Um estudo do espectro de Fucik para o operador p-laplaciano
Carregando...
Data
2022-10-03
Autores
Ambrosio, Enrico Sonvenso
Título da Revista
ISSN da Revista
Título de Volume
Editor
Universidade Estadual Paulista (Unesp)
Resumo
Neste trabalho estudaremos o Espectro de Fučik para o p-laplaciano Σ_p, o qual é definido pelos pontos (α,β)∈R^2 para os quais o Problema de Fučik tem solução não trivial. Partindo de resultados já conhecidos para o Problema de Autovalor do Operador p-Laplaciano provamos a existência de duas curvas triviais para o espectro de Fucik, bem como mostramos algumas de suas propriedades. Além disso, utilizando de uma versão do Teorema do Passo da Montanha para uma C^1 variedade conseguimos o primeiro ponto não trivial e partindo dele construímos uma curva no plano que pertence ao Espectro de Fučik. O resultado mais importante desse trabalho garante que essa curva é a primeira curva não-trivial do Espectro de Fučik. Estudamos também ao final do texto algumas propriedades e comportamentos da curva obtida como o fato dela ser contínua, estritamente decrescente e nos limite convergir para as linhas triviais.
In this work we going to be studying the Fučik Spectrum for the p-Laplacian Σ_p, which is defined as the points (α,β)∈R^2 that the Fučik problem has a non-trivial solution. Starting from the well-known results for the Eigenvalue Problem of the p-Laplacian Operator, we got the initial results for the Fučik Spectrum proving the existence of two trivial curves, as well as show some of their properties. Furthermore, using a version of the Mountain Pass Theorem for a C^1 manifold, we are able to get the first non-trivial point and starting from it we will build a curve in the plane that belongs to the Fučik Spectrum. The most important result of this work assures that this curve is the first non-trivial curve of the Fučik Spectrum. We also will have studied at the end of the text some properties and behaviors of the obtained curve, such as the fact that it is continuous, strictly decreasing and, in the limit, converges to trivial lines.
In this work we going to be studying the Fučik Spectrum for the p-Laplacian Σ_p, which is defined as the points (α,β)∈R^2 that the Fučik problem has a non-trivial solution. Starting from the well-known results for the Eigenvalue Problem of the p-Laplacian Operator, we got the initial results for the Fučik Spectrum proving the existence of two trivial curves, as well as show some of their properties. Furthermore, using a version of the Mountain Pass Theorem for a C^1 manifold, we are able to get the first non-trivial point and starting from it we will build a curve in the plane that belongs to the Fučik Spectrum. The most important result of this work assures that this curve is the first non-trivial curve of the Fučik Spectrum. We also will have studied at the end of the text some properties and behaviors of the obtained curve, such as the fact that it is continuous, strictly decreasing and, in the limit, converges to trivial lines.
Descrição
Palavras-chave
Equações diferenciais parciais, Operador p-Laplaciano, Espectro de Fučik, Teorema do passo da montanha, Partial differential equations, p-Laplacian operator, Fučik spectrum, Mountain pass theorem