Imbedding theorems
Theorems concerning a kind of problems involved in the study of inequalities between the norms of the same function in different classes (normed spaces). One is usually concerned with two classes and
, where
is a part of
(
), such that an inequality
![]() |
is satisfied for all , where
is a constant which is independent of
, and
,
are the norms in
and
, respectively. Under these conditions one speaks of an imbedding of
into
or one says that
is imbeddable in
, written as
(cf. also Imbedding of function spaces). The studies connected with imbedding theorems constitute a branch of the theory of functions, but their main paths of development concern the boundary value problems of mathematical physics, in particular direct variational methods. For this reason a systematic theory of imbeddings of classes of differentiable functions of several variables has been developed during the past three decades.
The following problems are examples of problems solved by imbedding theorems. Let a function be known to have, usually generalized (cf. Generalized derivative), partial derivatives of order
whose
-th powers are integrable on a given domain
of the
-dimensional space
. The questions are: 1) How many continuous derivatives does this function have on
? 2) If the domain
has a sufficiently smooth boundary
, is it possible to determine in some sense the trace
of the function
at the points
, i.e. the limit values of
as
tends to
, and what are the differentiability properties of this trace? Such properties should often be known exactly enough such that a function
given on
and possessing these properties can be extended from
to
in such a way that the extended function has generalized derivatives of order
whose
-th powers are integrable on
. It will be seen from the facts given below that these limits (in the sense of almost-everywhere convergence) for the determination of the trace
of
and of the extension of
can be accompanied by inequalities between the norms of
on
and
, which are used in the theory of boundary value problems.
The multi-dimensional theory of imbeddings of classes of differentiable functions originated in the 1930s in the studies of S.L. Sobolev in the context of problems in mathematical physics. He is to be credited with fundamental imbedding theorems for the classes (the Sobolev spaces, cf. Sobolev space) which play an important role in analysis. A function
belongs to
,
,
if it is defined on
and has a finite norm
![]() | (1) |
where
![]() | (2) |
and the sum is extended over all possible (Sobolev-generalized) partial derivatives
![]() | (3) |
![]() |
of order .
Sobolev's fundamental theorem (with completions by V.I. Kondrashov and V.P. Il'in) for the case : If
,
,
, the following imbedding is valid:
![]() | (4) |
where is the integer part of
.
If , this means that a function
has a trace (see below) on any coordinate hyperplane
of dimension
,
![]() |
and
![]() |
where does not depend on
[6], [7].
A function defined on
has a trace on
, where
is an
-dimensional (coordinate) subspace of points
with fixed
, if
can be modified on some set of
-dimensional measure zero, so that
![]() | (5) |
![]() |
![]() |
holds for the modified function (which is again denoted by ).
If is a set of functions
defined on
, the problem of describing the properties of the traces of these functions on a subspace
(
) is said to be the trace problem for the class
.
Theorem (4) is a final theorem in terms of the classes . Its further strengthening is possible only if new classes are introduced.
In the one-dimensional case , where the trace problem does not occur, theorem (4) is due to G.H. Hardy and J.E. Littlewood.
The next stages in the development of this theory were Nikol'skii's imbedding theorems for generalized Hölder classes (cf. Hölder space) (-classes). These classes form a scale with continuously varying parameters which characterize the smoothness of the functions. They are anisotropic in the sense that their functions usually display different differentiability properties in different directions. Let
be the set of points
at distance from the boundary of
greater than
, and let
be a positive vector (
;
),
, where
is an integer and
.
A function belongs to the class
,
, if
and if for an arbitrary
a generalized partial derivative
![]() | (6) |
exists which satisfies the inequality
![]() | (7) |
where denotes the second-difference operator of the function with respect to the variable
, with step
, and
is a constant which is independent of
.
The class forms a Banach space with norm
![]() |
where is the smallest constant
for which the inequalities (7) are satisfied. If
, the respective (isotropic) class is denoted by
. If
is an integer, the class
is close to the Sobolev class
, with an accuracy of
, in the sense that
![]() | (8) |
Nikol'skii's imbedding theorems are valid:
![]() | (9) |
where
![]() |
![]() |
![]() |
![]() | (10) |
where ,
,
,
![]() |
(cf. [5]).
Theorem (9) is the anisotropic analogue of theorem (4), with the advantage that the (vectorial) superscripts of the classes appearing in it may vary in a continuous manner. Moreover, it includes the cases
. However, for
it is not valid, unlike (4). Hardy and Littlewood demonstrated the theorem for the one-variable case
with non-integer
and
.
