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A. A. Murach

Publications and source records attributed to A. A. Murach.

9 recordsLinked to original sources
arXiv2024

Elliptic operators and boundary problems in spaces of generalized smoothness

The paper contains a survey of the results obtained during the last ten years in the theory of elliptic boundary problems in Hörmander function spaces, developed by the authors, and other related results of modern analysis. The basics of this theory and some of its applications are systematically expounded in the monograph "Hörmander Spaces, Interpolation, and Elliptic Problems" (De Gruyter, Berlin/Boston, 2014) by the first two authors of the survey.

math.AP↗
arXiv2021

Parabolic boundary-value problems and generalized Sobolev spaces

The research monograph expounds the foundation of a new theory of parabolic initial-boundary-value problems in scales of generalized anisotropic Sobolev spaces. These scales are calibrated essentially more finely with the help of a function parameter of regularity than the classes of spaces used earlier in the theory of parabolic problems. The monograph is featured by the systematic application of the method of quadratic interpolation (with function parameter) between abstract and Sobolev inner product spaces. The monograph is intended for the researches, professors, PhD students, and senior students specializing in differential equations and mathematical analysis.

math.AP↗
arXiv2021

Elliptic problems with rough boundary data in Nikolskii spaces

We investigate a general elliptic problem given in a bounded Euclidean domain with boundary data in Nikolskii spaces of low, specifically, negative order. The right-hand side of the elliptic differential equation is supposed to be an integrable function. We establish the Fredholm property of the problem, the maximal regularity and a priori estimate of its generalized solutions in the spaces indicated. We give an application of these results to some elliptic problems with boundary conditions induced by a Gaussian white noise.

math.AP↗
arXiv2021

Approximation properties of solutions to multipoint boundary-value problems

We consider a wide class of linear boundary-value problems for systems of $m$ ordinary differential equations of order $r$, known as general boundary-value problems. Their solutions $y:[a,b]\to \mathbb{C}^{m}$ belong to the Sobolev space $(W_1^{r})^m$, and the boundary conditions are given in the form $By=q$ where $B:(C^{(r-1)})^{m}\to\mathbb{C}^{rm}$ is an arbitrary continuous linear operator. We prove that a solution to such a problem can be approximated with an arbitrary precision in $(W_1^{r})^m$ by solutions to multipoint boundary-value problems with the same right-hand sides. These multipoint problems are built explicitly and do not depend on the right-hand sides of the general boundary-value problem. For these problems, we obtain estimates of errors of solutions in the normed spaces $(W_1^{r})^m$ and $(C^{(r-1)})^{m}$.

math.CA↗
arXiv2020

Elliptic problems with nonclassical boundary conditions in an extended Sobolev scale

We consider elliptic problems with nonclassical boundary conditions that contain additional unknown functions on the border of the domain of the elliptic equation and also contain boundary operators of higher orders with respect to the order of this equation. We investigate the solvability of the indicated problems and properties of their solutions in an extended Sobolev scale. It consists of Hilbert generalized Sobolev spaces for which the order of regularity is a general radial function $\mathrm{RO}$-varying in the sense of Avakumović at infinity. We establish a theorem on the Fredholm property of the indicated problems on appropriate pairs of these spaces and theorems on the regularity and a priori estimate of the generalized solutions to the problems. We obtain exact sufficient conditions for components of these solutions to be continuously differentiable.

math.AP↗
arXiv2018

Hörmander spaces on manifolds, and their application to elliptic boundary-value problems

We introduce an extended Sobolev scale on a smooth compact manifold with boundary. The scale is formed by inner-product Hörmander spaces for which an RO-varying radial function serves as a regularity index. These spaces do not depend on a choice of local charts on the manifold. The scale consists of all Hilbert spaces that are interpolation ones for pairs of inner-product Sobolev spaces, is obtained by the interpolation with a function parameter of these pairs, and is closed with respect to this interpolation. As an application of the scale introduced, we give a theorem on the Fredholm property of a general elliptic boundary-value problem on appropriate Hörmander spaces and find sufficient conditions under which its generalized solutions belong to the space of $p\geq0$ times continuously differential functions.

math.FA↗
arXiv2016

General parabolic initial-boundary value problems in Hörmander spaces

We investigate a general nonhomogeneous parabolic initial-boundary value problem in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions to the problem.

math.AP↗
arXiv2011

Hörmander Spaces, Interpolation, and Elliptic Problems

The research monograph gives the first systematic exposition of the elliptic (scalar and matrix) operators theory and elliptic boundary-value problems in the scales of Hilbert spaces of Hörmander of the functions/distributions of arbitrary positive or negative smoothness. The book is based on the method of interpolation with a functional parameter for the abstract and Sobolev unitary spaces. Some results are also new for the Sobolev scales. The monograph is intended for the researches, professors and PhD students.

math.FA↗