- Research Article
- Open Access
Degenerate Anisotropic Differential Operators and Applications
© Ravi Agarwal et al. 2011
Received: 2 December 2010
Accepted: 18 January 2011
Published: 23 February 2011
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].
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].
The Banach space is called a UMD space if the Hilbert operator is bounded in , (see, e.g., ). 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].
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 .
The embedding theorems play a key role in the perturbation theory of DOEs. For estimating lower order derivatives, we use following embedding theorems from .
From [18, Theorem 3.4.1], we have the following.
(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 ,
From [15, Theorem 2.8], we have the following.
3. Statement of the Problem
4. BVPs for Partial DOE
By applying the trace theorem [27, Section 1.8.2], we have the following.
Then, by applying the trace theorem [27, Section 1.8.2] to the space , we obtain the assertion.
Assume that the following conditions are satisfied:
Let denote the operator in generated by BVP (4.1). In [15, Theorem 5.1] the following result is proved.
From Theorems A5 and A6 we have.
Suppose the following conditions are satisfied:
Assume that Condition 2 is satisfied and the following hold:
for the solution of problem (4.13).
Consider the problem (3.11). Reasoning as in the proof of Lemma 4.2, we obtain.
Assume Condition 3 hold and suppose that
for the solution of problem (3.11).
Assume that Condition 3 is satisfied and that the following hold:
Assume all the conditions of Theorem 4.4 hold. Then,
5. The Spectral Properties of Anisotropic Differential Operators
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 .
Then, from (4.52) and (5.6), we obtain assertion (b).
From Theorem 5.2 and Remark 3.1, we get the following.
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  for smooth domains and in  for nonsmooth domains.
Let the following conditions be satisfied:
Then, Results 3 and 4 imply assertions (c), (d), (e).
7. Boundary Value Problems for Infinite Systems of Degenerate PDE
From the above estimate, we obtain assertions (a) and (b). The assertion (c) is obtained from Result 4.
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