Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 268032, 27 pages doi:10.1155/2011/268032 Research Article Degenerate Anisotropic Differential Operators and Applications Ravi Agarwal, 1 Donal O’Regan, 2 and Veli Shakhmurov 3 1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA 2 Department of Mathematics, National University of Ireland, Galway, Ireland 3 Department of Electronics Engineering and Communication, Okan University, Akfirat, Tuzla 34959 Istanbul, Turkey Correspondence should be addressed to Veli Shakhmurov, veli.sahmurov@okan.edu.tr Received 2 December 2010; Accepted 18 January 2011 Academic Editor: Gary Lieberman Copyright q 2011 Ravi Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 L p 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 H-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, n  k1 a k xD l k  k ux Axux  |α:l|<1 A α xD α ux f x, 1.1 where D i k uxγ k x k ∂/∂x k  i ux, γ k are weighted functions, A and A α are linear operators in a Banach Space E. The above DOE is a generalized form of an elliptic equation. In fact, the special case l k  2m, k  1,...,n reduces 1.1 to elliptic form.