arXiv subjects

Aleksandr A. Murach

Publications and source records attributed to Aleksandr A. Murach.

15 recordsLinked to original sources

arXiv2016

Continuity in a parameter of solutions to generic boundary-value problems

We introduce the most general class of linear boundary-value problems for systems of first-order ordinary differential equations whose solutions belong to the complex Hölder space $C^{n+1,α}$, with $0\leq n\in\mathbb{Z}$ and $0\leqα\leq1$. The boundary conditions can contain derivatives $y^{(r)}$, with $1\leq r\leq n+1$, of the solution $y$ to the system. For parameter-dependent problems from this class, we obtain constructive criterion under which their solutions are continuous in the normed space $C^{n+1,α}$ with respect to the parameter.

math.CA↗

arXiv2015

An isomorphism theorem for parabolic problems in Hörmander spaces and its applications

We investigate a general parabolic initial-boundary value problem with zero Cauchy data 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. We also obtain new sufficient conditions under which the generalized derivatives, of a given order, of the solutions should be continuous.

math.AP↗

arXiv2015

Elliptic problems in the sense of B. Lawruk on two-sided refined scales of spaces

We investigate elliptic boundary-value problems with additional unknown functions on the boundary of a Euclidean domain. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on two-sided refined scales built on the base of the isotropic Hörmander inner product spaces. The regularity of the distributions forming these spaces are characterized by a real number and an arbitrary function that varies slowly at infinity in the sense of Karamata. For the generalized solutions to the problem, we prove theorems on a priori estimates and local regularity in these scales. As applications, we find new sufficient conditions under which the solutions have continuous classical derivatives of a prescribed order.

math.AP↗

arXiv2015

Elliptic boundary-value problems in the sense of Lawruk on Sobolev and Hörmander spaces

We investigate elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on appropriate couples of the inner product isotropic Hörmander spaces $H^{s,φ}$, which form the refined Sobolev scale. The order of differentiation for these spaces is given by the real number $s$ and positive function $φ$ that varies slowly at infinity in the sense of Karamata. We consider this problem for an arbitrary elliptic equation $Au=f$ on a bounded Euclidean domain $Ω$ under the condition that $u\in H^{s,φ}(Ω)$, $s<\mathrm{ord}\,A$, and $f\in L_{2}(Ω)$. We prove theorems on the a priori estimate and regularity of the generalized solutions to this problem.

math.AP↗

arXiv2014

Interpolation Hilbert spaces between Sobolev spaces

We explicitly describe all Hilbert function spaces that are interpolation spaces with respect to a given couple of Sobolev inner product spaces considered over $\mathbb{R}^{n}$ or a half-space in $\mathbb{R}^{n}$ or a bounded Euclidean domain with Lipschitz boundary. We prove that these interpolation spaces form a subclass of isotropic Hörmander spaces. They are parametrized with a radial function parameter which is OR-varying at $+\infty$ and satisfies some additional conditions. We give explicit examples of intermediate but not interpolation spaces.

math.FA↗

arXiv2014

Parameter-elliptic problems and interpolation with a function parameter

Parameter-elliptic boundary-value problems are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. The latter are the Hörmander spaces $B_{2,k}$ for which the smoothness index $k$ is an arbitrary radial function RO-varying at infinity. We prove that the operator corresponding to this problem sets isomorphisms between appropriate Hörmander spaces provided that the absolute value of the parameter is large enough. For solutions to the problem, we establish two-sided estimates, in which the constants are independent of the parameter.

math.AP↗

arXiv2013

Regular elliptic boundary-value problems in the extended Sobolev scale

We investigate an arbitrary regular elliptic boundary-value problem given in a bounded Euclidean domain with infinitely smooth boundary. We prove that the operator of the problem is bounded and Fredholm in appropriate pairs of Hörmander inner product spaces. They are parametrized with the help of an arbitrary radial function RO-varying at infinity and form the extended Sobolev scale. We establish a priori estimates for solutions to the problem and investigate their local regularity on this scale. We find new sufficient conditions for generalized partial derivatives of the solutions to be continuous.

