# Murach A. A.

### Elliptic Boundary-Value Problems in the Sense of Lawruk on Sobolev and Hörmander Spaces

Chepurukhina I. S., Murach A. A.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 672–691

We study elliptic boundary-value problems with additional unknown functions in boundary conditions. These problems were introduced by Lawruk. We prove that the operator corresponding to a problem of this kind is bounded and Fredholm in 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 a real number $s$ and a positive function $φ$ slowly varying at infinity in Karamata’s sense. We consider this problem for an arbitrary elliptic equation $Au = f$ in a bounded Euclidean domain $Ω$ under the condition that $u ϵ H^{s,φ} (Ω),\; s < \text{ord} A$, and $f ϵ L_2 (Ω)$. We prove theorems on the a priori estimate and regularity of the generalized solutions to this problem.

### Regular Elliptic Boundary-Value Problems in the Extended Sobolev Scale

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 867–883

We investigate an arbitrary regular elliptic boundary-value problem given in a bounded Euclidean *C* ^{∞}- domain. It is shown 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 ∞ and form the extended Sobolev scale. We establish *a priori* estimates for the solutions of the problem and study their local regularity on this scale. New sufficient conditions for the generalized partial derivatives of the solutions to be continuous are obtained.

### Extended Sobolev Scale and Elliptic Operators

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 392-404

We obtain a constructive description of all Hilbert function spaces that are interpolation spaces with respect to a couple of Sobolev spaces $[H^{(s_0)}(\mathbb{R}^n), H^{(s_1)}(\mathbb{R}^n)]$ of some integer orders $s_0$ and $s_1$ and that form an extended Sobolev scale. We find equivalent definitions of these spaces with the use of uniformly elliptic pseudodifferential operators positive definite in $L_2(\mathbb{R}^n)$. Possible applications of the introduced scale of spaces are indicated.

### Douglis-Nirenberg elliptic systems in Hörmander spaces

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1477-1476

We investigate Douglis-Nirenberg uniformly elliptic systems in $\mathbb{R}^n$ on the class of Hormander Hilbert spaces $H^{\varphi}$, where $\varphi$ is an $RO$-varying function of scalar argument. An a priori estimate for solutions is proved, and their interior regularity is studied. A sufficient condition for these systems to have the Fredholm property is given.

### On the unconditional almost-everywhere convergence of general orthogonal series

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2011. - 63, № 10. - pp. 1360-1367

The Orlicz and Tandori theorems on the unconditional almost-everywhere convergence, with respect to Lebesgue measure, of real orthogonal series defined on the interval (0; 1) are extended to general complex orthogonal series defined on an arbitrary measure space.

### On elliptic systems in Hörmander spaces

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 391-399

We study a linear system of pseudodifferential equations uniformly elliptic in Petrovskii’s sense in the Hilbert scale of Hörmander functional spaces defined in $ℝ_n$. An a priori estimate is proved for the solution of the system and its interior smoothness in this scale of spaces is investigated. As an application, we establish a sufficient condition for the existence of continuous bounded derivatives of the solution.

### Elliptic boundary-value problem in a two-sided improved scale of spaces

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 497–520

We study a regular elliptic boundary-value problem in a bounded domain with smooth boundary. We prove that the operator of this problem is a Fredholm one in a two-sided improved scale of functional Hilbert spaces and that it generates there a complete collection of isomorphisms. Elements of this scale are Hörmander-Volevich-Paneyakh isotropic spaces and some their modi.cations. An *a priori* estimate for a solution is obtained and its regularity is investigated.

### Elliptic pseudodifferential operators in the improved scale of spaces on a closed manifold

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 798–814

We study linear elliptic pseudodifferential operators in the improved scale of functional Hilbert spaces on a smooth closed manifold. Elements of this scale are isotropic Hörmander-Volevich-Paneyakh spaces. We investigate the local smoothness of a solution of an elliptic equation in the improved scale. We also study elliptic pseudodifferential operators with parameter.

### Improved scales of spaces and elliptic boundary-value problems. III

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 679–701

We study elliptic boundary-value problems in improved scales of functional Hilbert spaces on smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The local smoothness of a solution of an elliptic problem in an improved scale is investigated. We establish a sufficient condition under which this solution is classical. Elliptic boundary-value problems with parameter are also studied.

### Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1536–1555

We study a regular elliptic boundary-value problem for a homogeneous differential equation in a bounded domain. We prove that the operator of this problem is a Fredholm (Noether) operator in a two-sided improved scale of functional Hilbert spaces. The elements of this scale are Hörmander-Volevich-Paneyakh isotropic spaces. We establish an *a priori* estimate for a solution and investigate its regularity.

### Improved scales of spaces and elliptic boundary-value problems. II

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 352–370

We study improved scales of functional Hilbert spaces over $\mathbb{R}^n$ and smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The theory of elliptic boundary-value problems in these spaces is developed.

### Improved scales of spaces and elliptic boundary-value problems. I

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 217–235

We study improved scales of functional Hilbert spaces over *R ^{n}* and smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The theory of elliptic boundary-value problems in these spaces is developed.

### Elliptic Operators in a Refined Scale of Functional Spaces

Mikhailets V. A., Murach A. A.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 689–696

We study the theory of elliptic boundary-value problems in the refined two-sided scale of the Hormander spaces $H^{s, \varphi}$, where $s \in R,\quad \varphi$ is a functional parameter slowly varying on $+\infty$. In the case of the Sobolev spaces $H^{s}$, the function $\varphi(|\xi|) \equiv 1$. We establish that the considered operators possess the properties of the Fredholm operators, and the solutions are globally and locally regular.