Dissertations / Theses on the topic 'Liouville theorems'

This thesis which is a compendium of seven papers, focuses on the study of the semilinear elliptic equations and systems, on both bounded and unbounded domains of dimension n, most importantly the Allen-Cahn equation and the De Giorgi’s conjecture (1978). This conjecture brings together two groups of mathematicians: one specializing in nonlinear partial differential equations and another in differential geometry, more specifically on minimal surfaces and constant mean curvature surfaces. De Giorgi conjectured that the monotone and bounded solutions of the Allen-Cahn equation on the whole space of dimension n ≤ 8 must be 1-dimensional solutions. This is known to be true for n ≤ 3 and with extra (natural) assumptions for 4 ≤ n ≤ 8. Motivated by this conjecture, I have introduced two main concepts: The first concept is the “H-monotone solutions” that allows us to formulate a counterpart of the De Giorgi’s conjecture for system of equations stating that the H-monotone and bounded solutions of the gradient systems on the whole space of dimension n ≤ 8 must be 1-dimensional solutions. This seems to be in the right track to extend the De Giorgi's conjecture to systems. The second concept is the “m-Liouville theorem” for m = 0, · · · , n − 1 that allows us to formulate a counterpart of the De Giorgi’s conjecture for equations but this time for higher-dimensional solutions as opposed to 1-dimensional solutions. We use the induction idea that is to use 0-Liouville theorem (0- dimensional solutions) to prove 1-Liouville theorem (1-dimensional solutions) and then to prove (n − 1)- Liouville theorem ((n − 1)-dimensional solutions). The reason that we call this “m-Liouville theorem” is because of the great mathematician Joseph Liouville (1809-1882) who proved a classical theorem in complex analysis stating that "bounded harmonic functions on the whole space must be constant" and constants are 0-dimensional objects. 0-Liouville theorem is at the heart of this thesis and it includes various 0-Liouville theorems for various equations and system. In particular, we give a positive answer to the Henon-Lane-Emden conjecture in dimension three under an extra boundedness assumption. On the other hand, it is well known that there is a close relationship between the regularity of solutions on bounded domains and 0-Liouville theorem for related “limiting equations” on the whole space, via rescaling and blow up procedures. In this direction, we present regularity of solutions for gradient and twisted-gradient systems as well as the uniqueness results for nonlocal eigenvalue problems. The novelty here is a stability inequality for both gradient and twisted-gradient systems that gives us the chance to adjust the known techniques and ideas (for equations) to systems.

The aim of this work is twofold. The first main concern, the analytical one, is to study, using the method of gradient estimates, various Liouville-type theorems which are extensions of the classical Liouville Theorem for harmonic functions. We generalize the setting - from the Euclidean space to complete Riemannian manifolds - and the relevant operator - from the Laplacian to a general diffusion operator - and we also consider ``relaxed'' boundedness conditions (such as non-negativity, controlled growth and so on). The second main concern is geometrical, and is deeply related to the first: we prove some triviality results for Einstein warped products and quasi-Einstein manifolds studying a specific Poisson equation for a particular, and geometrically relevant, diffusion operator.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior
Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico
In this work we approach four research lines, where we began with the study of isometrically immersed hypersurfaces in a horoball. Next we studied Liouville type theorems in a complete Riemannian manifold for general operators. After we studied hypersurfaces f-minimal closed on a manifold with density, and nally we studied properly embedded minimal hypersurfaces with free boundary in a n-dimensional compact Riemannian manifold. Continuing, we obtain under a more general class operator than '-Laplacian, a Liouville type theorem for a complete Riemannian manifold, so that, prove a classication theorem for Killing graph of a foliation. Firstly, we are going to assume a weak maximum principle and that immersion is contained in a horoball, i.e., the set of bounded above Bussemann functions . We obtain an estimate for the highest quotient of r-curvatures. Moreover, under certain conditions on sectional curvature and assuming that the immersion is contained in a horoball, we forced the validity of the weak maximum principle and obtain the same estimates. Next, we establish a Choi-Wang type estimate for the rst eigenvalue of the weighter Laplacian on spaces with density in responding partially to Yau's conjecture for the rst eigenvalue weighter Laplacian for spaces with density, and moreover, we obtain an inequality Poincare type. With the estimates obtained, we establish an estimate of volume for a closed surface immersed in a space with density. Still following the study of spaces with density, we obtain a type Hientze-Karcher inequality for a compact manifold with nonempty boundary , so that, we obtain that if holds the equality than the manifold is isometric to a Euclidian ball. As consequence, we obtain under same conditions that if the f-mean curvature satisfy a bounded below than the manifold is isometric to a Euclidian ball. Finally, we obtain an estimate for the nonzero rst Steklov eigenvalue, where we are giving a answer partial to a conjecture by Fraser and Li. Moreover, as a consequence we establish an estimate for the total length of the boundary of the properly embedded minimal surfaces with free boundary in terms of its topology, thus, we proved the same when the surface is embedded in the Euclidean ball 3-dimensional.