math.AP↗

arXiv2013

Parabolic problems and interpolation with a function parameter

We give an application of interpolation with a function parameter to parabolic differential operators. We introduce the refined anisotropic Sobolev scale that consists of some Hilbert function spaces of generalized smoothness. The latter is characterized by a real number and a function varying slowly at infinity in Karamata's sense. This scale is connected with anisotropic Sobolev spaces by means of interpolation with a function parameter. We investigate a general initial--boundary value parabolic problem in the refined Sobolev scale. We prove that the operator corresponding to this problem sets isomorphisms between appropriate spaces pertaining to this scale.

math.AP↗

arXiv2012

Parameter-elliptic operators on the extended Sobolev scale

Parameter--elliptic pseudodifferential operators given on a closed smooth manifold are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. We prove that these operators set isomorphisms between appropriate spaces of the scale provided the parameter is modulo large enough. For solutions to the corresponding parameter--elliptic equations, we establish two-sided a priori estimates, in which the constants are independent of the parameter.

math.AP↗

arXiv2012

The Refined Sobolev Scale, Interpolation, and Elliptic Problems

The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic Hörmander spaces $H^{s,φ}:=B_{2,μ}$, with $μ(ξ)=<ξ>^{s}φ(<ξ>)$ for $ξ\in\mathbb{R}^{n}$. They are parametrized by both the real number $s$ and the positive function $φ$ varying slowly at $+\infty$ in the Karamata sense. These spaces form the refined Sobolev scale, which is much finer than the Sobolev scale ${H^{s}}\equiv{H^{s,1}}$ and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this new scale. Theorems of various type about a solvability of elliptic problems are given. A local refined smoothness is investigated for solutions to elliptic equations. New sufficient conditions for the solutions to have continuous derivatives are found. Some applications to the spectral theory of elliptic operators are given.

math.FA↗

arXiv2012

Douglis--Nirenberg elliptic systems in Hörmander spaces

We investigate Douglis--Nirenberg uniformly elliptic systems in $\mathbb{R}^{n}$ on a class of Hörmander inner product spaces. They are parametrized with a radial function parameter which is RO-varying at $+\infty$, considered as a function of $(1+|ξ|^{2})^{1/2}$ with $ξ\in\mathbb{R}^{n}$. An a'priori estimate for solutions is proved, and their interior regularity is studied. A sufficient condition for the systems to have the Fredholm property is given.

math.AP↗

arXiv2011

General forms of the Menshov-Rademacher, Orlicz, and Tandori theorems on orthogonal series

We prove that the classical Menshov-Rademacher, Orlicz, and Tandori theorems remain true for orthogonal series given in the direct integrals of measurable collections of Hilbert spaces. In particular, these theorems are true for the spaces L_{2}(X,dμ;H) of vector-valued functions, where (X,μ) is an arbitrary measure space, and H is a real or complex Hilbert space of an arbitrary dimension.

math.FA↗

arXiv2009

Elliptic problems and Hörmander spaces

The paper gives a survey of the modern results on elliptic problems on the Hörmander function spaces. More precisely, elliptic problems are studied on a Hilbert scale of the isotropic Hörmander spaces parametrized by a real number and a function slowly varying at $+\infty$ in the Karamata sense. This refined scale is finer than the Sobolev scale and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this scale. A local refined smoothness of the elliptic problem solution is studied. An abstract construction of classes of function spaces in which the elliptic problem is a Fredholm one is found. In particular, some generalizations of the Lions-Magenes theorems are given.

math.AP↗

arXiv2009

Extension of some Lions-Magenes theorems

A general form of the Lions-Magenes theorems on solvability of an elliptic boundary-value problem in the spaces of nonregular distributions is proved. We find a general condition on the space of right-hand sides of the elliptic equation under which the operator of the problem is bounded and has a finite index on the corresponding couple of Hilbert spaces. Extensive classes of the spaces satisfying this condition are constructed. They contain the spaces used by Lions and Magenes.

math.AP↗

arXiv2007

Regular elliptic boundary-value problem in a two-sided refined scale of spaces

A regular elliptic boundary-value problem over a bounded domain with a smooth boundary is studied. We prove that the operator of this problem is a Fredholm one in the two-sided refined scale of the functional Hilbert spaces and generates a complete collection of isomorphisms. Elements of this scale are the isotropic spaces of Hormander-Volevich-Paneah and some its modifications. A priori estimate for the solution is established and its regularity is investigated.

math.AP↗