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| Author(s): |
Rafael Antonio Rossato
Total Authors: 1
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| Document type: | Master's Dissertation |
| Press: | São Carlos. |
| Institution: | Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) |
| Defense date: | 2010-04-09 |
| Examining board members: |
Eugenio Tommaso Massa;
Rodrigo da Silva Rodrigues;
Sérgio Henrique Monari Soares
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| Advisor: | Eugenio Tommaso Massa |
| Abstract | |
We study the Fucik Spectrum for the Laplacian operator, that is, the set \'SIGMA\' of the couples (\'mü\', \'nü\') \'ARE THIS ESTA CONTAINED\' \'R POT. 2\', for which the problem { - \'DELTA\' u(x) = \'\'\'mü \'nü\' POT. + (x); \'EPSILON\' \' OMEGA\', Bu = 0; x \'EPSILON\' \'PARTIAL\' \' OMEGA\', admits a nontrivial solution, where \'OMEGA\' \'EPSILON\' \'R POT. n\' is a bounded domain, \'u POT. + (x) = max {0, u(x)}, \'u POT. -\'(x) = {0, - u(x)} and B represents some boundary condition. We first show abstract results about the Fucik Spectrum and then we compute it explicitly in the one dimensional case for the Dirichlet and Neumann problems. These results one applied at the study of the solvability of the problem. { - \'DELTA\'u(x) = f (x, u(x)), x \'EPSILON\' \'OMEGA\', Bu = 0; x \'EPSILON\' \'PARTIAL\'\'OMEGA\', whe3n the nonlinearity f is a suitable pertubation of \'mü\'\'u POT. + - \'\'nü\' u+ - \'\'nü\' u POT. n\'; we describe different behaviors depending on the parameters (\'mü\', \'nü\'). Finally, we consider the Fucik Spectrum in higher dimension. In this case it is not possible to compute it explicitly, so we will show a variational characterization of the first nontrivial curve. This characterization will allow to obtain some information on the properties of this curve and also further results on the solvability of (2) (AU) | |
| FAPESP's process: | 07/06106-4 - Fucik Spectrum and elliptic equations with jumping nonlinearities |
| Grantee: | Rafael Antônio Rossato |
| Support Opportunities: | Scholarships in Brazil - Master |