<p>We study Ginzburg–Landau equations for a complex vector order parameter to a two-component system. We discuss the existence, uniqueness, asymptotics, monotonicity and stability of solutions by extending Alama-Bronsard-Mironescu's results in a more general case.</p> / Doctor of Philosophy (PhD)

Analisamos o comportamento assintótico de uma família de equilíbrios de uma equação de reação-difusão com a condição de Neumann homegênea definido num domínio fino bidimensional com termos de reação concentradas em uma vizinhança oscilante da fronteira. Assumimos que o domínio e portanto a vizinhança oscilante se degeneram em um intervalo quando o parâmetro positivo tende a zero. O objetivo principal foi mostrar que essa família de soluções, converge para uma equação limite unidimensional, que captura a geometria e o comportamento assintótico dos conjuntos abertos onde o problema é estabelecido. De fato mostramos a continuidade da família de equilíbrios / In this work we analyze the behavior of a family of stead state solutions of a semilinear reaction-diffusion equation with homogeneous Neumann boundary condition, posed in a two-dimensional thin domain with reaction term concentrated in an narrow oscillating neighborhood. Indeed, we assume that the domain of definition of the solutions degenerates into an interval as a small parameter $\\epsilon$ goes to zero. Our main result is that this family of solutions converge to the solution of a one-dimensional limit equation capturing the geometry and oscillatory behaviour of the open sets where the problem is estabilished.

Analisamos o comportamento assintótico de uma família de equilíbrios de uma equação de reação-difusão com a condição de Neumann homegênea definido num domínio fino bidimensional com termos de reação concentradas em uma vizinhança oscilante da fronteira. Assumimos que o domínio e portanto a vizinhança oscilante se degeneram em um intervalo quando o parâmetro positivo tende a zero. O objetivo principal foi mostrar que essa família de soluções, converge para uma equação limite unidimensional, que captura a geometria e o comportamento assintótico dos conjuntos abertos onde o problema é estabelecido. De fato mostramos a continuidade da família de equilíbrios / In this work we analyze the behavior of a family of stead state solutions of a semilinear reaction-diffusion equation with homogeneous Neumann boundary condition, posed in a two-dimensional thin domain with reaction term concentrated in an narrow oscillating neighborhood. Indeed, we assume that the domain of definition of the solutions degenerates into an interval as a small parameter $\\epsilon$ goes to zero. Our main result is that this family of solutions converge to the solution of a one-dimensional limit equation capturing the geometry and oscillatory behaviour of the open sets where the problem is estabilished.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-25T17:20:25Z No. of bitstreams: 1 ANNAXSUEL ARAÚJO DE LIMA - DISSERTAÇÃO PPGMAT 2011..pdf: 581866 bytes, checksum: cc44cd422d4a48ddad0354f215805918 (MD5) / Made available in DSpace on 2018-07-25T17:20:25Z (GMT). No. of bitstreams: 1 ANNAXSUEL ARAÚJO DE LIMA - DISSERTAÇÃO PPGMAT 2011..pdf: 581866 bytes, checksum: cc44cd422d4a48ddad0354f215805918 (MD5) Previous issue date: 2011-04 / Neste trabalho, apresentamos métodos envolvendo sub e supersolução para estudar a existência de solução de certas equações elípticas. / In this work, we present methods involving sub and supersolution to study the existence of solution of certain elliptic equations.

In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H1-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, we connect these large-scale estimates with local regulartity results at the microscopic-level to achieve interior Lipschitz regularity at every scale.

We studied some semilinear elliptic equations on the entire space R^N. Our approach was variational, and the major obstacle was the breakdown in compactness due to the unboundedness of the domain. First, we considered an asymptotically linear Scltrodinger equation under the presence of a steep potential well. Using Lusternik-Schnirelmann theory, we obtained multiple solutions depending on the interplay between the linear, and nonlinear parts. We also exploited the nodal structure of the solutions. For periodic potentials, we constructed infinitely many homoclinic-type multibump solutions. This recovers the analogues result for the superlinear case. Finally, we introduced weights on the linear and nonlinear parts, and studied how their interact ion affects the local and global compactness of the problem. Our approach is based on the Caffarelli-Kohn-Nirenberg inequalities.

We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain for a second order elliptic differential operator A. The differential operator is assumed to be of divergent form and the boundary operator B is of Robin type. The boundary is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset of the boundary and control the growth of solutions near this set. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set. Moreover, we prove the completeness of root functions related to L.

Οι ασθενείς λύσεις μιας διαφορικής εξίσωσης βρίσκονται σε μια ένα προς ένα αντιστοιχία με τα κρίσιμα σημεία ενός κατάλληλου συναρτησιοειδούς. Οι χώροι Sobolev που θα αναφερθούμε εδώ, για την ύπαρξη των ασθενών λύσεων, είναι υπόχωροι Χ του Η1=Η1(M,g). Οι άλλοι χώροι Ηqk αναφέρονται διότι θα χρησιμοποιηθούν για το πέρασμα των ασθενών λύσεων σε C2 κλασικές λύσεις. / --

Diss. Uppsala : Univ., 2002.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / In this paper we provide a universal solution for continuity module in the direction of the viscosity of fully nonlinear elliptic equations considering properties of the function f integrable in different situations. Established inner estimate for the solutions of these equations based on some conditions the norm of the function f. To obtain regularity in solutions of these inhomogeneous equations and coefficients of variables we use a method of compactness, which consists essentially of approximating solutions of inhomogeneous equations for a solution of a homogeneous equation in order to "inherit" the regularity that those equations possess. / Neste trabalho fornecemos mÃdulo de continuidade universal para soluÃÃes, no sentido da viscosidade,de equaÃÃes elÃpticas totalmente nÃo lineares, considerando propriedades de integrabilidade da funÃÃo f em diferentes situaÃÃes. Estabelecemos estimativa interior para as soluÃÃes dessas equaÃÃes baseadas em algumas condiÃÃes da norma da funÃÃo f. Para se obter regularidade nas soluÃÃes dessas equacÃes nÃo homogÃneas e de coeficientes variÃveis usamos um mÃtodo de compacidade, o qual consiste, essencialmente, em aproximar soluÃÃes de equaÃÃes nÃo homogÃneas por uma soluÃÃo de uma equaÃÃo homogÃnea com o objetivo de âherdarâ a regularidade que essas equaÃÃes possuem.