To the theory of semi-linear equations in the plane
In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z),whereas its reaction term f(u) is a continuous non-linear function. Assu...
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| Veröffentlicht in: | Український математичний вісник |
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| Datum: | 2019 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут прикладної математики і механіки НАН України
2019
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/169434 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | To the theory of semi-linear equations in the plane / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 105-140. — Бібліогр.: 74 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862732666880655360 |
|---|---|
| author | Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| author_facet | Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| citation_txt | To the theory of semi-linear equations in the plane / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 105-140. — Бібліогр.: 74 назв. — англ. |
| collection | DSpace DC |
| container_title | Український математичний вісник |
| description | In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z),whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak C(Ď )∩ W¹,² loc (D) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semi-linear equations of mathematical physics, arising from modelling processes in anisotropic and inhomogeneous media. With a view to further development of the theory of boundary value problems for the semi-linear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity.
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| first_indexed | 2025-12-07T19:32:52Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-169434 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-12-07T19:32:52Z |
| publishDate | 2019 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. 2020-06-13T08:36:40Z 2020-06-13T08:36:40Z 2019 To the theory of semi-linear equations in the plane / V.Ya. Gutlyanskii, O.V. Nesmelova, V.I. Ryazanov // Український математичний вісник. — 2019. — Т. 16, № 1. — С. 105-140. — Бібліогр.: 74 назв. — англ. 1810-3200 2010 MSC. Primary 30C62, 31A05, 31A20, 31A25, 31B25, 35J61 Secondary 30E25, 31C05, 34M50, 35Q15 https://nasplib.isofts.kiev.ua/handle/123456789/169434 In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z),whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak C(Ď )∩ W¹,² loc (D) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semi-linear equations of mathematical physics, arising from modelling processes in anisotropic and inhomogeneous media. With a view to further development of the theory of boundary value problems for the semi-linear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. This work was partially supported by grant of Ministry of Education and Science of Ukraine, project number is 0119U100421. en Інститут прикладної математики і механіки НАН України Український математичний вісник To the theory of semi-linear equations in the plane Article published earlier |
| spellingShingle | To the theory of semi-linear equations in the plane Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| title | To the theory of semi-linear equations in the plane |
| title_full | To the theory of semi-linear equations in the plane |
| title_fullStr | To the theory of semi-linear equations in the plane |
| title_full_unstemmed | To the theory of semi-linear equations in the plane |
| title_short | To the theory of semi-linear equations in the plane |
| title_sort | to the theory of semi-linear equations in the plane |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/169434 |
| work_keys_str_mv | AT gutlyanskiivya tothetheoryofsemilinearequationsintheplane AT nesmelovaov tothetheoryofsemilinearequationsintheplane AT ryazanovvi tothetheoryofsemilinearequationsintheplane |