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Vladimir A. Mikhailets

Publications and source records attributed to Vladimir A. Mikhailets.

9 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↗
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↗
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↗
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↗
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↗
arXiv2007

Elliptic systems of pseudodifferential equations in a refined scale on a closed manifold

We study a system of pseudodifferential equations that is elliptic in the sense of Petrovskii on a closed compact smooth manifold. We prove that the operator generated by the system is Fredholm one on a refined two-sided scale of the functional Hilbert spaces. Elements of this scale are the special isotropic spaces of Hörmander--Volevich--Paneah.

math.AP↗