The imbedding (10) with the upper arrow is also given by a special case of theorem (9), when . It states that a function
has a trace
on
and that also
![]() | (11) |
where is independent of
. The reverse statement, symbolized by the lower arrow, is also true, and should be understood in the following sense: Any function
defined on
may be extended to the entire space
so that the resulting function
(with trace on
equal to
) belongs to
and satisfies the inequality (reverse to (11)):
![]() |
where does not depend on
.
The mutually inverse imbeddings (10) represent a complete solution to the trace problem for -classes, in terms of these classes.
Theorem (9) is transitive, which means that the transition
![]() | (12) |
from the first class in the chain (12) to the second, and then from the second to the third, where the parameters are computed by the formulas in (9), may be replaced by a direct transition from the first to the third class,
being calculated by the same formulas.
Subsequently (cf. [14]) a solution was given for the trace problem in -classes, which are in general anisotropic. This resulted in the introduction of a new family of classes of differentiable functions of several variables,
, which depend on the vector parameter
and on two scalar parameters
which satisfy the inequalities
. This family was completely determined by O.V. Besov, who also studied its fundamental properties.
A function belongs to the class
, where
is an integer vector, if it has finite meaningful norm
![]() | (13) |
![]() |
A function belongs to the class
, where
is an arbitrary, not necessarily integer, vector,
,
, if it has finite norm
![]() |
![]() |
where the numbers and
were defined above.
It is natural to regard the class if
as identical with the class
(
). One usually writes
rather than
if
and
,
. The classes
are Banach spaces for any given
.
The imbedding theorems (9) and (10) are valid if the symbols in them are replaced by
. There also exist a mutually inverse imbedding:
![]() | (14) |
where is an integer,
,
,
, which completely solves the trace problem for
-classes, and does not interfere with mutually inverse imbeddings, completely expressed in the language of
-classes:
![]() | (15) |
The classes corresponding to the parameter values
are usually denoted by
(
). If
, the imbeddings (14) may also be written as
![]() | (16) |
Classes whose definition involves the concept of a Liouville fractional derivative (cf. Fractional integration and differentiation) are natural extensions of -classes.
Using the terminology of generalized functions (cf. Generalized function), it is possible to define a class of test functions such that the class
of generalized functions constructed over it will have the following properties: 1)
for any finite
; 2) for any
, not necessarily an integer, the operation
![]() | (17) |
where denote, respectively, the direct and the inverse Fourier transform of
, is meaningful; and 3) if
is an integer and the function
has a Sobolev-generalized derivative
, then equation (17) applies to it.
In case is a fraction, operation (17) on infinitely-differentiable functions of compact support is identical with the Liouville fractional differentiation operation. It is natural to call
the fractional derivative of
of order
with respect to
if
is not an integer.
If an arbitrary vector is given, one may introduce the space
,
, which is identical with
for integer
, by replacing
in (13) by
.
If , one puts
. The family of classes
,
,
, may be regarded as a natural extension of the family
to fractional
— "natural" , since from the point of view of the present circle of interest the classes
display "all the advantages and all the drawbacks of Wpl" . If
is substituted for
in formula (4) (where
may be substituted for
) or in (8) (where
may be a fraction) or in (14), (16) (where
may be a fraction), these formulas will remain valid. The same also applies to formula (9) if
is replaced by
, even under the wider condition
, but under the assumption that
.
In what follows the apparatus of generalized functions will be used, except that these now constitute the space . For any real number
the Bessel–Macdonald operation is meaningful:
![]() |
It has the following properties: ,
,
,
where
is the Laplace operator.
The isotropic class ,
, may also be defined as the set of functions
that can be represented in the form
where the functions
run through the space
(
); moreover, up to equivalence,
![]() |
This definition of the class is also suitable for negative
, but in such a case
is a set of (usually generalized) functions
. In particular,
.
The operation may also be employed as a tool in defining the classes
(
). To do this, one calls a generalized function
regular in the sense of
or belonging to
if there exists a
such that
. Any function
,
,
, can be defined as a function that is regular in the sense of
and that can be represented as a series
![]() |
weakly converging towards (in the sense of
), where
has spectrum (the support of
) in
, while
,
, has spectrum in
and
![]() |
and also
![]() |
In particular,
![]() |
This definition of the class is automatically extended to the case
, and the functions
belonging to classes with negative
will usually be generalized
. Here,
,
.
There also exist other, equivalent, definitions of the negative classes , based on the principle of interpolation of function spaces. The definition given above is constructive — each class defined by the parameters
is defined independently, and it is possible to define constructively linear operations with the aid of which a function
(of exponential type
if
and of type 1 if
) is defined in terms of a given function
.
