Sobolev space
A space of functions
on a set
(usually open) such that the
-th power of the absolute value of
and of its generalized derivatives (cf. Generalized derivative) up to and including order
are integrable (
).
The norm of a function is given by
![]() | (1) |
Here
![]() |
is the generalized partial derivative of of order
, and
![]() |
When , this norm is equal to the essential supremum:
![]() |
that is, to the greatest lower bound of the set of all for which
on a set of measure zero.
The space 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.
is considered in conjunction with the linear subspace
consisting of functions having partial derivatives of order
that are uniformly continuous on
.
has advantages over
, although it is not closed in the metric of
and is not a complete space. However, for a wide class of domains (those with a Lipschitz boundary, see below) the space
is dense in
for all
,
, that is, for such domains the space
acquires a new property in addition to completeness, in that every function belonging to it can be arbitrarily well approximated in the metric of
by functions from
.
It is sometimes convenient to replace the expression (1) for the norm of by the following:
![]() | (1prm) |
![]() |
The norm (1prm) is equivalent to the norm (1) i.e. , where
do not depend on
. When
, (1prm) 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) |
Sobolev space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sobolev_space&oldid=25912