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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| Дата: | 2019 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2019
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| Назва видання: | Український математичний вісник |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/169434 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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. |
Репозитарії
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irk-123456789-1694342020-06-15T01:26:40Z To the theory of semi-linear equations in the plane Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. 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. 2019 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/169434 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| spellingShingle |
Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. To the theory of semi-linear equations in the plane Український математичний вісник |
| author_facet |
Gutlyanskii, V.Ya. Nesmelova, O.V. Ryazanov, V.I. |
| author_sort |
Gutlyanskii, V.Ya. |
| title |
To the theory of semi-linear equations in the plane |
| title_short |
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_sort |
to the theory of semi-linear equations in the plane |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2019 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/169434 |
| 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. |
| series |
Український математичний вісник |
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AT gutlyanskiivya tothetheoryofsemilinearequationsintheplane AT nesmelovaov tothetheoryofsemilinearequationsintheplane AT ryazanovvi tothetheoryofsemilinearequationsintheplane |
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2023-10-18T22:25:11Z |
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