.46023}} {{ZBL|0727.46017}}
 +
|-
 +
|valign="top"|{{Ref|Ni}}||valign="top"| S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)   {{ZBL|0307.46024}}
 +
|-
 +
|valign="top"|{{Ref|So1}}||valign="top"|  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 
 +
|-
 +
|valign="top"|{{Ref|So2}}||valign="top"|  S.L. Sobolev,  "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963) (Translated from Russian)  {{MR|1125990}} {{MR|0986735}} {{MR|0052039}}  {{ZBL|0732.46001}}
 +
|-
 +
|valign="top"|{{Ref|Tr}}||valign="top"|  F. Trèves,  "Basic linear partial differential equations" , Acad. Press  (1975) pp. Sects. 24–26  {{MR|0447753}}  {{ZBL|0305.35001}}
 +
|-
 +
|}

Latest revision as of 17:47, 30 November 2012



2010 Mathematics Subject Classification: Primary: 46E35 [MSN][ZBL]

$\newcommand{\abs}[1]{\lvert #1\rvert} \newcommand{\norm}[1]{\lVert #1\rVert} \newcommand{\bfl}{\mathbf{l}}$


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 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} \norm{f}_{W^l_p(\Omega)}=\sum_{\abs{\alpha}\leq l} \norm{D^{\alpha}f}_{L_p(\Omega)}. \end{equation} Here \begin{equation*} D^{\alpha}f=\frac{\partial^{\lvert \alpha\rvert}f}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}},\qquad D^{0}f=f, \end{equation*} is the generalized partial derivative of $f$ of order $\abs{\alpha}=\sum_{j=1}^n \alpha_j$, and \begin{equation*} \norm{\psi}_{L_p(\Omega)} =\left( \int_\Omega \abs{\psi(x)}^p\,dx \right)^{1/p} \qquad (1\leq p< \infty). \end{equation*}

When $p=\infty$, this norm is equal to the essential supremum: \begin{equation*} \norm{\psi}_{L_\infty(\Omega)} =\operatorname*{ess sup}_{x\in\Omega}\abs{\psi(x)} \qquad (p=\infty), \end{equation*} that is, to the greatest lower bound of the set of all $A$ for which $A<\abs{\psi(x)}$ 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 [So1], [So2]).

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


Equivalent norm

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} \norm{f}^\prime_{W^l_p(\Omega)}=\left( \int_\Omega \sum_{\abs{\alpha}\leq l} \abs{D^{\alpha}f(x)}^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. $c_1 \norm{f}\leq\norm{f}^\prime\leq c_2\norm{f}$, 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.

Subspace $W^l_{pc}(\Omega)$

The 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 $\Omega$. The space $W^l_{pc}(\Omega)$ 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 $W^l_{pc}(\Omega)$.

The boundary $\Gamma$ of a bounded domain $\Omega$ is said to be Lipschitz if for any $x^0\in\Gamma$ there is a rectangular coordinate system $\xi=(\xi_1,\ldots,\xi_n)$ with origin $x^0$ so that the box \begin{equation*} \Delta=\{ \xi : \abs{\xi_j}<\delta,\ j=1,\ldots,n \} \end{equation*} is such that the intersection $\Gamma\cap\Delta$ is described by a function $\xi_n=\psi(\xi')$, with \begin{equation*} \xi'=(\xi_1,\ldots,\xi_n)\in\Delta'=\{\abs{\xi_j}<\delta,\ j=1,\ldots,n-1\}, \end{equation*} which satisfies on $\Delta'$ (the projection of $\Delta$ onto the plane $\xi_n=0$) the Lipschitz condition \begin{equation*} \abs{\psi(\xi'_1)-\psi(\xi'_2)}\leq M \abs{\xi'_1-\xi'_2},\quad \xi'_1,\xi'_2\in\Delta', \end{equation*} where the constant $M$ does not depend on the points $\xi'_1,\xi'_2$, and $\abs{\xi}^2=\sum_{j=1}^{n-1}\xi_j^2$. All smooth and many piecewise-smooth boundaries are Lipschitz boundaries.