Neste trabalho, abordamos quatro linhas de estudo, onde iniciamos com o estudo de hipersuperfcies isometricamente imersas sobre uma horobola. Em seguida estudamos Teoremas tipo Liouville para uma variedade Riemanniana completa em operadores mais gerais que o Laplaciano. Alem disso, estudamos hipersuperfcies f-mÃnimas fechadas em uma variedade com densidade e, por fim, estudamos hipersuperfÃcies mÃnimas com bordo livre, propriamente imersas em uma variedade Riemanniana compacta n-dimensional. Primeiramente, assumindo um princpio do maximo fraco e que a imersÃo està contida em uma horobola, i.e., um conjunto em que a funcÃo de Busemann à limitada superiormente, obtemos uma estimativa para o supremo do quociente das r-Ãsimas curvaturas. AlÃm disso, sob certas condiÃÃes sobre as curvaturas seccionais e assumindo que a imersÃo està contida em uma horobola, forÃamos a validade do princÃpio do mÃximo fraco e obtemos as mesmas estimativas. Prosseguindo, obtemos, para um operador mais geral que o '-Laplaciano, um teorema tipo-Liouville para uma variedade Riemanniana completa. Como aplicaÃÃo provamos um teorema de classificaÃÃo para grÃficos de Killing de uma folheaÃÃo. Em seguida, estabelecemos uma estimativa tipo Choi e Wang para o primeiro autovalor do f-Laplaciano em espaÃos com densidade, no sentido de responder parcialmente à conjectura de Yau para o primeiro autovalor do Laplaciano; alÃm disso, obtemos uma desigualdade tipo Poincarà para esse operador. Com a estimativa obtida, pudemos estabelecer uma estimativa de volume para uma superfÃcie fechada mergulhada em um espaÃo com densidade. Ainda seguindo o estudo de espaÃos com densidade, obtemos uma desigualdade tipo Heintze-Karcher para uma variedade compacta com bordo e verificamos que, se vale a igualdade, entÃo a variedade à isomÃtrica a uma bola Euclidiana. Como consequÃncia, obtemos que, nas mesmas condiÃÃes, e se a f-curvatura mÃdia satisfizer uma certa limitaÃÃo inferior, entÃo a variedade ainda à isometrica a uma bola Euclidiana. Finalmente, obtemos uma estimativa para o primeiro autovalor de Steklov, dando uma resposta parcial a uma conjectura devida a Fraser e Li. AlÃm disso, como consequÃncia, estabelecemos uma estimativa para o comprimento do bordo de uma superfÃcie mÃnima, compacta e propriamente megulhada com bordo livre em termos de sua topologia; assim, provamos o mesmo resultado quando a superfÃcie està mergulhada em uma bola Euclidiana 3-dimensional.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
In this dissertation, we study the Sturm Liouvile's theorems for the zeros of the solutions of linear differential equations of second order. These classical theorems are applied to analysis of the monotonicity of functions involving the zeros of classical orthogonal polynomials. in particular, Gegenbauer polynomials.
