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Doctoral Thesis
DOI
10.11606/T.55.2017.tde-25072017-111735
Document
Author
Full name
Camilo Campana
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2017
Supervisor
Committee
Bergamasco, Adalberto Panobianco (President)
Picon, Tiago Henrique
Santos Filho, José Ruidival Soares dos
Silva, Paulo Leandro Dattori da
Zani, Sergio Luis
Title in Portuguese
O problema de Riemann-Hilbert para campos vetoriais complexos
Keywords in Portuguese
Campos vetoriais complexos
Equações diferenciais parciais
Hipocomplexidade
Operadores integrais
Problema de Riemann-Hilbert
Abstract in Portuguese
Este trabalho trata de problemas de contorno definidos no plano. O problema central desta tese é chamado Problema de Riemann-Hilbert, o qual pode ser descrito como segue. Seja L um campo vetorial complexo não singular definido em uma vizinhança do fecho de um aberto simplesmente conexo do plano com fronteira suave. O Problema de Riemann-Hilbert para o campo L consiste em obter uma solução para a equação Lu = F(x, y, u) no aberto em estudo, sendo dada uma função F mensurável. Pede-se também que a solução tenha extensão contínua até a fronteira e que satisfaça lá uma condição adicional; trabalha-se aqui no contexto das funções Hölder contínuas. Foram obtidos resultados para o problema acima no caso em que L pertence a uma classe de campos hipocomplexos. O caso clássico conhecido é quando o campo vetorial é o operador de Cauchy-Riemann, ou, mais geralmente, quando é um campo elítico.
Title in English
The Riemann-Hilbert problem for complex vector fields
Keywords in English
Complex vector fields
Hipocomplexity
Integral operators
Partial differential equations
Riemann-Hilbert problem
Abstract in English
This work deals with boundary problems in the plane. The central problem in this thesis is the so-called Riemann-Hilbert problem, which may be described as follows. Let L be a non-singular complex vector field defined on a neighborhood of the closure of a simply connected open subset of the plane having smooth boundary. The Riemann-Hilbert problem for the vector field L consists in finding a solution to the equation Lu = F(x, y, u) on the open set under study, where the given function F is measurable. It is also required that the solution have a continuous extension up to the boundary and satisfy an additional condition there. Results were obtained for the above problem when L belongs to a class of hypocomplex vector fields. The well-known classical case is the one in which the vector field under study is the Cauchy-Riemann operator, or more generally when it is an elliptic vector field.
 
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Publishing Date
2017-07-25
 
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