The following imbedding theorem is valid:
![]() |
This theorem is of the same type as theorem (4), but with ; it is valid for all real
for
,
, or for
,
,
, or for
,
.
On the other hand, for , an arbitrary function
usually has no trace on
(
) unless additional conditions are imposed.
The imbedding theorems formulated above apply to classes of functions defined on the entire -dimensional space
[5]. In practical applications, however, it is important to have similar theorems for domains
which should be as general as possible. The geometrical structure of the domains
for which the above imbedding theorems are valid for the classes
,
and
, where
,
must be replaced, respectively, by
,
, has now been clarified. For the isotropic classes
,
the domain
must satisfy a cone condition or, which is equivalent to it, its boundary must locally satisfy a Lipschitz condition. For the anisotropic classes
,
the domain
must satisfy an
-horn condition or a bent cone condition, and this condition is, in a certain sense, necessary [2].
Another problem with important practical applications is the trace problem on -dimensional manifolds
.
This problem has been completely solved for the isotropic classes ,
,
(see [2], ). If
is sufficiently differentiable and
,
can be substituted for
in (14), (15) and (16), and
can be substituted for
in . If
is piecewise smooth the problem has also been solved completely , [22]. The conditions for the solution of the problem are expressed by mutually inverse imbeddings on individual smooth pieces of
on one hand, and, on the other hand, by special additional conditions on the behaviour of the functions of the respective classes at the points of contact of these smooth pieces. The solution of the trace problem for anisotropic classes [9], [21] is also in an advanced stage. Here major difficulties arise, concerning the characteristics of the trace at the points of
where the tangent planes to
are parallel to the coordinate axes.
One more problem follows. Given a function
![]() |
where denotes one of the classes considered above. What mixed partial derivatives
does this function have and what are their properties? A positive answer to this question depends on the magnitude
![]() |
In fact, for there exists a partial derivative
which belongs to
if
. This condition may be generalized to the case of the spaces
if
(see [5]).
Yet follows another characteristic theorem, which may perhaps be called a theorem on weak compactness, and which has applications in the theory of direct methods of variational calculus.
Out of the infinite set of functions
which satisfy the inequality
![]() |
where is a known constant and
is one of the classes discussed above, it is possible to separate a sequence
of functions and to indicate a function
with norm
![]() |
such that, for all bounded domains and all vectors
,
![]() |
[5]. In this formulation may be replaced by a domain
if the latter has a sufficiently nice boundary. Only the typical function classes (and the imbedding theorems connected with them) which are most often encountered in practical applications were discussed above. In modern investigations stress is laid [2] on more general classes, in which more or less arbitrary differential operators play the role of the starting partial derivatives
,
.
Other classes under study comprise the so-called weight classes (cf. Weight space), a typical example of which is the class , defined as follows. Let
be the distance between a point
and the boundary
of a domain
. A function
belongs to
,
,
, if it has finite norm (see [4], [12])
![]() |
where
![]() |
One result is as follows. Let be a sufficiently smooth boundary of dimension
; then
![]() |
if ,
,
.
Example. The use of imbedding theorems presents a complete solution of the problem of conditions on the boundary function under which the Dirichlet principle is applicable. In fact, take the partial derivatives in the generalized sense and assume, for the sake of simplicity, that the surface (the boundary of a three-dimensional domain) is bounded and is twice differentiable, and that a function
on
has been given. For this function the Dirichlet integral
and also, in accordance with the imbedding theorem
![]() |
has a trace on
(the fact that a trace of
exists can be established with the aid of coarser imbedding theorems). Denoting by
the class of functions
with the same trace on
as
,
, the Dirichlet principle may be formulated as follows: The minimum of
over the functions
is attained for a unique function which is also harmonic on
. It follows from the imbedding theorem above that the Dirichlet principle is applicable if and only if the class
is non-empty, i.e. when the boundary function
.
In justifying the Dirichlet principle, the first step is to prove that the function exists and is unique, and the fact that
is a generalized solution of the Dirichlet problem. A special method is then used to successively establish that the generalized solution belongs to the classes
, where
and
is an arbitrary closed sphere. In particular, from the fact that
, applying the imbedding theorem
![]() |
(cf. [2], [5]) for ,
,
,
, one deduces that the function
may be modified on a set of three-dimensional measure zero so that the function thus obtained is twice continuously differentiable on
. It can then readily be proved that
is harmonic.
This example may be generalized to include certain functionals with partial derivatives of different orders, raised to a power usually distinct from 2 ; it is then necessary to use imbedding theorems for more general, usually anisotropic, classes.