Neste trabalho estudamos os Teoremas de Sturm Liouville para zeros de soluções de equações diferenciais lineares de segunda ordem. Estes teoremas clássicos são aplicados para análise do crescimento e decrescimento de certas funções que envolvem os zeros de Polinômios Ortogonais Clássicos, como os Polinômios de Gegenbauer.
Mestre em Matemática

Neste trabalho são apresentados alguns conceitos básicos da Teoria de Liouville e N=1 Super Liouville, enfatizando o cálculo das funções de três pontos dessas teorias.Uma introdução a Teoria de Campos Conformes (CFT) e a Supersimetria também sao incluídas, as quais constituem ferramentas básicas da presente pesquisa.
In this dissertation we present some basic features about Liouville and N=1 Super Liouville Theory, and focus in the computation of their three point functions. Additionally, we include an introduction to Conformal Field Theories (CFT) and Supersymmetry, which are the basic tools of the present research.

A linear differential operator L is called weakly hypoelliptic if any local solution u of Lu = 0 is smooth. We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which covers all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p solution must vanish.

Recently, it has become increasingly clear that boundaries play a significant role in the understanding of the non-perturbative phase of the dynamics of strings. In this thesis we propose to study the effects of boundaries in non-critical string theory. We thus analyse boundary conformal field theories on random surfaces using the conformal gauge approach of David, Distler and Kawai. The crucial point is the choice of boundary conditions on the Liouville field. We discuss the Weyl anomaly cancellation for Polyakov's non-critical open bosonic string with Neumann, Dirichlet and free boundary conditions. Dirichlet boundary conditions on the Liouville field imply that the metric is discontinuous as the boundary is approached. We consider the semi-classical limit and argue how it singles out the free boundary conditions for the Liouville held. We define the open string susceptibility, the anomalous gravitational scaling dimensions and a new Yang-Mills Feynman mass critical exponent. Finally, we consider an application to the theory of non-critical dual membranes. We show that the strength of the leading stringy non-perturbative effects is of the order e(^-o(1/βst)), a result that mimics those found in critical string theory and in matrix models. We show how this restricts the space of consistent theories. We also identify non-critical one dimensional D-instantons as dynamical objects which exchange closed string states and calculate the order of their size. The extension to the minimal c ≤ 1 boundary conformal models is also briefly discussed.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this work, we will do some applications of the Moving Spheres method, a variant of the method of Moving Planes, in order to obtain some Liouville-type theorems and some Harnack-type inequalities in Rn, as well as in the Euclidian half space Rn +. Our study focuses on, mostly, in the article written by Yan Yan Li and Lei Zhan [32], as well as some references of the same article. We concentrate in studying some properties of positive solutions of some semilinear elliptic partial differential equations with critical exponent and giving different proofs, improvements, and extensions of some previously established Liouville-type theorems and Harnack-type inequalities.
Neste trabalho, faremos algumas aplicações do método Moving Spheres, uma variante do método Moving Planes, na obtenção de alguns teoremas tipo Liouville e de algumas desigualdades tipo Harnack em Rn, bem como no semi-espaço euclidiano Rn +. Nosso estudo se concentra, marjoritariamente, no artigo do Yan Yan Li e Lei Zhang [32], bem como algumas referências do mesmo. Nos concentramos em estudar propriedades de soluções positivas de algumas equações diferenciais parciais elípticas semilineares com expoente crítico e dar provas diversificadas, refinamentos e extensões de alguns Teoremas tipo Liouville e desigualdades tipo Harnack já estabelecidos.

