# Difference between revisions of "Sobolev space"

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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*} | ||

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<!-- 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 | + | 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]]. | ||

− | < | + | <!-- 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> | ||

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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/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