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Degenerate Anisotropic Differential Operators and Applications
Boundary Value Problems volume 2011, Article number: 268032 (2011)
Abstract
The boundary value problems for degenerate anisotropic differential operator equations with variable coefficients are studied. Several conditions for the separability and Fredholmness in Banachvalued spaces are given. Sharp estimates for resolvent, discreetness of spectrum, and completeness of root elements of the corresponding differential operators are obtained. In the last section, some applications of the main results are given.
1. Introduction and Notations
It is well known that many classes of PDEs, pseudoDes, and integroDEs can be expressed as differentialoperator equations (DOEs). As a result, many authors investigated PDEs as a result of single DOEs. DOEs in valued (Hilbert space valued) function spaces have been studied extensively in the literature (see [1–14] and the references therein). Maximal regularity properties for higherorder degenerate anisotropic DOEs with constant coefficients and nondegenerate equations with variable coefficients were studied in [15, 16].
The main aim of the present paper is to discuss the separability properties of BVPs for higherorder degenerate DOEs; that is,
where , are weighted functions, and are linear operators in a Banach Space . The above DOE is a generalized form of an elliptic equation. In fact, the special case , reduces (1.1) to elliptic form.
Note, the principal part of the corresponding differential operator is nonselfadjoint. Nevertheless, the sharp uniform coercive estimate for the resolvent, Fredholmness, discreetness of the spectrum, and completeness of root elements of this operator are established.
We prove that the corresponding differential operator is separable in ; that is, it has a bounded inverse from to the anisotropic weighted space . This fact allows us to derive some significant spectral properties of the differential operator. For the exposition of differential equations with bounded or unbounded operator coefficients in Banachvalued function spaces, we refer the reader to [8, 15–25].
Let be a positive measurable weighted function on the region . Let denote the space of all strongly measurable valued functions that are defined on with the norm
For , the space will be denoted by .
The weight we will consider satisfies an condition; that is, , if there is a positive constant such that
for all cubes .
The Banach space is called a UMD space if the Hilbert operator is bounded in , (see, e.g., [26]). UMD spaces include, for example, , spaces, and Lorentz spaces , , .
Let be the set of complex numbers and
A linear operator is said to be positive in a Banach space with bound if is dense on and
for all , is an identity operator in , and is the space of bounded linear operators in . Sometimes will be written as and denoted by . It is known [27, Section 1.15.1] that there exists fractional powers of the sectorial operator . Let denote the space with graphical norm
Let and be two Banach spaces. Now, , , will denote interpolation spaces obtained from by the method [27, Section 1.3.1].
A set is called bounded (see [3, 25, 26]) if there is a constant such that for all and ,
where is a sequence of independent symmetric valued random variables on .
The smallest for which the above estimate holds is called an bound of the collection and is denoted by .
Let denote the Schwartz class, that is, the space of all valued rapidly decreasing smooth functions on . Let be the Fourier transformation. A function is called a Fourier multiplier in if the map , is well defined and extends to a bounded linear operator in . The set of all multipliers in will denoted by .
Let
Definition 1.1.
A Banach space is said to be a space satisfying a multiplier condition if, for any , the boundedness of the set implies that is a Fourier multiplier in , that is, for any .
Let be a multiplier function dependent on the parameter . The uniform boundedness of the set ; that is,
implies that is a uniform collection of Fourier multipliers.
Definition 1.2.
The positive operator is said to be positive in a Banach space if there exists such that the set is bounded.
A linear operator is said to be positive in uniformly in if is independent of , is dense in and for any , .
The positive operator , is said to be uniformly positive in a Banach space if there exists such that the set is uniformly bounded; that is,
Let denote the space of all compact operators from to . For , it is denoted by .
For two sequences and of positive numbers, the expression means that there exist positive numbers and such that
Let denote the space of all compact operators from to . For , it is denoted by .
Now, denotes the approximation numbers of operator (see, e.g., [27, Section 1.16.1]). Let
Let and be two Banach spaces and continuously and densely embedded into and .
We let denote the space of all functions possessing generalized derivatives such that with the norm
Let . Consider the following weighted spaces of functions:
2. Background
The embedding theorems play a key role in the perturbation theory of DOEs. For estimating lower order derivatives, we use following embedding theorems from [24].
Theorem A1.
