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Emerson Alves Mendonça de Abreuhttp://lattes.cnpq.br/0989407026771712Ederson Moreira dos SantosEzequiel Rodrigues BarbosaLucas Catão de Freitas FerreiraMarcos da Silva Montenegrohttp://lattes.cnpq.br/5761417192608218Leandro Gonzaga Fernandes Junior2019-11-23T00:21:13Z2019-11-23T00:21:13Z2019-09-19http://hdl.handle.net/1843/31236O objetivo geral da tese é o estudo de Equações Diferenciais Parciais Elípticas. A tese é dividida em duas Partes: (I) Desigualdade do Tipo Trudinger-Moser sobre espaços de Sobolev com pesos; e (II) A existência e não-existência de desigualdades isoperimétricas com pesos monomiais diferentes. Na Parte I, estabelecemos uma desigualdade do tipo Trudinger-Moser sobre espaços de Sobolev com pesos sobre o intervalo $(0,+\infty)$, relacionada com a classe de operadores elípticos quasilineares cuja forma radial é dada por $\displaystyle Lu:=-r^{-\theta} (r^{\alpha}\vert u'(r)\vert^{\beta}u'(r))',$ onde $\theta, \beta\geq 0$ e $\alpha>0$, são constantes satisfazendo algumas condições de existência. Vale enfatizar que esses operadores generalizam o $p$-Laplaceano e $k$- Hessiana, no caso radial. Os resultados envolvem dimensão fracionária, um princípio de P\'olya-Szeg{\"o} com pesos e uma limitação para a constante ótima associada com a desigualdade do tipo Gagliardo-Nirenberg. Na Parte II, consideramos pesos monomiais $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N}}$, onde $a_{i}$ é um número real não negativo para cada $i\in\{1,\ldots,N\}$, e estabelecemos a existência e não-existência de desigualdades isoperimétricas com pesos monomiais diferentes. Estudamos minimizadores positivos de $\int_{\partial\Omega}x^{A}\mathcal{H}^{N-1}(x)$ sobre todos os conjuntos abertos, limitados e suaves cujo volume $\int_{\Omega}x^{B}dx$ é fixo.The main topic of the thesis is the study of Elliptic Partial Differential Equations. The thesis is divided into two Parts: (I) Trudinger-Moser Type inequality on weighted Sobolev spaces; and (II) on existence and nonexistence of isoperimetric inequalities with different monomial weights. In part I, we establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $\displaystyle Lu:=-r^{-\theta} (r^{\alpha}\vert u'(r)\vert^{\beta}u'(r))',$ where $\theta, \beta\geq 0$ and $\alpha>0$, are constants satisfying some existence conditions. It is worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted P\'olya-Szeg{\"o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality. In part II, we consider the monomial weight $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N}}$, where $a_{i}$ is a nonnegative real number for each $i\in\{1,\ldots,N\}$, and we establish the existence and nonexistence of isoperimetric inequalities with different monomial weights. We study positive minimizers of $\int_{\partial\Omega}x^{A}\mathcal{H}^{N-1}(x)$ among all smooth bounded open sets $\Omega$ in $\mathbb{R}^{N}$ with fixed Lebesgue measure with monomial weight $\int_{\Omega}x^{B}dx$.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorengUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em MatemáticaUFMGBrasilICX - DEPARTAMENTO DE MATEMÁTICAhttp://creativecommons.org/licenses/by/3.0/pt/info:eu-repo/semantics/openAccessMatemática – TesesEquações diferenciais Elípticas – Teses.Sobolev, Espaço de - Tesesweighted Trudinger-Moser inequalityweighted rearrangementSchwarz symmetrizationisoperimetric inequalitiesSobolev Inequalitiesmonomial weightsOn weighted Sobolev spaces: Trudinger-Moser and isoperimetric inequalitiesinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGLICENSElicense.txtlicense.txttext/plain; charset=utf-82119https://repositorio.ufmg.br/bitstream/1843/31236/3/license.txt34badce4be7e31e3adb4575ae96af679MD53ORIGINALThesis- L. G. Fernandes JR .pdfThesis- L. G. Fernandes JR .pdfThesis- L. G. Fernandes JRapplication/pdf1034909https://repositorio.ufmg.br/bitstream/1843/31236/1/Thesis-%20L.%20G.%20Fernandes%20JR%20.pdfdda7486064891a407d059bded4d498d3MD51CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8914https://repositorio.ufmg.br/bitstream/1843/31236/2/license_rdff9944a358a0c32770bd9bed185bb5395MD52TEXTThesis- L. G. Fernandes JR .pdf.txtThesis- L. G. Fernandes JR .pdf.txtExtracted texttext/plain154796https://repositorio.ufmg.br/bitstream/1843/31236/4/Thesis-%20L.%20G.%20Fernandes%20JR%20.pdf.txtbe957b7caca6fd02fabb9b7fbd942a8dMD541843/312362019-11-23 03:25:14.721oai:repositorio.ufmg.br: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Repositório InstitucionalPUBhttps://repositorio.ufmg.br/oaiopendoar:2019-11-23T06:25:14Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false
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