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Showing 51 to 56 of 56 (0.08 seconds)| 51 |
Desigualdades universais para autovalores do polidrifting laplaciano em dominios compactos do R^n e S^n / Universal bounds for eigenvalues of the poli-drifting laplaciano operators ìn compact domains in the R^n and S^nPereira, Rosane Gomes 08 March 2016 (has links)
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Previous issue date: 2016-03-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we study eigenvalues of poly-drifting laplacian on compact
Riemannian manifolds with boundary (possibly empty). Here, we bring a
universal inequality for the eigenvalues of the poly-drifting operator on compact
domains in an Euclidean spaceRn. Besides,weintroduce universal inequalities for
eigenvalues of poly-drifting operator on compact domains in a unit n-sphere Sn.
We give an universal inequality for lower order eigenvalues of the poly-drifting
operator inRn and Sn. Moreover, we prove an universal inequality type Ashbaugh
and Benguria for the drifting Laplacian on Riemannian manifold immersed in an
unit sphere or a projective space. Let
be a bounded domain in a n-dimensional
Euclidean space Rn. We study eigenvalues of an eigenvalue problem of a system
of elliptic equations of the drifting laplacian
8>><>>:
L u+ (r(divu)r divu) = ¯ u; in
;
uj@
= 0
Estimates for eigenvalues of the above eigenvalue problem are obtained. Furthermore,
a universal inequality for lower order eigenvalues of the problem is also
derived. / Neste trabalho, estudamos autovalores do polidrifting Laplaciano em variedades
Riemannianas compactas com fronteira (possivelmente vazia). Aqui, trazemos
uma desigualdade universal para autovalores do polidrifting operador em
domínios compactos no espaço Euclidiano Rn. Além disso, introduzimos desigualdades
universais para autovalores do polidrifting operador em domínios
compactos na n-esfera unitária Sn. Fornecemos uma estimativa para autovalores
de ordem inferior do polidrifting operador emRn e Sn. Mais ainda, provamos uma
desigualdade universal do tipo Ashbaugh-Benguria para o drifting Laplacianoem
variedades Riemannianas imersas em uma esfera unitária ou no espaço projetivo.
Seja
um domínio limitado no n-dimensional espaço Euclidiano Rn. Estudamos
autovalores de um problema de autovalores de um sistema de equações elípticas
do drifting Laplaciano
8>><>>:
L u+ (r(divu)r divu) = ¯ u; in
;
uj@
= 0
Estimativas para autovalores do problema de autovalores acima são obtidas. Além
disso, uma desigualdade universal de ordem inferior também é encontrada.
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Two Problems in non-linear PDE’s with Phase TransitionsJonsson, Karl January 2018 (has links)
This thesis is in the field of non-linear partial differential equations (PDE), focusing on problems which show some type of phase-transition. A single phase Hele-Shaw flow models a Newtoninan fluid which is being injected in the space between two narrowly separated parallel planes. The time evolution of the space that the fluid occupies can be modelled by a semi-linear PDE. This is a problem within the field of free boundary problems. In the multi-phase problem we consider the time-evolution of a system of phases which interact according to the principle that the joint boundary which emerges when two phases meet is fixed for all future times. The problem is handled by introducing a parameterized equation which is regularized and penalized. The penalization is non-local in time and tracks the history of the system, penalizing the joint support of two different phases in space-time. The main result in the first paper is the existence theory of a weak solution to the parameterized equations in a Bochner space using the implicit function theorem. The family of solutions to the parameterized problem is uniformly bounded allowing us to extract a weakly convergent subsequence for the case when the penalization tends to infinity. The second problem deals with a parameterized highly oscillatory quasi-linear elliptic equation in divergence form. As the regularization parameter tends to zero the equation gets a jump in the conductivity which occur at the level set of a locally periodic function, the obstacle. As the oscillations in the problem data increases the solution to the equation experiences high frequency jumps in the conductivity, resulting in the corresponding solutions showing an effective global behaviour. The global behavior is related to the so called homogenized solution. We show that the parameterized equation has a weak solution in a Sobolev space and derive bounds on the solutions used in the analysis for the case when the regularization is lost. Surprisingly, the limiting problem in this case includes an extra term describing the interaction between the solution and the obstacle, not appearing in the case when obstacle is the zero level-set. The oscillatory nature of the problem makes standard numerical algorithms computationally expensive, since the global domain needs to be resolved on the micro scale. We develop a multi scale method for this problem based on the heterogeneous multiscale method (HMM) framework and using a finite element (FE) approach to capture the macroscopic variations of the solutions at a significantly lower cost. We numerically investigate the effect of the obstacle on the homogenized solution, finding empirical proof that certain choices of obstacles make the limiting problem have a form structurally different from that of the parameterized problem. / <p>QC 20180222</p>
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[pt] TEORIA DE REGULARIDADE PARA MODELOS COMPLETAMENTE NÃO-LINEARES / [en] TOWARDS A REGULARITY THEORY FOR FULLY NONLINEAR MODELSPEDRA DARICLEA SANTOS ANDRADE 28 December 2020 (has links)
[pt] Neste trabalho examinamos equações completamente não-lineares em dois contextos distintos. A princípio, estudamos jogos de campo médio completamente não-lineares. Aqui, examinamos ganhos de regularidade para as soluções do problema, existência de soluções, resultados de relaxação e aspectos particulares de um example explícito. A segunda metade da tese dedica-se à regularidade ótima das soluções de um modelo completamente não-linear que degenera-se com respeito ao gradiente das soluções. A pergunta fundamental subjacente a ambos os tópicos diz respeito aos efeitos da elipticidade sobre propriedades intrínsecas das soluções de equações não-lineares. Mais precisamente, no caso dos jogos de campo médio, a elipticidade parece magnificada pelos efeitos do acoplamento, enquanto no caso dos problemas degenerados, esta quantidade colapsa em sub-regiões do domínio, dando origem a delicados fenômenos. Nossa análise inclui um
breve contexto da inserção do trabalho. / [en] In this thesis, we examine fully nonlinear problems in two distinct contexts. The first part of our work focuses on fully nonlinear mean-field games. In this context, we examine gains of regularity, the existence of solutions, relaxation results, and particular aspects of a one-dimensional problem. The second half of the thesis concerns a (sharp) regularity theory for fully nonlinear equations degenerating with respect to the gradient of the solutions. The fundamental question underlying both topics regards the effects of ellipticity on the intrinsic properties of solutions to nonlinear equations. To be more precise, in the case of mean-field game systems, ellipticity seems to be magnified through the coupling structure. On the other hand, in the degenerate setting, ellipticity collapses, giving rise to intricate regularity phenomena. Our analysis is preceded by some context on both topics.