For a domain with a Lipschitz boundary, \eqref{eq:1} is equivalent to the following: \begin{equation*} \norm{f}_{W^l_p(\Omega)}=\norm{f}_{L_p(\Omega)}+\norm{f}'_{w^l_p(\Omega)}, \end{equation*} where \begin{equation*} \norm{f}'_{w^l_p(\Omega)}=\sum_{\abs{\alpha}=l}\norm{D^{\alpha}f}_{L_p(\Omega)}. \end{equation*}

Anisotropic spaces

One can consider more general anisotropic spaces (classes) $W^\bfl_p(\Omega)$, where $\bfl=(l_1,\ldots,l_n)$ is a positive vector (see Imbedding theorems). For every such vector $\bfl$ one can define, effectively and to a known extent exhaustively, a class of domains $\mathfrak{M}^{(\bfl)}$ with the property that if $\Omega\in\mathfrak{M}^{(\bfl)}$, then any function $f\in W^\bfl_p(\Omega)$ can be extended to $\R^n$ within the same class. More precisely, it is possible to define a function $\bar{f}$ on $\R^n$ with the properties \begin{equation*} \bar{f}(x)=f(x),\quad x\in\Omega, \quad \norm{\bar{f}}_{W^\bfl_p(\R^n)}\leq c \norm{f}_{W^\bfl_p(\R^n)}, \end{equation*} where $c$ does not depend on $f$ (see [BeIlNi]).

In virtue of this property, inequalities of the type found in imbedding theorems for functions $f\in W^\bfl_p(\R^n)$ automatically carry over to functions $f\in W^\bfl_p(\Omega)$, $\Omega\in\mathfrak{M}^{(\bfl)}$.

For vectors $\bfl=(l,\ldots,l)$, the domains $\Omega\in\mathfrak{M}^{(\bfl)}$ have Lipschitz boundaries, and $W^\bfl_p(\Omega)=W^l_p(\Omega)$.

The investigation of the spaces (classes) $W^\bfl_p(\Omega)$ ($\Omega\in\mathfrak{M}^{(\bfl)}$) is based on special integral representations for functions belonging to these classes. The first such representation was obtained (see [So1], [So2]) for an isotropic space $W^\bfl_p(\Omega)$ of a domain $\Omega$, star-shaped with respect to some sphere. For the further development of this method see, for example, [BeIlNi].

The classes $W^\bfl_p$ and $W^l_p$ can be generalized to the case of fractional $l$, or vectors $\bfl=(l_1,\ldots,l_n)$ with fractional components $l_j$.

Adjoint space

The space $W^l_p(\Omega)$ can also be defined for negative integers $l$. Its elements are usually generalized functions, that is, linear functionals $(f,\phi)$ on infinitely-differentiable functions $\phi$ with compact support in $\Omega$.

By definition, a generalized function $f$ belongs to the class $W^{-l}_p(\Omega)$ ($l=1,2,\ldots$) if \begin{equation*} \norm{f}_{W^{-l}_p(\Omega)}=\sup(f,\phi) \end{equation*} is finite, where the supremum is taken over all functions $\phi\in W^l_q(\Omega)$ with norm at most one ($1/p+1/q=1$). The functions $f\in W^{-l}_p(\Omega)$ form the adjoint space to the Banach space $W^l_q(\Omega)$.

References

[Ad] R.A. Adams, "Sobolev spaces" , Acad. Press (1975) MR0450957 Zbl 0314.46030
[BeIlNi] 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) MR0519341 MR0521808 Zbl 0392.46022
[Ma] V.G. Maz'ja, "Sobolev spaces" , Springer (1985) Zbl 0692.46023 Zbl 0727.46017
[Ni] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) Zbl 0307.46024
[So1] 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
[So2] S.L. Sobolev, "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) MR1125990 MR0986735 MR0052039 Zbl 0732.46001
[Tr] F. Trèves, "Basic linear partial differential equations" , Acad. Press (1975) pp. Sects. 24–26 MR0447753 Zbl 0305.35001
How to Cite This Entry:
Sobolev space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sobolev_space&oldid=17396
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