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.
- Agranovich MS:Spectral problems in Lipschitz domains for strongly elliptic systems in the Banach spaces and . Functional Analysis and Its Applications 2008, 42(4):249-267. 10.1007/s10688-008-0039-xView ArticleMathSciNetGoogle Scholar
- Ashyralyev A: On well-posedness of the nonlocal boundary value problems for elliptic equations. Numerical Functional Analysis and Optimization 2003, 24(1-2):1-15. 10.1081/NFA-120020240View ArticleMathSciNetGoogle Scholar
- Denk R, Hieber M, Prüss J: R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Mem. Amer. Math. Soc.. American Mathematical Society; 2003.Google Scholar
- Favini A, Shakhmurov V, Yakubov Y: Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. Semigroup Forum 2009, 79(1):22-54. 10.1007/s00233-009-9138-0View ArticleMathSciNetGoogle Scholar
- Gorbachuk VI, Gorbachuk ML: Boundary Value Problems for Differential-Operator Equations. "Naukova Dumka", Kiev, Ukraine; 1984:284.Google Scholar
- Goldstein JA: Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York, NY, USA; 1985:x+245.Google Scholar
- Lions J-L, Magenes E: Nonhomogenous Boundary Value Problems. Mir, Moscow, Russia; 1971.Google Scholar
- Shklyar AYa: Complete Second Order Linear Differential Equations in Hilbert Spaces, Operator Theory: Advances and Applications. Volume 92. Birkhäuser, Basel, Switzerland; 1997:xii+219.Google Scholar
- Shakhmurov VB: Nonlinear abstract boundary-value problems in vector-valued function spaces and applications. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(3):745-762. 10.1016/j.na.2006.06.027View ArticleMathSciNetGoogle Scholar
- Shakhmurov VB: Theorems on the embedding of abstract function spaces and their applications. Matematicheskiĭ Sbornik. Novaya Seriya 1987, 134(176)(2):260-273.Google Scholar
- Shakhmurov VB: Embedding theorems and their applications to degenerate equations. Differentsial'nye Uravneniya 1988, 24(4):672-682.MathSciNetGoogle Scholar
- Yakubov S: Completeness of Root Functions of Regular Differential Operators, Pitman Monographs and Surveys in Pure and Applied Mathematics. Volume 71. Longman Scientific & Technical, Harlow, UK; 1994:x+245.Google Scholar
- Yakubov S: A nonlocal boundary value problem for elliptic differential-operator equations and applications. Integral Equations and Operator Theory 1999, 35(4):485-506. 10.1007/BF01228044View ArticleMathSciNetGoogle Scholar
- Yakubov S, Yakubov Y: Differential-Operator Equations: Ordinary and Partial Differential Equations, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Volume 103. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2000:xxvi+541.Google Scholar
- Agarwal RP, Bohner R, Shakhmurov VB: Maximal regular boundary value problems in Banach-valued weighted spaces. Boundary Value Problems 2005, 2005(1):9-42. 10.1155/BVP.2005.9View ArticleMathSciNetGoogle Scholar
- Shakhmurov VB: Separable anisotropic differential operators and applications. Journal of Mathematical Analysis and Applications 2007, 327(2):1182-1201. 10.1016/j.jmaa.2006.05.007View ArticleMathSciNetGoogle Scholar
- Amann H: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Mathematische Nachrichten 1997, 186: 5-56.View ArticleMathSciNetGoogle Scholar
- Amann H: Linear and Quasi-Linear Equations. Birkhauser; 1995.Google Scholar
- Krein SG: Linear Differential Equations in Banach Space. American Mathematical Society, Providence, RI, USA; 1971.Google Scholar
- Lunardi A: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser, Basel, Switzerland; 1995:xviii+424.Google Scholar
- Orlov VP: Regular degenerate differential operators of arbitrary order with unbounded operators coefficients. Proceedings of Voronej State University 1974, 2: 33-41.Google Scholar
- Sobolevskiĭ PE: Inequalities coerciveness for abstract parabolic equations. Doklady Akademii Nauk SSSR 1964, 57: 27-40.Google Scholar
- Shakhmurov VB: Coercive boundary value problems for regular degenerate differential-operator equations. Journal of Mathematical Analysis and Applications 2004, 292(2):605-620. 10.1016/j.jmaa.2003.12.032View ArticleMathSciNetGoogle Scholar
- Shakhmurov VB: Degenerate differential operators with parameters. Abstract and Applied Analysis 2007, 2007:-27.Google Scholar
- Weis Lutz:Operator-valued Fourier multiplier theorems and maximal -regularity. Mathematische Annalen 2001, 319(4):735-758. 10.1007/PL00004457View ArticleMathSciNetGoogle Scholar
- Burkholder DL: A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. Proceedings of the Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II, 1981, Chicago, Ill, USA, Wadsworth Math. Ser. 270-286.Google Scholar
- Triebel H: Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library. Volume 18. North-Holland, Amsterdam, The Netherlands; 1978:528.Google Scholar
- Grisvard P: Elliptic Problems in Non Smooth Domains. Pitman; 1985.Google Scholar
- Besov OV, Ilin VP, Nikolskii SM: Integral Representations of Functions and Embedding Theorems. Wiley, New York, NY, USA; 1978.Google Scholar
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