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Équations polyharmoniques sur les variétés et études asymptotiques dans une équation de Hardy-Sobolev / Some Polyharmonic equations on Manifolds and Blow-up Analysis of a Hardy-Sobolev equationMazumdar, Saikat 27 June 2016 (has links)
Ce mémoire est divisé en deux parties : Partie 1 : Nous obtenons des résultats d'existence pour des problèmes au limite mettant en jeu des opérateurs polyharmoniques conformément invariants. Nous nous plaçons indifféremment dans le cas d'une variété riemannienne avec ou sans bord. En particulier, nous montrons que la meilleure constante de Sobolev sur les variétés est exactement la constante euclidienne. En conséquence, nous montrons l'existence d'une solution d'énergie minimale lorsque la fonctionnelle descend en-dessous d'un seuil quantifié. Puis nous montrons l'existence de solutions de haute énergie en utilisant la méthode topologique de Coron. Nous généralisons la décomposition des suites de Palais-Smale comme somme de bulles sur une variété avec ou sans bord : il s'agit d'un résultat dans l'esprit du célèbre théorème de Struwe en 1984. Nous obtenons aussi une version du lemme de compacité-concentration de Pierre-Louis Lions sur les variétés. Partie 2 : Dans cette partie, nous effectuons une analyse de blow-up pour une équation de Hardy-Sobolev à croissance critique et à singularité évanescente au bord. En supposant que l'équation limite n'admet pas de solution minimisante, nous étudions le comportement asymptotique d’une suite de solutions de l'équation perturbée. Ici, la perturbation est la singularité à l'origine. Dans un premier temps, nous obtenons un contrôle ponctuel optimal de la suite de solutions. Dans un second temps, nous obtenons des informations précises sur le point d'explosion en utilisant une identité de Pohozaev / This memoir can be divided into two parts: Part 1: In this part we obtain some existence results for conformally invariant polyharmonic boundary value problems on a compact Riemannian manifold with or without boundary. In particular we show that the best constant of the Sobolev embedding on manifolds is same as the euclidean one, and as a consequence prove the existence of minimum energy solutions when the energy functionnal goes below a quantified threshold. Next we show the existence of high energy solution using the topological method of Coron. We generalize the decomposition of Palais Smale sequences as a sum of bubble on manifolds with or without boundary, a result in the spirit of Struwe's celebrated 1984 result and also an extension of PL Lions concentration compactness result on manifolds. Part2: In this part we do a blow-up analysis of the nonlinear elliptic Hardy-Sobolev equation with critical growth and vanishing boundary singularity. We assume that our equation does not admit minimising solutions, and study the asymptotic behaviour of a sequence of solution to the perturbed equation. Here the perturbation is the singularity at the origin. First we obtain optimal pointwise controlon the sequence and then obtain more precise informations on the localization of the blow-up point using the Pohozaev identity
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Theory and Numerics for Shape Optimization in Superconductivity / Theorie und Numerik für ein Formoptimierungsproblem aus der SupraleitungHeese, Harald 21 July 2006 (has links)
No description available.
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Théorie non linéaire du potentiel et équations quasilinéaires avec données mesures / Nonlinear potential theory and quasilinear equations with measure dataNguyen, Quoc-Hung 25 September 2014 (has links)
Cette thèse concerne l’existence et la régularité de solutions d’équations non-linéaires elliptiques, d’équations paraboliques et d’équations de Hesse avec mesures, et les critères de l’existence de solutions grandes d’équations elliptiques et paraboliques non-linéaires. / This thesis is concerned to the existence and regularity of solutions to nonlinear elliptic, parabolic and Hessian equations with measure, and criteria for the existence of large solutions to some nonlinear elliptic and parabolic equations.
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