DSpace Communidade:
http://www.repositorio.ufc.br/handle/riufc/59
Sat, 21 May 2022 08:42:09 GMT2022-05-21T08:42:09ZTeoria de fronteira livre para equações não lineares singulares/degeneradas não homogêneas: uma abordagem não variacional
http://www.repositorio.ufc.br/handle/riufc/65724
Título: Teoria de fronteira livre para equações não lineares singulares/degeneradas não homogêneas: uma abordagem não variacional
Autor(es): Oliveira, José Erivamberto Lima
Abstract: Free boundary problems are related to questions involving solutions to partial differential equations. Its applicability extends to several areas of knowledge. Our goal is study the free boundary regularity of the one phase problem to the g-laplacian. Deal with a sofisticate machinery, we get sucess. Getting, the C 1,α regularity to free boundary in two results: the fi rst, heuristically, the free boundary is in between two planes and second, when its has Lipschitz regularity in a neiborhood of point. In this last one, the C 1,α regularity occurs in a smaller (possibly) neighborhood of the point in question. Although the operator is of the divergent form, where we traditionally use a variational approach, we were able to use results that guarantee the equivalence between solutions in the sense of distributions and in the sense of viscosity to proceed with our intentions.Mon, 28 Feb 2022 00:00:00 GMThttp://www.repositorio.ufc.br/handle/riufc/657242022-02-28T00:00:00ZRegularidade Hölder em equações elípticas na forma divergente
http://www.repositorio.ufc.br/handle/riufc/64692
Título: Regularidade Hölder em equações elípticas na forma divergente
Autor(es): Saboya, Pedro Medeiros
Abstract: Elliptic partial differential equations are essential objects of study for modern Mathematics, particularly in the area of analysis, but also in Physics. We initially aim to study the weak solutions of such equations. For this we will deﬁne such solutions and obtain a minimum condition for them to be studied. We will analyze, before
delving into the solutions of such equations, the Hölder continuity of functions from the local growth of its integral. Then we will obtain the John-Nirenberg Inequality through the study of Diadic Cubes together with the Calderon-Zygmund Lemma. Having ﬁnished the study of the bounded mean oscillation functions, we will in fact turn to the solutions of the homogeneous equations, thus passing through the Caccioppoli Inequality and also approaching the Harmonic Functions. Using these estimates we will arrive at Hölder continuity of the solutions and their gradient, assuming the coefﬁcients of the equations are at least continuous.
Then we will approach more general coefﬁcients, and for that we will initially obtain the local limitation of the subsolutions of the equation by the approach of De Giorgi. Having done that, we will analyze both the subsolutions and the supersolutions of the equation in the homogeneous case, passing through Density and Oscillation Theorems, and ﬁnally arriving at De Giorgi’s Theorem, from which it is also possible to obtain the Hölder continuity of the solutions. Finally, we will approach the Weak Harnack Inequality and enunciate some consequences of it, among which the Moser’s Harnack Inequality, the Hölder continuity of Solutions, and the Liouville Theorem.Thu, 10 Feb 2022 00:00:00 GMThttp://www.repositorio.ufc.br/handle/riufc/646922022-02-10T00:00:00ZSemialgebraic Lipschitz equivalence of polynomial functions.
http://www.repositorio.ufc.br/handle/riufc/64519
Título: Semialgebraic Lipschitz equivalence of polynomial functions.
Autor(es): Correia, Sergio Alvarez Araujo
Abstract: We show how to determine, under fairly general conditions, whether two given β-quasi-homogeneous polynomials in two variables, with real coeﬃcients, are R-semialgebraically Lipschitz equivalent. Following the strategy used in BIRBRAIR, FERNANDES, and PANAZZOLO (2009), we ﬁrst show how to determine whether two given real polynomial functions of a single variable are Lipschitz equivalent by comparing the values and
also the multiplicities of the given polynomial functions at their critical points, and then we show how to reduce, under fairly general conditions, the problem of R-semialgebraic Lipschitz equivalence of β-quasihomogeneous polynomials in two variables, with real coeﬃcients, to the problem of Lipschitz equivalence of real polynomial functions of a single variable. As an application of our main results on R-semialgebraic Lipschitz equivalence of β-quasihomogeneous polynomials in two variables, we investigate the properties, in the context of R-semialgebraic Lipschitz equivalence, of a speciﬁc family of quasihomogeneous polynomials, which has been used before in HENRY and PARUSINSKI (2004), to show that the bi-Lipschitz equivalence of analytic function germs ( R2, 0) → ( R , 0) admits continuous moduli. As a byproduct, our conclusions show that the R-semialgebraic Lipschitz equivalence of real β-quasihomogeneous polynomials in two variables also admits continuous moduli.Thu, 08 Apr 2021 00:00:00 GMThttp://www.repositorio.ufc.br/handle/riufc/645192021-04-08T00:00:00ZO teorema de comparação do hessiano e aplicações sobre variedades completas com curvatura de Ricci nao-negativa
http://www.repositorio.ufc.br/handle/riufc/64503
Título: O teorema de comparação do hessiano e aplicações sobre variedades completas com curvatura de Ricci nao-negativa
Autor(es): Fonteles, José Nazareno Cardeal
Abstract: n this dissertation, as the title implies, we present the Hessian Comparison Theorem and Laplacian-related applications of the distance function over complete manifolds with non-negative Ricci curvature. In Chapter II, the Hessian Comparison Theorem and the Lemma of the Index, which is used in the proof of the referred theorem. On the other hand, in chapter III, five propositions are demonstrated, the first of which is the Laplacian Comparison Theorem. The second is a consequence of the first and the third generalizes the second in the sense of distributions. Finally, the last two propositions deal with the comparison with volumes, ending with the following corollary of proposition 5: A complete and non-compact Riemannian manifold M, with Ri,c(M) > 0, has infinite volume.Thu, 01 Jan 1998 00:00:00 GMThttp://www.repositorio.ufc.br/handle/riufc/645031998-01-01T00:00:00Z