2019
Том 71
№ 11

# Bogdanskii Yu. V.

Articles: 19
Article (Russian)

### Infinite-dimensional version of the Friedrichs inequality

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1476-1483

Two infinite-dimensional versions of the classical Friedrichs inequality are proposed.

Article (Russian)

### Divergence theorem in the $L_2$ -version. Application to the Dirichlet problem

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 611-624

We propose the $L_2$ -version of the divergence theorem. The Green and Poisson operators associated with the infinitedimensional version of the Dirichlet problem are investigated.

Article (Russian)

### Transitivity of the surface measures on Banach manifolds with uniform structure

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1299-1309

We perform the analysis of transitivity of associated measures on the surfaces with finite codimension imbedded in a Banach manifold with uniform atlas.

Article (Russian)

### Surface measures on Banach manifolds with uniform structure

Ukr. Mat. Zh. - 2017. - 69, № 8. - pp. 1030-1048

We propose a method for the construction of associated measures on the surfaces of finite codimension embedded in a Banach manifold with uniform atlas.

Article (Russian)

### Laplacian with respect to the measure on a Riemannian manifold and the Dirichlet problem. II

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1443-1449

We propose the $L^2$ -version of Laplacian with respect to measure on an (infinite-dimensional) Riemannian manifold. The Dirichlet problem for equations with proposed Laplacian is solved in a part of the Rimannian manifold of a certain class.

Article (Russian)

### Laplacian with respect to measure on a Riemannian manifold and Dirichlet problem. I

Ukr. Mat. Zh. - 2016. - 68, № 7. - pp. 897-907

We propose an $L^2$ -version of the Laplacian with respect to measure on an infinite-dimensional Riemannian manifold. The Dirichlet problem for equations with proposed Laplacian is solved in the region of a Rimannian manifold from a certain class.

Article (Russian)

### Maximum principle for the Laplacian with respect to the measure in a domain of the Hilbert space

Ukr. Mat. Zh. - 2016. - 68, № 4. - pp. 460-468

We obtain the maximum principle for two versions of the Laplacian with respect to the measure, namely, for the “classical” and “$L^2$” versions in a domain of the Hilbert space.

Article (Russian)

### Boundary Trace Operator in a Domain of Hilbert Space and the Characteristic Property of its Kernel

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1450-1460

We prove an infinite-dimensional analog of the classical theorem on density of the set $C_0^1 (G)$ of finite smooth functions in the kernel of the boundary trace operator $γ: H_1(G) → L_2(∂G)$.

Article (Russian)

### Laplacian Generated by the Gaussian Measure and Ergodic Theorem

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1172-1180

We consider the Laplacian generated by the Gaussian measure on a separable Hilbert space and prove the ergodic theorem for the corresponding one-parameter semigroup.

Article (Russian)

### The Dirichlet Problem with Laplacian with Respect to a Measure in the Hilbert Space

Ukr. Mat. Zh. - 2014. - 66, № 6. - pp. 733–739

We study the Dirichlet problem for a specified class of elliptic equations in a region of the Hilbert space consistent with a given Borel measure.

Article (Russian)

### Banach Manifolds with Bounded Structure and the Gauss?Ostrogradskii Formula

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1299-1313

We propose a version of the Gauss - Ostrogradskii formula for a Banach manifold with uniform atlas.

Article (Russian)

### Laplacian with respect to a measure on a Hilbert space and an L 2-version of the Dirichlet problem for the Poisson equation

Ukr. Mat. Zh. - 2011. - 63, № 9. - pp. 1169-1178

We propose a version of the Laplace operator for functions on a Hilbert space with measure. In terms of this operator, we investigate the Dirichlet problem for the Poisson equation.

Brief Communications (Ukrainian)

### Nonlinear equations with essentially infinite-dimensional differential operators

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1571–1576

We consider nonlinear differential equations and boundary-value problems with essentially infinite-dimensional operators (of the Laplace–Lévy type). An analog of the Picard theorem is proved.

Article (English)

### Cauchy problem for the essentially infinite-dimensional heat equation on a surface in a Hilbert space

Ukr. Mat. Zh. - 1995. - 47, № 6. - pp. 737–746

It is proved that the Cauchy problem for a simple parabolic equation with essentially infinite-dimensional coefficients on bounded level surfaces of smooth functions in a Hilbert space is uniformly well posed.

Article (Russian)

### Dirichlet problem for the Poisson equation with an essentially infinite-dimensional elliptic operator

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 803–808

In a special class of domains in an infinite-dimensional Hilbert space, the solvability of the Dirichlet problem for the Poisson equation with an elliptic operator of the form $(Lu)(x)=j(x)(u''(x))$ vanishing on cylindrical functions is proved.

Article (Russian)

### Cauchy problem for an essentially infinite-dimensional parabolic equation with variable coefficients

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 663–670

The Cauchy problem for the equation $\cfrac{\partial u}{\partial t} = \mathcal{L}_x u = j(x) (u_x'')$ with positive essentially infinite-dimensional functionals $j(x)$ is studied in a certain Banach space of functions on an infinite-dimensional separable real Hilbert space.

Article (Ukrainian)

### A maximum principle for nonregular elliptic differential equations in a Hilbert space of countable dimension

Ukr. Mat. Zh. - 1988. - 40, № 1. - pp. 21-25

Article (Ukrainian)

### Cauchy's problem for an essentially infinite-dimensional parabolic equation on an infinite-dimensional sphere

Ukr. Mat. Zh. - 1983. - 35, № 1. - pp. 18—22

Article (Ukrainian)

### The Cauchy problem for parabolic equations with essentially infinite-dimensional elliptic operators

Ukr. Mat. Zh. - 1977. - 29, № 6. - pp. 781–784