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
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.
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:
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:
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 ).
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 , ) for an isotropic space of a domain , star-shaped with respect to some sphere. For the further development of this method see, for example, .
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 .
|||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|
|||S.L. Sobolev, "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)|
|||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)|
|||S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)|
|[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=17396