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Master's Dissertation
DOI
10.11606/D.55.2013.tde-19032013-094256
Document
Author
Full name
Camilo Campana
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2013
Supervisor
Committee
Bergamasco, Adalberto Panobianco (President)
Ebert, Marcelo Rempel
Zani, Sergio Luis
Title in Portuguese
Campos hipoelíticos no plano
Keywords in Portuguese
Campos vetoriais complexos
Condição (P)
Hipoeliticidade
Integrabilidade global
Integrabilidade local
Resolubilidade local
Uniformização
Abstract in Portuguese
Seja L um campo vetorial complexo não singular definido em um aberto do plano. Treves provou que se L é localmente resolúvel então L é localmente integrável. Para campos planares hipoelíticos, vale uma propriedade adicional, a saber, toda integral primeira (restrita a um aberto suficientemente pequeno) é uma aplicação injetiva (e aberta); isto, por sua vez, implica que toda solução da equação homogênea Lu = 0 é localmente da forma u = h 0 Z, com h holomorfa, sendo Z uma integral primeira do campo. O problema central de interesse desta dissertação é a questão global correspondente, ou seja, a exisatência de integrais primeiras globais injetoras e a representação dde soluções globais por composições da integral primeira com uma função holomorfa
Title in English
Hypoelliptic planar vector fields
Keywords in English
Complex vector fields
Condition (P)
Global integrability
Hypoellipticity
Local integrability
Local solvability
Uniformization
Abstract in English
Let L be a nonsingular complex vector field defined on an open subset of the plane. Treves proved that if L is locally solvable then L is locally integrable. For hypoelliptic planar vector fields an additional property holds, namely, every first integral (restricted to a sufficiently small open set) is an injective (and open) mapping; this, on its turn, implies that each solution of the homogeneous equation Lu = 0 is locally of the form u = h Z, where h is holomorphic and Z is a first integral of the vector eld. The central problem of interest in this work is the corresponding global question, that is, the existence of global, injective first integrals and the representation of global solutions as compositions of the first integral with a holomorphic function
 
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disscamilo.pdf (1.16 Mbytes)
Publishing Date
2013-03-19
 
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