Alexander Kozhevnikov

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.

Parameter-ellipticity for mixed-order elliptic boundary problems

Abstract The aim of this Note is to formulate a parameter-ellipticity condition for the mixed order (Douglis-Nirenberg) boundary-value problems and to prove a theorem on the minimal growth of the resolvent.

Power series solutions to basic stationary boundary value problems of elasticity

The four basic stationary boundary value problems of elasticity for the Lamé equation in a bounded domain of ℝ3 are under consideration. Their solutions are represented in the form of a power series with non-positive degrees of the parameter ω=1/(1–2σ), depending on the Poisson ratio σ. The “coefficients” of the series are solutions of the stationary linearized non-homogeneous Stokes boundary value problems. It is proved that the series converges for any values of ω outside of the minimal interval with the center at the origin and of radiusr≥1, which contains all of the Cosserat eigenvalues.

A history of the Cosserat spectrum

The paper is a brief survey of the 100-year history of the Cosserat spectrum in elastostatics which was first studied by Eugene and Francois Cosserat between 1898 and 1901 and later by Vladimir Maz’ya and Solomon Mikhlin.

On explicit solvability of an elliptic boundary value problem and its application

A homogeneous boundary condition is constructed for the equation (I − Δ)u = f in an arbitrary bounded or exterior domain Ω ⊆  (I and Δ being the identity operator and the Laplacian), which generates a boundary value problem with an explicit formula of the solution u. The problem creates an isomorphism between the appropriate Sobolev spaces with an explicitly written inverse operator. In the article, all results are obtained not just for the operator I − Δ but also for an arbitrary elliptic differential operator in of an even order with constant coefficients. As an application, the usual Dirichlet boundary value problem for the homogeneous equation (I − Δ)u = 0 in a bounded or exterior domain is reduced to an integral equation in a thin boundary layer. An approximate solution of the integral equation generates a rather simple new numerical algorithm solving the 2D and 3D Dirichlet problem.A homogeneous boundary condition is constructed for the equation (I − Δ)u = f in an arbitrary bounded or exterior domain Ω ⊆  (I and Δ being the identity operator and the Laplacian), which generates a boundary value problem with an explicit formula of the solution u. The problem creates an isomorphism between the appropriate Sobolev spaces with an explicitly written inverse operator. In the article, all results are obtained not just for the operator I − Δ but also for an arbitrary elliptic differential operator in of an even order with constant coefficients. As an application, the usual Dirichlet boundary value problem for the homogeneous equation (I − Δ)u = 0 in a bounded or exterior domain is reduced to an integral equation in a thin boundary layer. An approximate solution of the integral equation generates a rather simple new numerical algorithm solving the 2D and 3D Dirichlet problem.

On isomorphism of ordinary differential operators

The article deals with an ordinary linear differential operator L of even order 2m with constant coefficients which defines a natural mapping of the space to itself. The operator L is considered under an extra condition that its characteristic polynomial has no real roots and exactly m roots with strictly positive imaginary part. This work prepares the treatment of properly elliptic partial differential operators in bounded domains of It turns out that there exists an extension operator to L which is an isomorphism between the standard Sobolev spaces Hm ( a , b ) and and such that the inverse operator can be represented in an extremely simple form using the Fourier transform.The article deals with an ordinary linear differential operator L of even order 2m with constant coefficients which defines a natural mapping of the space to itself. The operator L is considered under an extra condition that its characteristic polynomial has no real roots and exactly m roots with strictly positive imaginary part. This work prepares the treatment of properly elliptic partial differential operators in bounded domains of It turns out that there exists an extension operator to L which is an isomorphism between the standard Sobolev spaces Hm ( a , b ) and and such that the inverse operator can be represented in an extremely simple form using the Fourier transform.

Analyticity of the semigroup generated by the operator of a viscous compressible fluid

The linearized initial-boundary value problem describing small motions of the viscous, barotropic compressible fluid in a bounded vessel is studied under various boundary conditions (Dirichlet, Neumann and intermediate). It is shown that the corresponding operator generates an analytic semigroup in the space L 2 .

On explicit and numerical solvability of parabolic initial-boundary value problems

A homogeneous boundary condition is constructed for the parabolic equation in an arbitrary cylindrical domain ( being a bounded domain, and being the identity operator and the Laplacian) which generates an initial-boundary value problem with an explicit formula of the solution. In the paper, the result is obtained not just for the operator, but also for an arbitrary parabolic differential operator, where is an elliptic operator in of an even order with constant coefficients. As an application, the usual Cauchy-Dirichlet boundary value problem for the homogeneous equation in is reduced to an integral equation in a thin lateral boundary layer. An approximate solution to the integral equation generates a rather simple numerical algorithm called boundary layer element method which solves the 3D Cauchy-Dirichlet problem (with three spatial variables).

On a Lower Bound of the Cosserat Spectrum for the Second Boundary Value Problem of Elastostatics

The boundary value problem with given stresses on the boundary for the Navier (Lamé) equation is under consideration. The Cosserat eigenvalues are those values of a spectral parameter related to the Poisson ratio σ which admit non-trivial solution to the homogeneous boundary value problem. It is known that all finite-multiple Cosserat eigenvalues belong to the ray (-∞, 1/3]. The aim of the paper is to prove that for any convex domain the Cosserat eigenvalues are contained in the interval(-1,1/3]