arXiv subjects

Tetiana Kasirenko

Publications and source records attributed to Tetiana Kasirenko.

6 recordsLinked to original sources

arXiv2018

Sobolev-like Hilbert spaces induced by elliptic operators

We investigate properties of function spaces induced by the inner product Sobolev spaces $H^{s}(Ω)$ over a bounded Euclidean domain $Ω$ and by an elliptic differential operator $A$ on $\overlineΩ$. The domain and the coefficients of $A$ are of the class $C^{\infty}$. These spaces consist of all distributions $u\in H^{s}(Ω)$ such that $Au\in H^λ(Ω)$ and are endowed with the corresponding graph norm, with $s,λ\in\mathbb{R}$. We prove an interpolation formula for these spaces and discuss their application to elliptic boundary-value problems.

math.AP↗

arXiv2018

Elliptic problems in the sense of Lawruk with boundary operators of higher orders in refined Sobolev scale

In a refined Sobolev scale, we investigate an elliptic boundary-value problem with additional unknown functions in boundary conditions for which the maximum of orders of boundary operators is grater than or equal to the order of the elliptic equation. This scale consists of inner product Hörmander spaces whose order of regularity is given by a real number and a function varying slowly at infinity in the sense of Karamata. We prove a theorem on the Fredholm property of a bounded operator corresponding to this problem in the refined Sobolev scale. For the generalized solutions to the problem, we establish a local a priory estimate and prove a theorem about their regularity in Hörmander spaces. We find sufficient conditions under which given generalized derivatives of the solutions are continuous.

math.AP↗

arXiv2018

Nonregular elliptic boundary-value problems and Hörmander spaces

We investigate nonregular elliptic problems with boundary conditions of higher orders. We prove that these problems are Fredholm on appropriate pairs of inner product Hörmander spaces that form a two-sided refined Sobolev scale. We also prove a theorem on the regularity of generalized solutions to the problems in these spaces.

math.AP↗

arXiv2018

Elliptic problems with boundary operators of higher orders in Hörmander-Roitberg spaces

We investigate elliptic boundary-value problems for which the maximum of the orders of the boundary operators is equal to or greater than the order of the elliptic differential equation. We prove that the operator corresponding to an arbitrary problem of this kind is bounded and Fredholm between appropriate Hilbert spaces which form certain two-sided scales and are built on the base of isotropic Hörmander spaces. The differentiation order for these spaces is given by an arbitrary real number and positive function which varies slowly at infinity in the sense of Karamata. We establish a local a priori estimate for the generalized solutions to the problem and investigate their local regularity (up to the boundary) on these scales. As an application, we find sufficient conditions under which the solutions have continuous classical derivatives of a given order.

math.AP↗

arXiv2017

Elliptic problems with boundary conditions of high orders in Hörmander spaces

In a class of inner product Hörmander spaces, we investigate a general elliptic problem for which the maximum of orders of boundary conditions is grater than or equal to the order of elliptic equation. The order of regularity for these spaces is an arbitrary radial positive function RO-varying at infinity in the sense of Avakumović. We prove that the operator of the problem under investigation is bounded and Fredholm on appropriate pairs of Hörmander spaces indicated. A theorem on isomorphism generated by this operator is proved. For generalized solutions to this problem, we establish a local a priory estimate and prove a theorem about their local regularity in Hörmander spaces. As application, we obtain new sufficient conditions under which given derivatives of the solutions are continuous.

math.AP↗

arXiv2016

Elliptic boundary-value problems in Hörmander spaces

We investigate general elliptic boundary-value problems in Hörmander inner product spaces that form the extended Sobolev scale. The latter consists of all Hilbert spaces that are interpolation spaces with respect to the Sobolev Hilbert scale. We prove that the operator corresponding to an arbitrary elliptic problem is Fredholm in appropriate couples of the Hörmander spaces and induces a collection of isomorphisms on the extended Sobolev scale. We obtain a local a priory estimate for generalized solutions to this problem and prove a theorem on their local regularity in the Hörmander spaces. We find new sufficient conditions under which generalized derivatives (of a given order) of the solutions are continuous.

math.AP↗