Representação de soluções homogêneas contínuas de campos vetoriais no plano

• Ano de defesa: 2015
• Autor(a) principal: Menis, Alexandra Cristina
• Orientador(a): Hounie, Jorge Guillermo
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal de São Carlos; Câmpus São Carlos
• Programa de Pós-Graduação: Programa de Pós-Graduação em Matemática - PPGM
• Palavras-chave em Português: Equações diferenciais parciais; Campos vetoriais; Teorema de Baouendi-Treves
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: https://repositorio.ufscar.br/handle/20.500.14289/7525

In this work we study conditions for the validity of the analogue of Mergelyan’s theorem for continuous solutions of a type of locally integrable vector field. On a domain in the plane, we consider a vector field L that has a first integral on of the form Z(x, t) = x + i'(x, t), where '(x, t) is a smooth, realvalued function. Given a continuous solution u of Lu = 0 on , our first objective was to find conditions on and Z for the validity of the factorization u = U Z, where U 2 C0(Z ()) \ H(int{Z ()}). We will next study this factorization on the closure of . We assume that u 2 C0( ) and that the boundary of is real analytic, then we show in which cases the condition Z(p1) = Z(p2) implies that u(p1) = u(p2), for p1, p2 2 . The cases are divided according to the geometry of the boundary in the points p1 and p2. When is a compact set and u = U Z on , we obtain that u is uniformly approximated by polynomials of Z on .