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ETDs @PUC-Rio
Estatística
Título: UNIQUENESS OF LP-STRONG SOLUTIONS
Autor: GABRIEL GOMES FIGUEIREDO
Colaborador(es): BOYAN SLAVCHEV SIRAKOV - Orientador
PAMMELLA QUEIROZ DE SOUZA - Coorientador
Catalogação: 26/SET/2023 Língua(s): PORTUGUESE - BRAZIL
Tipo: TEXT Subtipo: THESIS
Notas: [pt] Todos os dados constantes dos documentos são de inteira responsabilidade de seus autores. Os dados utilizados nas descrições dos documentos estão em conformidade com os sistemas da administração da PUC-Rio.
[en] All data contained in the documents are the sole responsibility of the authors. The data used in the descriptions of the documents are in conformity with the systems of the administration of PUC-Rio.
Referência(s): [pt] https://www.maxwell.vrac.puc-rio.br/projetosEspeciais/ETDs/consultas/conteudo.php?strSecao=resultado&nrSeq=64105&idi=1
[en] https://www.maxwell.vrac.puc-rio.br/projetosEspeciais/ETDs/consultas/conteudo.php?strSecao=resultado&nrSeq=64105&idi=2
DOI: https://doi.org/10.17771/PUCRio.acad.64105
Resumo:
This master s thesis delves into an in-depth study of the article [2]. Chapter2 begins by introducing fundamental definitions and concepts essential forthe subsequent theoretical analysis. A proposition is then demonstrated,establishing the existence of a Taylor expansion for functions in a givenspace, emphasizing the role of the Escauriaza exponent.The chapter proceeds to present two lemmas that relate subsolutions andsupersolutions in terms of viscosity and properties of norms. The firstversion of the lemma considers the relationship between the dimension ofspace and the norm, while the second version uses the Escauriaza exponentto obtain more refined results. Two results are shown to explain that explainthe relationship between different notions of viscous solutions and theirconnection with Sobolev spaces.The properties of the Pucci operators are discussed at the conclusion of thischapter. Chapter 3 begins by establishing the definition of the boundarygeometry of the domain in question. An important lemma is demonstrated,which establishes the existence of strong solutions in a given space andexplores the regularity of the functions involved based on this lemma.The concepts of superdifferentiability and subdifferentiability areintroduced, playing a crucial role in understanding the behavior of viscoussolutions and their relationships with higher order derivatives. A generalresult that extends these definitions is presented. The dissertation discussestwo versions wherein the function u is twice super-differentiable, consideringthe space Ld and later the space Lp, so that p less than d.The dissertation goes on to demonstrate the relationship between Lp-viscosity sub-solution and Lp-strong sub-solution when u belongs to aspecific space. Next, it is shown that the uniform limits of solutions arealso solutions. Finally, the main result of the dissertation is presented,demonstrating the uniqueness of strong solutions.
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