References
[1] | O.V. Besov, et al., "The theory of imbedding classes of differentiable functions of several variables" , Partial differential equations , Moscow (1970) pp. 38–63 (In Russian) |
[2] | O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian) |
[3] | V.I. Burenkov, "Imbedding and extension theorems for classes of differentiable functions of several variables in the whole space" Itogi Nauk. Mat. Anal. 1965 (1966) (In Russian) |
[4] | S.M. Nikol'skii, "On imbedding, continuation and approximation theorems for differentiable functions of several variables" Russian Math. Surveys , 16 : 5 (1961) pp. 55–104 Uspekhi Mat. Nauk , 16 : 5 (1961) pp. 63–114 |
[5] | S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian) |
[6] | S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) |
[7] | S.L. Sobolev, "Introduction to the theory of cubature formulas" , Moscow (1974) (In Russian) |
[8] | O.V. Besov, "Investigation of a class of function spaces in connection with imbedding and embedding theorems" Trudy Mat. Inst. Steklov. , 60 (1961) pp. 42–81 (In Russian) |
[9] | Ya.S. Bugrov, "Boundary properties of functions of class ![]() |
[10] | V.P. Il'in, "On an inclusion theorem for a limiting exponent" Dokl. Akad. Nauk SSSR , 96 : 5 (1954) pp. 905–908 (In Russian) |
[11] | V.I. Kondrashov, "Sur certaines propriétés des fonctions dans l'espace" Dokl. Akad. Nauk SSSR , 48 (1945) pp. 535–538 |
[12] | L.D. Kudryavtsev, "Direct and inverse imbedding theorems. Applications to solutions of elliptic equations by variational methods" Trudy Mat. Inst. Steklov. , 55 (1959) (In Russian) |
[13] | P.I. Lizorkin, "Boundary properties of functions from "weight" classes" Soviet Math. Dokl. , 1 : 3 (1960) pp. 589–593 Dokl. Akad. Nauk SSSR , 132 : 3 (1960) pp. 514–517 |
[14] | P.I. Lizorkin, "Generalized Liouville differentiation and the function spaces ![]() |
[15] | S.M. Nikol'skii, "Inequalities for entire functions of finite degree and their application to the theory of differentiable functions in several variables" Trudy Mat. Inst. Steklov. , 38 (1951) pp. 244–278 (In Russian) |
[16a] | S.M. Nikol'skii, "Properties of certain classes of functions of several variables on a differentiable manifold" Mat. Sb. , 33 (75) : 2 (1953) pp. 261–326 (In Russian) |
[16b] | S.M. Nikol'skii, "Boundary properties of functions defined in a region with corner points" Mat. Sb. , 43 (85) : 1 (1957) pp. 127–144 (In Russian) |
[17] | S.L. Sobolev, "Le problème de Cauchy dans l'espace des fonctionnelles" Dokl. Akad. Nauk SSSR , 3 : 7 (1935) pp. 291–294 |
[18a] | S.L. Sobolev, "A new method for solving the Cauchy problem for partial differential equations of normal hyperbolic type" Mat. Sb. , 1 (43) : 1 (1936) pp. 39–72 (In Russian) |
[18b] | S.L. Sobolev, "On a theorem in functional analysis" Mat. Sb. , 4 (46) : 3 (1938) pp. 471–497 (In Russian) (French abstract) |
[19] | L.N. Slobodetskii, "S.L. Sobolev's spaces of fractional order and their application to boundary value problems for partial differential equations" Dokl. Akad. Nauk SSSR , 118 : 2 (1958) pp. 243–246 (In Russian) |
[20] | S.V. Uspenskii, "Imbedding theorems for classes with weights" Trudy Mat. Inst. Steklov. , 60 (1961) pp. 282–303 (In Russian) |
[21] | S.V. Uspenskii, "Boundary properties of a class of differentiable funtions in smooth regions" Soviet Math. Dokl. , 6 : 5 (1965) pp. 1299–1302 Dokl. Akad. Nauk SSSR , 164 : 4 (1965) pp. 750–752 |
[22] | G.N. Yakovlev, "Boundary properties of a class of functions" Trudy Mat. Inst. Steklov. , 60 (1961) pp. 325–349 (In Russian) |
[23] | E. Gagliardo, "Caratterizzazioni delle trace sulla frontiera relative ad alcune classi di funzioni in ![]() |
[24] | G.H. Hardy, J.E. Littlewood, "A convergence criterion for Fourier series" Math. Z. , 28 (1928) pp. 612–634 |
[25] | J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French) |
Imbedding theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Imbedding_theorems&oldid=14600