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Semilinear elliptic partial differential equations with the critical Sobolev exponent
We present how variational methods and results from linear and non-linear functional analysis are applied to solving certain types of semilinear elliptic partial differential equations (PDEs). The ultimate goal is to prove results on the existence and non-existence of solutions to the Semilinear Ell...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2017
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| _version_ | 1867614013760733184 |
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| access_status_str | Open Access |
| author | Mavuso, Melusi Manqoba |
| author2 | Ebobisse Bille, Francois |
| author_browse | Ebobisse Bille, Francois Mavuso, Melusi Manqoba |
| author_facet | Ebobisse Bille, Francois Mavuso, Melusi Manqoba |
| author_sort | Mavuso, Melusi Manqoba |
| collection | Thesis |
| description | We present how variational methods and results from linear and non-linear functional analysis are applied to solving certain types of semilinear elliptic partial differential equations (PDEs). The ultimate goal is to prove results on the existence and non-existence of solutions to the Semilinear Elliptic PDEs with the Critical Sobolev Exponent. To this end, we first recall some useful results from functional analysis, including the Sobolev spaces, which provide a natural setting for the idea of weak or generalised solutions. We then present linear PDE theory, including eigenvalues of the Dirichlet Laplacian operator. We discuss the Direct Methods of Calculus of Variations and Critical Point Theory, together with examples of how these techniques are applied to solving PDEs. We show how the existence of solutions to semilinear elliptic equations depends on the exponent of the growth of the non-linear term. This then naturally leads to the discussion of the critical Sobolev exponent, where we present both positive and negative results. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/25441 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:45:17.884Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2017 |
| publishDateRange | 2017 |
| publishDateSort | 2017 |
| 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:11427/25441 Semilinear elliptic partial differential equations with the critical Sobolev exponent Mavuso, Melusi Manqoba Ebobisse Bille, Francois Mathematics We present how variational methods and results from linear and non-linear functional analysis are applied to solving certain types of semilinear elliptic partial differential equations (PDEs). The ultimate goal is to prove results on the existence and non-existence of solutions to the Semilinear Elliptic PDEs with the Critical Sobolev Exponent. To this end, we first recall some useful results from functional analysis, including the Sobolev spaces, which provide a natural setting for the idea of weak or generalised solutions. We then present linear PDE theory, including eigenvalues of the Dirichlet Laplacian operator. We discuss the Direct Methods of Calculus of Variations and Critical Point Theory, together with examples of how these techniques are applied to solving PDEs. We show how the existence of solutions to semilinear elliptic equations depends on the exponent of the growth of the non-linear term. This then naturally leads to the discussion of the critical Sobolev exponent, where we present both positive and negative results. 2017-09-28T05:26:30Z 2017-09-28T05:26:30Z 2017 Master Thesis Masters MSc http://hdl.handle.net/11427/25441 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics Mavuso, Melusi Manqoba Semilinear elliptic partial differential equations with the critical Sobolev exponent |
| thesis_degree_str | Master's |
| title | Semilinear elliptic partial differential equations with the critical Sobolev exponent |
| title_full | Semilinear elliptic partial differential equations with the critical Sobolev exponent |
| title_fullStr | Semilinear elliptic partial differential equations with the critical Sobolev exponent |
| title_full_unstemmed | Semilinear elliptic partial differential equations with the critical Sobolev exponent |
| title_short | Semilinear elliptic partial differential equations with the critical Sobolev exponent |
| title_sort | semilinear elliptic partial differential equations with the critical sobolev exponent |
| topic | Mathematics |
| url | http://hdl.handle.net/11427/25441 |
| work_keys_str_mv | AT mavusomelusimanqoba semilinearellipticpartialdifferentialequationswiththecriticalsobolevexponent |