Degenerate Anisotropic Differential Operators and Applications

The boundary value problems for degenerate anisotropic differential operator equations with variable coefficients are studied. Several conditions for the separability and Fredholmness in Banach-valued 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.

It is well known that many classes of PDEs, pseudo-Des, and integro-DEs can be expressed as differential-operator 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 higher-order 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 higher-order 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 nonself-adjoint. 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 Banach-valued 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:

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,

Consider the BVPs for the degenerate anisotropic DOE

where

 , , are complex numbers, are complex-valued 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

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:

(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 real-valued 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 matrix-function with continuous components .

Then, we obtain the separability of the following BVPs for the system of anisotropic PDEs with varying coefficients:

in the vector-valued space .

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 .

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 scalar-valued 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).

Consider the infinity systems of BVP for the degenerate anisotropic PDE

where , are complex-valued 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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