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Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators
The purpose of this work is to explore some notions of monotonicity for operators between Banach spaces and the applications to the study of boundary value problems (BVPs) and initial boundary value problems (IBVPs) for partial differential equations (PDEs), with the possibility in the end to examin...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2021
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| _version_ | 1867613265060691968 |
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| access_status_str | Open Access |
| author | Lin, Tianyu |
| author2 | Ebobisse, Bille Francois |
| author_browse | Ebobisse, Bille Francois Lin, Tianyu |
| author_facet | Ebobisse, Bille Francois Lin, Tianyu |
| author_sort | Lin, Tianyu |
| collection | Thesis |
| description | The purpose of this work is to explore some notions of monotonicity for operators between Banach spaces and the applications to the study of boundary value problems (BVPs) and initial boundary value problems (IBVPs) for partial differential equations (PDEs), with the possibility in the end to examine new problems and provide some solutions. Variational approach will be used to reformulate these problems into stationary equations (in the case of BVPs) and evolution equations (in the case of IBVPs), where the underlined operators constructed as realizations of those problems in appropriate function spaces. This is known as weak formulation, which allows us to find weak solutions of the problems in a larger functions space rather than classical solutions that are sufficiently smooth. The theory of monotone and pseudomonotone operators will be applied to find existence theorems for stationary equations and evolution equations. In addition, the existence theorem for evolution equations with locally monotone operator will also be presented as a generalisation of the one with monotone operators. Another type of monotonicity so-called strict p-quasimonotonicity, which is defined in term of Young measures. This type of weaker, integrated version of monotonicity is directly applied in the study of elliptic and parabolic system of PDEs, the difficulty arises from dealing with this monotonicity is overcome by the theory of Young measures. The application of these monotonicity in the study of variational inequality will also be discussed. In particular, there is a new setting for strict p-quasimonotonicity in a particular type of elliptic variational inequalities, the proof of the new existence theorem will also be presented. Some open problems on the application of strict p-quasimonotonicity in the study of parabolic variational inequalities will also be discussed. Finally, we mention the theory of monotone and pseudomonotone operators in the study of second order evolution equations. A new setting of the local monotonicity in the second order evolution equations will be presented as well as the new existence theorem. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/32783 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:33:23.204Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2021 |
| publishDateRange | 2021 |
| publishDateSort | 2021 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/32783 Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators Lin, Tianyu Ebobisse, Bille Francois Mathematics The purpose of this work is to explore some notions of monotonicity for operators between Banach spaces and the applications to the study of boundary value problems (BVPs) and initial boundary value problems (IBVPs) for partial differential equations (PDEs), with the possibility in the end to examine new problems and provide some solutions. Variational approach will be used to reformulate these problems into stationary equations (in the case of BVPs) and evolution equations (in the case of IBVPs), where the underlined operators constructed as realizations of those problems in appropriate function spaces. This is known as weak formulation, which allows us to find weak solutions of the problems in a larger functions space rather than classical solutions that are sufficiently smooth. The theory of monotone and pseudomonotone operators will be applied to find existence theorems for stationary equations and evolution equations. In addition, the existence theorem for evolution equations with locally monotone operator will also be presented as a generalisation of the one with monotone operators. Another type of monotonicity so-called strict p-quasimonotonicity, which is defined in term of Young measures. This type of weaker, integrated version of monotonicity is directly applied in the study of elliptic and parabolic system of PDEs, the difficulty arises from dealing with this monotonicity is overcome by the theory of Young measures. The application of these monotonicity in the study of variational inequality will also be discussed. In particular, there is a new setting for strict p-quasimonotonicity in a particular type of elliptic variational inequalities, the proof of the new existence theorem will also be presented. Some open problems on the application of strict p-quasimonotonicity in the study of parabolic variational inequalities will also be discussed. Finally, we mention the theory of monotone and pseudomonotone operators in the study of second order evolution equations. A new setting of the local monotonicity in the second order evolution equations will be presented as well as the new existence theorem. 2021-02-04T13:56:07Z 2021-02-04T13:56:07Z 2020 2021-02-04T05:39:44Z Master Thesis Masters MSc http://hdl.handle.net/11427/32783 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science |
| spellingShingle | Mathematics Lin, Tianyu Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators |
| thesis_degree_str | Master's |
| title | Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators |
| title_full | Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators |
| title_fullStr | Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators |
| title_full_unstemmed | Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators |
| title_short | Some Stationary and Evolution Problems Governed by Various Notions of Monotone Operators |
| title_sort | some stationary and evolution problems governed by various notions of monotone operators |
| topic | Mathematics |
| url | http://hdl.handle.net/11427/32783 |
| work_keys_str_mv | AT lintianyu somestationaryandevolutionproblemsgovernedbyvariousnotionsofmonotoneoperators |