Yakov Yakubov

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal Lp-regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces Wp,q2,2.

Perturbation Results for Multivalued Linear Operators

We give some perturbation theorems for multivalued linear operators in a Banach space. Two different approaches are suggested: the resolvent approach and the modified resolvent approach. The results allow us to handle degenerate abstract Cauchy problems (inclusions). A very wide application of obtained abstract results to initial boundary value problems for degenerate parabolic (elliptic-parabolic) equations with lower-order terms is studied. In particular, integro-differential equations have been considered too.

Application of abstract differential equations to some mechanical problems

- Preface. - Introduction. 1. Laplace problem in a strip. 2. Laplace problem in a sector. 3. Presentation of the different chapters. - 1: General notions, definitions and results. 1. Introduction. 2. General notions from functional analysis. 3. Vector-value functions of Banach spaces. 4. Semigroup of linear bounded operators in a Banach space. 5. Differential-operator equations and fold completeness. 6. Isomorphism and coerciveness. 7. Interpolation of spaces. 8. Useful theorems. - 2: Thermal conduction in a half-strip and a sector. 1. Asymptotic expansion for the thermal conduction in a plate. 2. Completeness of a system of root functions for the thermal conduction in a half-strip and a sector with smooth coefficients. 3. Completeness of a system of root functions for the thermal conduction in a half-strip with piecewise smooth coefficients. - 3: Elasticity problems in a half-strip. 1. Asymptotic expansion for the elasticity in a plate. 2. Completeness of a system of root functions for elasticity problems in a half-strip. 3. Thermoelasticity systems in bounded domains with non-smooth boundaries. - 4: Completeness of elementary solutions of problems for second and fourth orders elliptic equations in semi-infinite tube domains. 1. Abstract results for second order elliptic equations. 2. Boundary value problems for second order elliptic equations. 3. Boundary value problems for fourth order elliptic equations. - 5: Basis property of elementary solutions for second order elliptic equations with a selfadjoint operator coefficient. 1. Abstract results for second order elliptic equations with a selfadjoint operator coefficient. 2. Boundary value problems for second order elliptic equations. - Problems. References. List of notations. - Subject index. Author index.

Hyperbolic differential-operator equations on a whole axis

We give an abstract interpretation of initial boundary value problems for hyperbolic equations such that a part of initial boundary value conditions contains also a differentiation on the time t of the same order as equations. The case of stable solutions of abstract hyperbolic equations is treated. Then we show applications of obtained abstract results to hyperbolic differential equations which, in particular, may represent the longitudinal displacements of an inhomogeneous rod under the action of forces at the two ends which are proportional to the acceleration.

Elliptic Differential-Operator Problems with the Spectral Parameter in Both the Equation and Boundary Conditions and the Corresponding Abstract Parabolic Initial Boundary Value Problems

We consider, in UMD Banach spaces, boundary value problems for second order elliptic differential-operator equations with the spectral parameter and boundary conditions containing the parameter in the same order as the equation. An isomorphism and the corresponding estimate of the solution (with respect to the space variable and the parameter) are obtained. Then, an application of the obtained abstract results is given to boundary value problems for second order elliptic differential equations with the parameter in non-smooth domains. Further, the corresponding abstract parabolic initial boundary value problem is treated and an application to initial boundary value problems with time differentiation in boundary conditions is demonstrated.

Completeness of elementary solutions of problems for second and fourth orders elliptic equations in semi-infinite tube domains

When nonstationary equations are solved, the question of completeness of a system of root vectors, corresponding to the whole spectrum, arises. However, in the case of stationary equations, the question applies to the completeness of a system of root vectors, corresponding to some part of the spectrum. For some general equations it is a problem. For example, for problem (2.2.11), (2.2.12) D it can be shown that the spectrum is symmetric with respect to the imaginary axis (replacing λ by iμ we get that three differential expressions of powers of μ are symmetric) but to prove the completeness of a system of root functions corresponding to the eigenvalues λ i with Reλ i < 0 is an open problem.

Basis property of elementary solutions for second order elliptic equations in semi-infinite tube domains

We consider the corresponding results of chapter 4 for second order equations in the selfadjoint case.

Elasticity problems in a half-strip

In chapter 2, we have studied the thermal conduction problem in a plate. This problem can be considered as a simple approach of the more general problem of elasticity. As a matter of fact, in chapter 2, the unknown function was a one component function. Now, in this chapter, we shall consider the elasticity problem where the unknown function is a three component function. There are similarities between problems of thermal conduction and elasticity and in order to intensify these similarities, we shall use the same notations.

General notions, definitions, and results

Below, a number of notions, terms and facts are given which will be used throughout the book. Many facts of functional analysis are already considered classical. However, we derive some of them not in the form of theorems, without reference of their authors. One can find the proofs of these facts in any textbook on functional analysis. For some special results we will refer the reader to the corresponding literature.

Thermal conduction in a half-strip and a sector

A plate is a three-dimensional body, a dimension of which is thinner than the other ones. That dimension is denoted thickness. If we want to use a finite elements method to compute the solution of a mechanical problem posed over a plate, it will cost a lot in memory. If we take, for instance, ten elements in the thickness, we have to take several thousands in the length and the mesh becomes quickly huge.