Let and and suppose that the following conditions are satisfied:
(1) is a Banach space satisfying the multiplier condition with respect to and ,
(2) is an positive operator in ,
(3) and are tuples of nonnegative integer such that
(4) is a region such that there exists a bounded linear extension operator from to .
Then, the embedding is continuous. Moreover, for all positive number and , the following estimate holds
Theorem A2.
Suppose that all conditions of Theorem A1 are satisfied. Moreover, let , be a bounded region and . Then, the embedding
is compact.
Let denote the closure of the linear span of the root vectors of the linear operator .
From [18, Theorem 3.4.1], we have the following.
Theorem A3.
Assume that
(1) is an UMD space and is an operator in , ,
(2) are non overlapping, differentiable arcs in the complex plane starting at the origin. Suppose that each of the regions into which the planes are divided by these arcs is contained in an angular sector of opening less then ,
(3) is an integer so that the resolvent of satisfies the inequality
as along any of the arcs .
Then, the subspace contains the space .
Let
Let
Let denote the embedding operator .
From [15, Theorem 2.8], we have the following.
Theorem A4.
Let and be two Banach spaces possessing bases. Suppose that
Then,
3. Statement of the Problem
Consider the BVPs for the degenerate anisotropic DOE
where
, , are complex numbers, are complexvalued functions on , , and are linear operators in . Moreover, and are such that
A function and satisfying (3.1) a.e. on is said to be solution of the problem (3.1)(3.2).
We say the problem (3.1)(3.2) is separable if for all , there exists a unique solution of the problem (3.1)(3.2) and a positive constant depending only such that the coercive estimate
holds.
Let be a differential operator generated by problem (3.1)(3.2) with ; that is,
We say the problem (3.1)(3.2) is Fredholm in if , where is a conjugate of .
Remark 3.1.
Under the substitutions
the spaces and are mapped isomorphically onto the weighted spaces and , where
Moreover, under the substitution (3.7) the problem (3.1)(3.2) reduces to the nondegenerate BVP
where
By denoting , , , , , , , again by , , , , , , , , respectively, we get
4. BVPs for Partial DOE
Let us first consider the BVP for the anisotropic type DOE with constant coefficients
where
are boundary conditions defined by (3.2), are complex numbers, is a complex parameter, and is a linear operator in a Banach space . Let be the roots of the characteristic equations
Now, let
By applying the trace theorem [27, Section 1.8.2], we have the following.
Theorem A5.
Let and be integer numbers, , , . Then, for any , the transformations are bounded linear from onto , and the following inequality holds:
Proof.
It is clear that
Then, by applying the trace theorem [27, Section 1.8.2] to the space , we obtain the assertion.
Condition 1.
Assume that the following conditions are satisfied:
(1) is a Banach space satisfying the multiplier condition with respect to and the weight function , ;
(2) is an positive operator in for ;
(3), and
for , .
Let denote the operator in generated by BVP (4.1). In [15, Theorem 5.1] the following result is proved.
Theorem A6.
Let Condition 1 be satisfied. Then,

(a)
the problem (4.1) for and with sufficiently large has a unique solution that belongs to and the following coercive uniform estimate holds:
(4.8)
(b)the operator is positive in .
From Theorems A5 and A6 we have.
Theorem A7.
Suppose that Condition 1 is satisfied. Then, for sufficiently large with the problem (4.1) has a unique solution for all and . Moreover, the following uniform coercive estimate holds:
Consider BVP (3.11). Let be roots of the characteristic equations
Condition 2.
Suppose the following conditions are satisfied:
(1) and
for
(2) is a Banach space satisfying the multiplier condition with respect to and the weighted function , .
Remark 4.1.
Let and , where are realvalued positive functions. Then, Condition 2 is satisfied for .
Consider the inhomogenous BVP (3.1)(3.2); that is,
Lemma 4.2.
Assume that Condition 2 is satisfied and the following hold:
(1) is a uniformly positive operator in for , and are continuous functions on , ,
(2) and for .
Then, for all and for sufficiently large the following coercive uniform estimate holds:
for the solution of problem (4.13).
Proof.
