Spectrality of ordinary differential operators

We prove the long standing conjecture in the theory of two-point boundary value problems that completeness and Dunford's spectrality imply Birkhoff regularity. In addition we establish the even order part of S.G.Krein's conjecture that dissipative differential operators are Birkhoff-regular and give sharp estimate of the norms of spectral projectors in the odd case. Considerations are based on a new direct method, exploiting \textit{almost orthogonality} of Birkhoff's solutions of the equation $l(y)=\lambda y$, which was discovered earlier by the author.Comment: AmsLaTeX, 26 pages, added section on dissipative operators and reference

Maximal regular boundary value problems in Banach-valued weighted space

This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banach-valued weighted Lp spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties

Fredholmness of an abstract differential equation of elliptic type

In this work, we obtain algebraic conditions which assure the Fredholm solvability of an abstract differential equation of elliptic type. In this respect, our work can be considered as an extension of Yakubov's results to the case of boundary conditions containing a linear operator. Although essential technical, this extension is not straight forward as we show it below. The obtained abstract result is applied to a non regular boundary value problem for a second order partial differential equation of an elliptic type in a cylindrical domain. It is interesting to note that the problems considered in cylindrical domains are not coercive

Singular perturbations for abstract elliptic operators and applications

Dirichlet problem for parameter depended elliptic differential-operator equation with variable coefficients in smooth domains is studied. The uniform maximal regularity, Fredholmness and the positivity of this problem in vector-valued Lp-spaces are obtained. It is proven that the corresponding differential operator is positive and is a generator of an analytic semigroup. In application, the maximal regularity properties of Cauchy problem for abstract parabolic equation and anisotropic elliptic equations with small parameters are established.Publisher's Versio

Anisotropic differential operators with parameters and applications

In the paper, we study the boundary-value problems for parameter-dependent anisotropic differential-operator equations with variable coefficients. Several conditions for the uniform separability and Fredholmness in Banach-valued L p -spaces are given. Sharp uniform estimates for the resolvent are established. It follows from these estimates that the indicated operator is uniformly positive. Moreover, it is also the generator of an analytic semigroup. As an application, the maximal regularity properties of the parameter-dependent abstract parabolic problem and infinite systems of parabolic equations are established in mixed L p -spaces.Вивчаються граничні задачi для анізотропних диференціально-операторних рівнянь зі змінними коефiцiєнтами, що залежать від параметрів. Наведено кілька умов рівномірної сепарабельності та фредгольмовості в банаховозначних L p -просторах. Встановлено точні рівномірні оцінки для резольвенти, з яких випливає, що вказаний оператор є рівномірно додатним. Більш того, він є також генератором деякої аналітичної напівгрупи. Як застосування, встановлено властивості максимальної регулярності абстрактної параболічної задачі, що залежить від параметра, та нескінченних систем рівнянь параболічного типу в L p -просторах

Eigenvalues and completeness for regular and simply irregular two-point differential operators

August 29, 2006.Includes bibliographical references (pages 305-307) and index.In this monograph the author develops the spectral theory for an nth order two-point differential operator L in the Hilbert space L2[0,1], where L is determined by an nth order formal differential operator ℓ having variable coefficients and by n linearly independent boundary values B1,…,Bn. Using the Birkhoff approximate solutions of the differential equation (ρnI−ℓ)u=0, the differential operator L is classified as belonging to one of three possible classes: regular, simply irregular, or degenerate irregular. For the regular and simply irregular classes, the author develops asymptotic expansions of solutions of the differential equation (ρnI−ℓ)u=0, constructs the characteristic determinant and Green's function, characterizes the eigenvalues and the corresponding algebraic multiplicities and ascents, and shows that the generalized eigenfunctions of L are complete in L2[0,1]. He also gives examples of degenerate irregular differential operators illustrating some of the unusual features of this class.1. Introduction -- 2. Birkhoff approximate solutions -- 3. The approximate characteristic determinant: classification -- 4. Asymptotic expansion of solutions -- 5. The characteristic determinant -- 6. The Green's function -- 7. The eigenvalues for n even -- 8. The eigenvalues for n odd -- 9. Completeness of the generalized eigenfunctions -- 10. The case L = T, degenerate irregular examples -- 11. Unsolved problems -- 12. Appendix