This thesis is concerned with the study of qualitative properties of solutions of the minimal surface equation and of a class of prescribed mean curvature equations on complete Riemannian manifolds. We derive global gradient bounds for non-negative solutions of such equations on manifolds satisfying a uniform Ricci lower bound and we obtain Liouville-type theorems and other rigidity results on Riemannian manifolds with non-negative Ricci curvature. The proof of the aforementioned global gradient bounds for non-negative solutions u is based on the application of the maximum principle to an elliptic differential inequality satisfied by a suitable auxiliary function z=f(u,|Du|), in the spirit of Bernstein’s method of a priori estimates for nonlinear PDEs and of Yau’s proof of global gradient bounds for harmonic functions on complete Riemannian manifolds. The particular choice of the auxiliary function z parallels the one in Korevaar’s proof of a priori gradient estimates for the prescribed mean curvature equation in Euclidean space. The rigidity results obtained in the last part of the thesis include a Liouville theorem for positive solutions of the minimal surface equation on complete Riemannian manifolds with non-negative Ricci curvature, a splitting theorem for complete parabolic manifolds of non-negative sectional curvature supporting non-constant solutions with linear growth of the minimal surface equation, and a splitting theorem for domains of complete parabolic manifolds with non-negative Ricci curvature supporting non-constant solutions of overdetermined problems involving the mean curvature operator.

This thesis studies the geometry of objects from 2-dimensional statistical physics in the continuum. Chapter 1 is an introduction to Schramm-Loewner evolutions (SLE). SLEs are the canonical family of non-self-intersecting, conformally invariant random curves with a domain-Markov property. The family is indexed by a parameter, usually denoted by κ, which controls the regularity of the curve. We give the definition of the SLEκ process, and summarise the proofs of some of its properties. We give particular attention to the Rohde-Schramm theorem which, in broad terms, tells us that an SLEκ is a curve. In Chapter 2 we introduce the Gaussian free field (GFF), a conformally invariant random surface with a domain-Markov property. We explain how to couple the GFF and an SLEκ process, in particular how a GFF can be unzipped along a reverse SLEκ to produce another GFF. We also look at how the GFF is used to define Liouville quantum gravity (LQG) surfaces, and how thick points of the GFF relate to the quantum gravity measure. Chapter 3 introduces a diffusion on LQG surfaces, the Liouville Brownian motion (LBM). The main goal of the chapter is to complete an estimate given by N. Berestycki, which gives an upper bound for the Hausdor dimension of times that a γ-LBM spends in α-thick points for γ, α ∈ [0, 2). We prove the corresponding, tight, lower bound. In Chapter 4 we give a new proof of the Rohde-Schramm theorem (which tells us that an SLEκ is a curve), which is valid for all values of κ except κ = 8. Our proof uses the coupling of the reverse SLEκ with the free boundary GFF to bound the derivative of the inverse of the Loewner flow close to the origin. Our knowledge of the structure of the GFF lets us find bounds which are tight enough to ensure continuity of the SLEκ trace.

Neste trabalho buscamos caracterizar o espectro de uma classe de operadores bloco--Jacobi limitados definidos em $l^2(\\Lambda,\\mathbb{C}^L)$ ($\\Lambda: \\mathbb{Z}_+\\times\\{0,1,\\ldots,L-1\\}$ representa uma faixa de largura $L\\ge 2$ no semi--plano $\\mathbb{Z}_+^2$) e sujeitos a perturbações esparsas (no sentido que as distâncias entre as ``barreiras\'\' crescem geometricamente à medida que estas se afastam da origem) distribuídas aleatoriamente. Tais operadores são construídos a partir da soma de Kronecker de matrizes de Jacobi $J$, cada qual atuando em uma direção do espaço. Demonstramos, por meio da bloco--diagonalização do operador, que %o estudo de suas principais propriedades espectrais dependem da %se limita à caracterização da ``medida de mistura\'\' $\\frac{1}{L}\\sum_{j=0}^{L-1}\\mu_j$, $\\mu_j$ a medida espectral associada à matriz de Jacobi $J^j=J+2\\cos(2\\pi j/L)I $. Para tanto, buscamos primeiramente caracterizar cada uma das medidas $\\mu_j$, explorando e aperfeiçoando algumas técnicas bastante conhecidas no estudo de operadores esparsos unidimensionais. Demonstramos, por exemplo, que a seqüência de ângulos de Prüfer (variáveis que, juntamente com os raios de Prüfer, parametrizam as soluções da equação de autovalores) é uniformemente distribuída no intervalo $[0,\\pi)$, o %que %resultado que nos permite determinar o comportamento assintótico médio das soluções da equação de autovalores. Tal resultado, aliado às técnicas desenvolvidas por Marchetti \\textit{et. al.