Let be regions covering and let be a corresponding partition of unity; that is, , and . Now, for and , we get
where
here, and are boundary operators which orders less than . Freezing the coefficients of (4.15), we have
where
It is clear that on neighborhoods of and
on neighborhoods of and on other parts of the domains , where are positive constants. Hence, the problems (4.17) are generated locally only on parts of the boundary. Then, by Theorem A7 problem (4.17) has a unique solution and for the following coercive estimate holds:
From the representation of , and in view of the boundedness of the coefficients, we get
Now, applying Theorem A1 and by using the smoothness of the coefficients of (4.16), (4.18) and choosing the diameters of so small, we see there is an and such that
Then, using Theorem A5 and using the smoothness of the coefficients of (4.16), (4.18), we get
Now, using Theorem A1, we get that there is an and such that
where
Using the above estimates, we get
Consequently, from (4.22)–(4.26), we have
Choosing from the above inequality, we obtain
Then, by using the equality and the above estimates, we get (4.14).
Condition 3.
Suppose that part (1.1) of Condition 1 is satisfied and that is a Banach space satisfying the multiplier condition with respect to and the weighted function , , .
Consider the problem (3.11). Reasoning as in the proof of Lemma 4.2, we obtain.
Proposition 4.3.
Assume Condition 3 hold and suppose that
(1) is a uniformly positive operator in for , and that are continuous functions on , ,
(2) and for .
Then, for all and for sufficiently large , the following coercive uniform estimate holds
for the solution of problem (3.11).
Let denote the operator generated by problem (3.11) for ; that is,
Theorem 4.4.
Assume that Condition 3 is satisfied and that the following hold:
(1) is a uniformly positive operator in , and are continuous functions on ,
(2), and for .
Then, problem (3.11) has a unique solution for and with large enough . Moreover, the following coercive uniform estimate holds:
Proof.
By Proposition 4.3 for , we have
It is clear that
Hence, by using the definition of and applying Theorem A1, we obtain
From the above estimate, we have
The estimate (4.35) implies that problem (3.11) has a unique solution and that the operator has a bounded inverse in its rank space. We need to show that this rank space coincides with the space ; that is, we have to show that for all , there is a unique solution of the problem (3.11). We consider the smooth functions with respect to a partition of unity on the region that equals one on , where supp and . Let us construct for all the functions that are defined on the regions and satisfying problem (3.11). The problem (3.11) can be expressed as
Consider operators in that are generated by the BVPs (4.17); that is,
By virtue of Theorem A6, the operators have inverses for and for sufficiently large . Moreover, the operators are bounded from to , and for all , we have
Extending to zero outside of in the equalities (4.36), and using the substitutions , we obtain the operator equations
where are bounded linear operators in defined by
In fact, because of the smoothness of the coefficients of the expression and from the estimate (4.38), for with sufficiently large , there is a sufficiently small such that
Moreover, from assumption (2.2) of Theorem 4.4 and Theorem A1 for , there is a constant such that
Hence, for with sufficiently large , there is a such that . Consequently, (4.39) for all have a unique solution . Moreover,
Thus, are bounded linear operators from to . Thus, the functions
are solutions of (4.38). Consider the following linear operator in defined by
It is clear from the constructions and from the estimate (4.39) that the operators are bounded linear from to , and for with sufficiently large , we have
Therefore, is a bounded linear operator in . Since the operators coincide with the inverse of the operator in , then acting on to gives
where are bounded linear operators defined by
Indeed, from Theorem A1 and estimate (4.46) and from the expression , we obtain that the operators are bounded linear from to , and for with sufficiently large , there is an such that . Therefore, there exists a bounded linear invertible operator ; that is, we infer for all that the BVP (3.11) has a unique solution
Result 1.
Theorem 4.4 implies that the resolvent satisfies the following anisotropic type sharp estimate:
for , .
Let denote the operator generated by BVP (3.1)(3.2). From Theorem 4.4 and Remark 3.1, we get the following.
Result 2.
Assume all the conditions of Theorem 4.4 hold. Then,
(a)the problem (3.1)(3.2) for , and for sufficiently large has a unique solution , and the following coercive uniform estimate holds
(b)if , then the operator is Fredholm from into .
Example 4.5.
Now, let us consider a special case of (3.1)(3.2). Let , and , , and ; that is, consider the problem
where
Theorem 4.4 implies that for each , problem (4.52) has a unique solution satisfying the following coercive estimate:
Example 4.6.
Let and , where are positive continuous function on , and is a diagonal matrixfunction with continuous components .
Then, we obtain the separability of the following BVPs for the system of anisotropic PDEs with varying coefficients:
in the vectorvalued space .
5. The Spectral Properties of Anisotropic Differential Operators
Consider the following degenerated BVP:
where
Consider the operator generated by problem (5.1).
