2018
Том 70
№ 7

All Issues

Bogdanskii Yu. V.

Articles: 17
Article (Russian)

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

Bogdanskii Yu. V.

↓ Abstract

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

Bogdanskii Yu. V., Moravets’ka E. V.

↓ Abstract

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

Bogdanskii Yu. V., Moravets’ka E. V.

↓ Abstract

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

Bogdanskii Yu. V., Potapenko A. Yu.

↓ Abstract

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

Bogdanskii Yu. V., Potapenko A. Yu.

↓ Abstract

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

Bogdanskii Yu. V.

↓ Abstract

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

Bogdanskii Yu. V.

↓ Abstract

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

Bogdanskii Yu. V., Sanzharevskii Ya. Yu.

↓ Abstract

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

Bogdanskii Yu. V., Sanzharevskii Ya. Yu.

↓ Abstract   |   Full text (.pdf)

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

Bogdanskii Yu. V.

↓ Abstract   |   Full text (.pdf)

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

Bogdanskii Yu. V.

↓ Abstract   |   Full text (.pdf)

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

Bogdanskii Yu. V., Statkevych V. M.

↓ Abstract   |   Full text (.pdf)

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 (Ukrainian)

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

Bogdanskii Yu. V.

↓ Abstract   |   Full text (.pdf)

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

We prove that 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 solvable for a special class of domains in an infinite-dimensional Hilbert space.

Article (Ukrainian)

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

Bogdanskii Yu. V.

↓ Abstract   |   Full text (.pdf)

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

The Cauchy problem for the equation\(\partial u/\partial t = \mathcal{L}_x u = j(x) (u''_x )\) with positive essentially infinite-dimensional functionalsj(x) is studied in a properly chosen 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

Bogdanskii Yu. V.

Full text (.pdf)

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

Bogdanskii Yu. V.

Full text (.pdf)

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

Article (Ukrainian)

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

Bogdanskii Yu. V.

Full text (.pdf)

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