} em \\cite{MarWre} e a uma adaptação dos critérios de Last e Simon \\cite{LS} para operadores esparsos, nos permitem demonstrar a existência de uma transição aguda (pontual) entre os espectros singular--contínuo e puramente pontual. Empregamos em seguida os resultados de Jitomirskaya e Last presentes em \\cite{JitLast} e obtemos a dimensão Hausdorff exata associada à medida $\\mu_j$, dada por $\\alpha_j=1+\\frac{4(1-p^2)^2}{p^2(4- (\\lambda-2\\cos(2\\pi j/L))^2)}$ ($\\lambda\\in[-2,2]$), recuperando um resultado análogo obtido por Zlato\\v s em \\cite{Zla}. Por fim, adaptamos tais resultados à situação da medida de mistura associada à matriz bloco--Jacobi, obtendo $\\alpha=\\min_{j\\in\\mathcal{I}(\\lambda)}\\alpha_j$, $\\mathcal{I}(\\lambda):\\{m \\in\\{0,1,\\ldots,L-1\\}:\\lambda\\in[-2+2\\cos(2\\pi j/L),2+2\\cos(2\\pi j/L)]\\}$, como sua dimensão Hausdorff exata. Estudamos modelos idênticos com esparsidades sub e super-geométricas, obtendo na primeira situação um espectro puramente pontual (de dimensão Hausdorff nula) e na segunda um espectro puramente singular--contínuo (de dimensão Hausdorff 1). Finalmente, verificamos a existência de transição entre os espectros puramente pontual e singular--contínuo em um modelo com esparsidade super-geométrica cuja dimensão Hausdorff associada à medida espectral é nula.
In this work we attempt to caracterize the spectrum of a class of limited block--Jacobi operators defined in $l^2(\\Lambda,\\mathbb{C}^L)$ ($\\Lambda: \\mathbb{Z}_+\\times\\{0,1,\\ldots,L-1\\}$ represents a strip of width $L\\ge 2$ on the semi--plane $\\mathbb{Z}_+^2$) subject to a sparse perturbation (which means that the distance between the ``barries\'\' grow geometrically with their distance to the origin) randomly distributed. Such operators are defined as Kronecker sums of unidimensional Jacobi matrices $J$, each one acting in different directions of the space. We prove, by means of a block--diagonalization of the operator, that %the study of its most relevant spectral properties depend on %is related to the caracterization of the ``mixture measure\'\' $\\frac{1}{L}\\sum_{j=0}^{L-1}\\mu_j$, $\\mu_j$ the spectral measure of the Jacobi matrix $J^j=J+2\\cos(2\\pi j/L)I$. For this, we must characterize at first each one of the measures $\\mu_j$, exploiting and improving some well known techniques developed in the study of unidimensional sparse operators. We prove, for instance, that the sequence of Prüfer angles (variables which parametrize the solutions of the eigenvalue equation) are uniform distributed on the interval $[0,\\pi)$, a result which gives us condition to determine the average asymptotic behavior of the solutions of the eigenvalue equation. Such result, in association with the techniques developed by Marchetti \\textit{et. al.} in \\cite{MarWre} and with an adaptation of Last--Simon \\cite{LS} criteria for sparse operator, permit us to prove the existence of a sharp transition between singular continuous and pure point spectra. Following on, we use the results from Jitomirskaya--Last of \\cite{JitLast} and obtain the exact Hausdorff dimension of the measure $\\mu_j$, given by $\\alpha_j=1+\\frac{4(1-p^2)^2}{p^2(4-(\\lambda-2\\cos(2\\pi j/L))^2)}$ ($\\lambda\\in[- 2,2]$), recovering an analogous result due to Zlato\\v s in \\cite{Zla}. At last, we adapt these results to the mixture measure of the block--Jacobi matrix, obtaining $\\alpha=\\min_{j\\in\\mathcal{I}(\\lambda)}\\alpha_j$, $\\mathcal{I}(\\lambda):\\{m \\in\\{0,1,\\ldots,L-1\\}:\\lambda\\in[-2+2\\cos(2\\pi j/L),2+2\\cos(2\\pi j/L)]\\}$, as its exact Hausdorff dimension. We study as well identical models with sub and super geometric sparsities conditions, obtaining a pure point spectrum (with null Hausdorff dimension) in the first case, and a purely singular continuous spectrum (such that its Hausdorff dimension is 1) in the second. Finally, we prove the existence of a transition between pure point and singular continuous spectra in a model with sub--geometric sparsity whose Hausdorff dimension related to the spectral measure is null.

