Sasun Yakubov

Problems for Ordinary Differential Equations with Transmission Conditions

We investigate a boundary-functional problem with transmission conditions for ordinary differential-operator equation in Sobolev spaces with a weight. We prove an isomorphism, coerciveness with respect to the spectral parameter, completeness and Abel basis of a system of root functions of the problem. Obtained results in the article are new even in case of Sobolev spaces without the weight.

A nonlocal boundary value problem for elliptic differential-operator equations and applications

In this paper we give, for the first time, an abstract interpretation of nonlocal boundary value problems for elliptic differential equations of the second order. We prove coerciveness and Fredholmness of nonlocal boundary value problems for the second order elliptic differential-operator equations. We apply then, in section 6, these results for investigation of nonlocal boundary value problems for the second order elliptic differential equations (one can find the references on the subject in the introduction and Chapter V in the book by A. L. Skubachevskii [27]). Abstract results obtained in this paper can be used for study of nonlocal boundary value problems for quasielliptic differential equations.

Degenerate elliptic boundary value problems

In this paper we find conditions that guarantee that regular boundary value problems for elliptic differential-operator equations of the second order in an interval are coercive and Fredholm, and we prove the compactness of a resolvent. We apply this result to find some algebraic conditions that guarantee that regular boundary value problems for degenerate elliptic differential equations of the second order in cylindrical domains have the same properties. Note that considered boundary value conditions are nonlocal and are differential only in their principal part, and a domain is nonsmooth.

On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions

General elliptic boundary value problems with the spectral parameter appearing linearly both in the elliptic equation and in boundary conditions are considered. It is proved that the corresponding matrix operator from the Boutet de Monvel algebra is similar to an almost diagonal operator. This result is applied to prove the completeness and the summability (in the sense of Abel) of the root vectors of this operator.

Problems for elliptic equations with operator-boundary conditions

We consider, in a Hilbert space H, a problem on [0,1] for a second order elliptic operator-differential equation with operator-boundary conditions. We also consider second order elliptic differential equations with operatorboundary conditions in cylindrical domains in the case when operator-boundary conditions contain integral terms over the whole domain. In this case, the proof of the density of the domain of definition of operators in a space is difficult. When boundary conditions are local, this fact is a simple corollary of the density ofC0∞ (Ω) inLp(Ω).

Fredholm property of elliptic boundary value problems for partial differential and differential- operator equations

In this paper we find conditions on boundary value problems for elliptic differential-operator equations of the 4-th order in an interval to be fredholm. Apparently, this is the first publication for elliptic differential-operator equations of the 4-th order, when the principal part of the equation has the form u′‴n(t) + Au″(t) + Bu(t), where AB-1/2 is a bounded operator and is not compact. As an application we find some algebraic conditions on boundary value problems for elliptic partial equations of the 4-th order in cylindrical domains to be fredholm. Note that a new method has actually been suggested here for investigation of boundary value problems for elliptic partial equations of the 4-th order.

A Boundary Value Problem for Elliptic Differential-Operator Equations

In this paper we give, for the first time, an abstract interpretation of such boundary value problems for elliptic equations of the second order that a part of boundary value conditions contains also a differentiation of the second order. Boundary value problems for elliptic equations are reduced to the boundary value problem for a system of differential- operator equations (see below problem (1)− (3)). A solution of this system is not a vector- function but one function. At the same time, the system is not overdetermined. Boundary value problems for elliptic equations (see section 3), which we consider in this paper, do not satisfy the Lopatinskii condition (so, the problems are irregular) since the tangent derivative is given on the domain boundary.