An example of researching boundary value problems correctness using diffeomorphism method

Authors

  • Oleksii Yu. Potapenko P.C. "ISTA group", Irpin, Ukraine

DOI:

https://doi.org/10.20535/SRIT.2308-8893.2018.3.08

Keywords:

Hilbert space, Riemannian manifold, diffeomorphism, Borel measure, derivation of measures, Laplacian, Dirichlet problem

Abstract

The search for methods for checking correctness of boundary value problems in spaces of an infinite-dimensional argument is one of the problems of the infinite-dimensional analysis. In this paper, the author proposed a method to broaden the class of correct problems by reducing them to already previously considered "canonical type" problems. The reduction process consists of searching for a special class diffeomorphism between Riemannian manifolds, areas in Hilbert’s space among them, which allows to reduce the problem to a simpler one. Boundary value problems are considered in "L2-version". This paper provides an example of such a problem. To fulfill the example, Fréchet derivatives of the diffeomorphism and the inverse mapping are found; diffeomorphism boundedness — a condition of the theorem about a boundary value problem associated with diffeomorphism applicability — is proved.

Author Biography

Oleksii Yu. Potapenko, P.C. "ISTA group", Irpin

Potapenko Oleksii Iiuriiovich,

a Ph.D. student at IASA "Igor Sikorsky KPI", a systems analyst at P.C. "ISTA group", Ukraine.

Research interests: infinite-dimensional spaces and manifolds, boundary value problems on infinite-dimensional spaces and manifolds.

References

Potapenko A.Ju. Kraevaja zadacha, assotsiirovannaja s diffeomorfizmom mezhdu rimanovymi mnogoobrazijami / A.Ju. Potapenko // Sistemnye issledovanija i informatsionnye tehnologii. — 2018. — № 1. — S. 132–140.

Bogdanskij Ju.V. Laplasian po mere na gil'bertovom prostranstve i zadacha Dirihle dlja uravnenija Puassona v -versii / Ju.V. Bogdanskij // Ukr. mat. zhurn. — 2011. — 63, № 9. — C. 1169–1178.

Potapenko O.Ju. Neskinchennovymirni rimanovi mnohovody z rivnomirnoju strukturoju / O.Ju. Potapenko // Naukovi visti NTUU "KPI". — 2016. — T. 108, № 4. — S. 73–79.

Daletskij Ju.L. Stohasticheskie uravnenija i differentsial'naja geometrija / Ju.L. Daletskyj, Ja.Y. Belopol'skaja. — K.: Vyshcha shk., 1989. — 296 c.

Bogdanskij Ju.V. Granichnyj operator sleda v oblasti gil'bertova prostranstva i harakteristicheskoe svojstvo ego jadra / Ju.V. Bogdanskij // Ukr. mat. zhurn. — 2015. — 67, № 11. — C. 1450–1460.

Bogdanskij Ju.V. Zadacha Dirihle s laplasianom po mere na gil'bertovom prostranstve / Ju. V. Bogdanskij, Ja.Ju. Sanzharevskij // Ukr. mat. zhurn. — 2014. — 66, № 6. — C. 733–739.

Bogdanskij Ju.V. Laplasian po mere na rimanovom mnogoobrazii i zadacha Dirihle. I / Ju.V. Bogdanskij, A.Ju. Potapenko // Ukr. mat. zhurn. — 2016. — 68, № 7. — C. 897–907.

Bogdanskij Ju.V. Laplasian po mere na rimanovom mnogoobrazii i zadacha Dirihle. II / Ju.V. Bogdanskij, A.Ju. Potapenko // Ukr. mat. zhurn. — 2016. — 68, № 11. — C. 1443–1449.

Published

2018-10-16

Issue

Section

Mathematical methods, models, problems and technologies for complex systems research