View source for Sobolev space

{{MSC|46E35}}
{{TEX|done}}

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

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|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\leq \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 {{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]].

$W^l_p(\Omega)$ 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}
   \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.

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*}

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}}).

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 {{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}}.

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 $l_j$.

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 space adjoint to the Banach space $W^l_q(\Omega)$.

====References====
{|
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|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}} 
|-
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