Theorem 5.1.
Let all the conditions of Theorem 4.4 hold for and . Then, the operator is Fredholm from into .
Proof.
Theorem 4.4 implies that the operator for sufficiently large has a bounded inverse from to ; that is, the operator is Fredholm from into . Then, from Theorem A2 and the perturbation theory of linear operators, we obtain that the operator is Fredholm from into .
Theorem 5.2.
Suppose that all the conditions of Theorem 5.1 are satisfied with . Assume that is a Banach space with a basis and
Then,
(a)for a sufficiently large positive
(b)the system of root functions of the differential operator is complete in .
Proof.
Let denote the embedding operator from to . From Result 2, there exists a resolvent operator which is bounded from to . Moreover, from Theorem A4 and Remark 3.1, we get that the embedding operator
is compact and
It is clear that
Hence, from relations (5.6) and (5.7), we obtain (5.4). Now, Result 1 implies that the operator is positive in and
Then, from (4.52) and (5.6), we obtain assertion (b).
Consider now the operator in generated by the nondegenerate BVP obtained from (5.1) under the mapping (3.7); that is,
From Theorem 5.2 and Remark 3.1, we get the following.
Result 3.
Let all the conditions of Theorem 5.1 hold. Then, the operator is Fredholm from into .
Result 4.
Then,
(a)for a sufficiently large positive
(b)the system of root functions of the differential operator is complete in .
6. BVPs for Degenerate Quasielliptic PDE
In this section, maximal regularity properties of degenerate anisotropic differential equations are studied. Maximal regularity properties for PDEs have been studied, for example, in [3] for smooth domains and in [28] for nonsmooth domains.
Consider the BVP
where , , are complex number, and
Let , . Now, will denote the space of all summable scalarvalued functions with mixed norm (see, e.g., [29, Section 1, page 6]), that is, the space of all measurable functions defined on , for which
Analogously, denotes the Sobolev space with corresponding mixed norm.
Let , , denote the roots of the equations
Let denote the operator generated by BVP (6.1). Let
Theorem 6.1.
Let the following conditions be satisfied:
(1) for each and for each with , and , ,
(2) for each , , , , ,
(3)for , , , , let
(4)for each , the local BVPs in local coordinates corresponding to
has a unique solution for all and for with ,
(5), and
Then,
(a)the following coercive estimate
holds for the solution of problem (6.1),
(b)for and for sufficiently large , there exists a resolvent and
(c)the problem (6.1) for is Fredholm in ,
(d)the relation with
holds,
(e)for the system of root functions of the BVP (6.1) is complete in .
Proof.
Let . Then, from [3, Theorem 3.6], part (1.1) of Condition 1 is satisfied. Consider the operator which is defined by
For , we also consider operators
The problem (6.1) can be rewritten as the form of (3.1)(3.2), where and are functions with values in . From [3, Theorem 8.2] problem
has a unique solution for and arg , . Moreover, the operator , generated by (5.8) is positive in ; that is, part (2.2) of Condition 1 holds. From (2.2), (3.7), and by [29, Section 18], we have
that is, all the conditions of Theorem 5.2 and Result 4 are fulfilled. As a result, we obtain assertion (a) and (b) of the theorem. Also, it is known (e.g., [27, Theorem 3.2.5, Section 4.10]) that the embedding is compact and
Then, Results 3 and 4 imply assertions (c), (d), (e).
7. Boundary Value Problems for Infinite Systems of Degenerate PDE
Consider the infinity systems of BVP for the degenerate anisotropic PDE
where , are complexvalued functions, , are complex numbers. Let
Let denote the operator in generated by problem (7.1). Let
Theorem 7.1.
Let , , , , , , and , , , , , , such that
Then,
(a)for all , for and sufficiently large , the problem (7.1) has a unique solution that belongs to the space and the following coercive estimate holds:
(b)there exists a resolvent of the operator and
(c)for , the system of root functions of the BVP (7.1) is complete in .
Proof.
Let , and be infinite matrices such that
It is clear that the operator is positive in . The problem (7.1) can be rewritten in the form (1.1). From Theorem 4.4, we obtain that problem (7.1) has a unique solution for all and
From the above estimate, we obtain assertions (a) and (b). The assertion (c) is obtained from Result 4.
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Keywords
 Banach Space
 Unique Solution
 Linear Operator
 Bounded Linear Operator
 Weighted Space