An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations
This work addresses the uniqueness and regularity of solutions to integral equations associated with elliptic boundary value problems in irregular domains. Traditional results often assume smooth (Lipschitz) boundaries, but this study extends these results to more general domains with irregular bou...
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| Format: | Article |
| Language: | English |
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Universidade Federal de Viçosa (UFV)
2024-07-01
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| Series: | The Journal of Engineering and Exact Sciences |
| Subjects: | |
| Online Access: | https://periodicos.ufv.br/jcec/article/view/18895 |
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| _version_ | 1832569805925777408 |
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| author | Rômulo Damasclin Chaves dos Santos Jorge Henrique de Oliveira Sales |
| author_facet | Rômulo Damasclin Chaves dos Santos Jorge Henrique de Oliveira Sales |
| author_sort | Rômulo Damasclin Chaves dos Santos |
| collection | DOAJ |
| description |
This work addresses the uniqueness and regularity of solutions to integral equations associated with elliptic boundary value problems in irregular domains. Traditional results often assume smooth (Lipschitz) boundaries, but this study extends these results to more general domains with irregular boundaries. By leveraging Sobolev spaces, particularly fractional Sobolev spaces , and the properties of the Slobodetskii norm, we develop a robust theoretical framework. Our main theorem demonstrates that, under suitable conditions, has a unique solution in , and this solution inherits the regularity properties from the function . The results provide significant advancements in the mathematical understanding of boundary value problems in non-smooth domains, with potential applications in various fields of physics and engineering.
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| format | Article |
| id | doaj-art-78862b8d1c824e7995cdda1c86f69aa1 |
| institution | Kabale University |
| issn | 2527-1075 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | Universidade Federal de Viçosa (UFV) |
| record_format | Article |
| series | The Journal of Engineering and Exact Sciences |
| spelling | doaj-art-78862b8d1c824e7995cdda1c86f69aa12025-02-02T19:53:32ZengUniversidade Federal de Viçosa (UFV)The Journal of Engineering and Exact Sciences2527-10752024-07-0110710.18540/jcecvl10iss7pp18895An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equationsRômulo Damasclin Chaves dos Santos0Jorge Henrique de Oliveira Sales1Department of Physics, Technological Institute of Aeronautics, São Paulo, BrazilState University of Santa Cruz – Department of Exact Sciences, Ilhéus, Bahia, Brazil This work addresses the uniqueness and regularity of solutions to integral equations associated with elliptic boundary value problems in irregular domains. Traditional results often assume smooth (Lipschitz) boundaries, but this study extends these results to more general domains with irregular boundaries. By leveraging Sobolev spaces, particularly fractional Sobolev spaces , and the properties of the Slobodetskii norm, we develop a robust theoretical framework. Our main theorem demonstrates that, under suitable conditions, has a unique solution in , and this solution inherits the regularity properties from the function . The results provide significant advancements in the mathematical understanding of boundary value problems in non-smooth domains, with potential applications in various fields of physics and engineering. https://periodicos.ufv.br/jcec/article/view/18895Irregular DomainIntegral EquationsSobolev SpacesUniqueness and Regularity |
| spellingShingle | Rômulo Damasclin Chaves dos Santos Jorge Henrique de Oliveira Sales An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations The Journal of Engineering and Exact Sciences Irregular Domain Integral Equations Sobolev Spaces Uniqueness and Regularity |
| title | An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations |
| title_full | An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations |
| title_fullStr | An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations |
| title_full_unstemmed | An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations |
| title_short | An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations |
| title_sort | essay on the foundations of variational methods exploring sobolev spaces for boundary integral equations |
| topic | Irregular Domain Integral Equations Sobolev Spaces Uniqueness and Regularity |
| url | https://periodicos.ufv.br/jcec/article/view/18895 |
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