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Universidade de Bras´ılia
Departamento de Matem´atica
Exame de Qualifica¸c˜ao ao Doutorado
Bras´ılia, 20 de maio de 2016

Aluno: Bruno Marino Xavier
Orientador: Pedro Roitman

Banca Examinadora:
Marcelo Fernandes Furtado (UnB) - Presidente
Lu´ıs Henrique de Miranda (UnB)
Ricardo Ruviaro (UnB)

´
Ementa de An´
alise (2a Area)
1. Operadores Lineares Cont´ınuos
1.1 Caracteriza¸c˜oes de operadores lineares cont´ınuos
1.2 Teorema de Banach-Steinhaus
1.3 Teorema da Aplica¸c˜ao Aberta
1.4 Teorema do Gr´afico Fechado
2. Teoremas de Hahn-Banach
2.1 Forma anal´ıtica do Teorema de Hahn-Banach
2.2 Formas geom´etricas do Teorema de Hahn-Banach
2.3 Aplica¸c˜oes
3. Topologias fraca e fraca*
3.1 Conceito de topologia fraca e fraca*
3.2 Compacidade fraca e reflexividade
3.3 Metrizabilidade e separabilidade
3.4 Espa¸cos uniformemente convexos
4. Espa¸cos de Hilbert
1

4.1 Espa¸cos com produto interno
4.2 Ortogonalidade e proje¸co˜es
4.3 Conjuntos ortonormais em espa¸cos de Hilbert
4.4 Teorema da Representa¸ca˜o de Riesz
4.5 Teorema de Lax-Milgram
5. Teoria Espectral de Operadores Compactos e Autoadjuntos
5.1 Espectro de um operador cont´ınuo
5.2 Operadores compactos
5.3 Espectro de operadores compactos
5.4 Operadores autoadjuntos em espa¸cos de Hilbert
5.5 Espectro de operadores compactos e autoadjuntos
6. Operadores lineares de 2a ordem
6.1 Princ´ıpio de M´aximo
6.2 M´etodo da continua¸ca˜o
6.3 Espa¸cos de H¨older
6.4 Teoremas de imers˜ao cont´ınua ou compacta
6.5 Teorema da existˆencia de Schauder
7. Espa¸cos de Sobolev
7.1 Derivada Fraca
7.2 Espa¸cos de Sobolev
7.3 Aproxima¸ca˜o por fun¸co˜es suaves
7.4 Imers˜oes cont´ınua e compactas de W k,p
8. Solu¸c˜oes fracas para equa¸c˜oes lineares de 2a ordem
8.1 Existˆencia de solu¸c˜ao
i. Alternativa de Fredholm
ii. Autovalores do operador
8.2 Espectro do laplaciano
8.3 Regularidade de solu¸co˜es

2

References
[1] GILBARG, D.; TRUDINGER, N. S. Elliptical Partial Differential Equations of Second
Order. Springer, Berlim (2001).
[2] KREYSZIG, E. Introductory Functional Analysis with Applications . John Wiley and
Sons, Nova Iorque (1989).
[3] BREZIS, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations .
Springer, Nova Iorque (2011).
[4] EVANS, L. C. Partial Differential Equations. American Mathematical Society, Providence (2002).
[5] ADAMS, R. Sobolev Spaces . Academic Press (1975).
[6] FURTADO, M. Notas de aula do curso ’Equa¸c˜oes Diferenciais Parciais II ’ 2/2015
UnB, Bras´ılia.

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