Sobolev space

2020 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 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} \norm{f}_{W^l_p(\Omega)}=\sum_{\abs{k}\leq l} \norm{f^{(k)}}_{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 $\abs{k}=\sum_{j=1}^n k_j$, 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 [So1], [So2]).

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} \norm{f}^\prime_{W^l_p(\Omega)}=\left( \int_\Omega \sum_{\abs{k}\leq l} \abs{f^{(k)}(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{k}=l}\norm{f^{(k)}}_{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). 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 $\overline{f}$ on $\R^n$ with the properties \begin{equation*} \overline{f}(x)=f(x),\quad x\in\Omega, \quad \norm{\overline{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 $l_j$.

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

References

[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)
[Ni]	S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[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)

References

[Ad]	R.A. Adams, "Sobolev spaces" , Acad. Press (1975)
[Ma]	V.G. Maz'ja, "Sobolev spaces" , Springer (1985)
[Tr]	F. Trèves, "Basic linear partial differential equations" , Acad. Press (1975) pp. Sects. 24–26