# Degenerate Anisotropic Differential Operators and Applications

- Ravi Agarwal
^{1}, - Donal O'Regan
^{2}and - Veli Shakhmurov
^{3}Email author

**Received: **2 December 2010

**Accepted: **18 January 2011

**Published: **23 February 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 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.

## 1. Introduction and Notations

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].

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].

For , the space will be denoted by .

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 and be two Banach spaces. Now, , , will denote interpolation spaces obtained from by the method [27, Section 1.3.1].

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 .

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 .

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 , .

Let denote the space of all compact operators from to . For , it is denoted by .

Let denote the space of all compact operators from to . For , it is denoted by .

Let and be two Banach spaces and continuously and densely embedded into and .

## 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 ,

(4) is a region such that there exists a bounded linear extension operator from to .

Theorem A2.

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 ,

Then, the subspace contains the space .

Let denote the embedding operator .

From [15, Theorem 2.8], we have the following.

Theorem A4.

## 3. Statement of the Problem

A function and satisfying (3.1) a.e. on is said to be solution of the problem (3.1)-(3.2).

holds.

We say the problem (3.1)-(3.2) is Fredholm in if , where is a conjugate of .

Remark 3.1.

## 4. BVPs for Partial DOE

By applying the trace theorem [27, Section 1.8.2], we have the following.

Theorem A5.

Proof.

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 ;

Let denote the operator in generated by BVP (4.1). In [15, Theorem 5.1] the following result is proved.

Theorem A6.

- (a)

(b)the operator is -positive in .

From Theorems A5 and A6 we have.

Theorem A7.

Condition 2.

Suppose the following conditions are satisfied:

(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 .

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 , ,

for the solution of problem (4.13).

Proof.

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 , ,

for the solution of problem (3.11).

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 ,

Proof.

Result 1.

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,

(b)if , then the operator is Fredholm from into .

Example 4.5.

Example 4.6.

Let and , where are positive continuous function on , and is a diagonal matrix-function with continuous components .

## 5. The Spectral Properties of Anisotropic Differential Operators

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.

Then,

(b)the system of root functions of the differential operator is complete in .

Proof.

Then, from (4.52) and (5.6), we obtain assertion (b).

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,

(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.

Analogously, denotes the Sobolev space with corresponding mixed norm.

Theorem 6.1.

Let the following conditions be satisfied:

(1) for each and for each with , and , ,

has a unique solution for all and for with ,

Then,

holds for the solution of problem (6.1),

(c)the problem (6.1) for is Fredholm in ,

holds,

(e)for the system of root functions of the BVP (6.1) is complete in .

Proof.

Then, Results 3 and 4 imply assertions (c), (d), (e).

## 7. Boundary Value Problems for Infinite Systems of Degenerate PDE

Theorem 7.1.

Then,

(c)for , the system of root functions of the BVP (7.1) is complete in .

Proof.

From the above estimate, we obtain assertions (a) and (b). The assertion (c) is obtained from Result 4.

## Authors’ Affiliations

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