Cette thèse porte principalement sur l'étude des solutions de certaines équations aux dérivées partielles elliptiques via l'indice de Morse, y compris des solutions stables, i.e. quand l'indice de Morse est égal à zéro. Elle comporte deux parties indépendantes.Dans la première partie, sous des hypothèses sur-linéaires et sous-critiques sur f, on établit d'abord une estimation explicite de la norme L [infini] des solutions de -Δu = f(u) avec u = 0 sur le bord, via leurs indices de Morse. On propose une approche plus transparente et plus souple que le travail de Yang [1998], ce qui nous permet de traiter des non linéarités très proches de la croissance critique. Les résultats obtenus nous ont motivé de travailler sur des équations polyharmoniques (-Δ)ku = f(x; u) avec notamment k = 2 et 3. Avec des hypothèses semblables à Yang [1998] sur f et des conditions au bord convenables, on obtient pour la première fois des estimations explicites de solution des équations polyhamoniques, via l'indice de Morse. Dans la seconde partie, on considère un système de Lane-Emden-Δu = ρ(x)vp; -Δv = ρ(x)u θ ; u; v > 0; dans RN; avec 1 < p< θ et un poids radial ρ strictement positif. Nous montrons la non-existence de solution stable en petites dimensions N. Nos résultats améliorent les travaux précédents de Cowan & Fazly [2012]; Fazly [2012]; Hu [2015], et fournissent notamment des résultats du type Liouville pour solution stable, en petites dimensions N, valables pour tout 1 < ρ min(4 3 ; θ)
The main concern of this thesis deals with the study of solutions of several elliptic partial differential equations via the Morse index, including the stable solutions, i.e. when the Morse index is zero. The thesis has two independent parts. In the first part, under suplinear and subcritical assumptions on f, we establish firstly some explicit estimation for the L1 norms of solutions to -Δu = f(u) avec u = 0 on the boundary, via its Morse index. We propose an approach more transparent and easier than the work of Yang [1998], which allow us to treat some nonlinearities very close to the critical growth. These results motivated us to consider the polyharmonic equations (-Δ)ku = f(x; u) with especially k = 2 and 3. With the hypothesis on f similar to Yang [1998] and appropriate boundary conditions, we obtain for the _rst time some explicit estimations of solution via its Morse index, for the polyharmonic equations.In the second part, we consider a Lane-Emden system -Δu = ρ(x)vp; -Δv = ρ(x)u_; u; v > 0; in RN; with 1 < p< θ and a radial positive weight ρ. We prove the non-existence of stable solution in small dimension case. Our results improve the previous works Cowan & Fazly [2012]; Fazly [2012]; Hu [2015], especially we prove some general Liouville type results for stable solutions in small dimension which hold true for any 1 < ρ min(4 3 ; θ)

The monograph is devoted to the study of nonlinear first order systems in the plane where the principal term is the gradient of a positive and positively 2-homogeneous Hamiltonian (or the convex combination of two of such gradients). After some preliminaries about positively 2-homogeneous autonomous systems, some results of existence and multiplicity of T-periodic solutions are presented in case of bounded or sublinear nonlinear perturbations. Our attention is mainly focused on the occurrence of resonance phenomena, and the corresponding results rely essentially on conditions of Landesman-Lazer or Ahmad-Lazer-Paul type. The techniques used are predominantly topological, exploiting the theory of coincidence degree and the use of the Poincaré-Birkhoff fixed point theorem. At the end, other boundary conditions, including the Sturm-Liouville ones, are taken into account, giving the corresponding existence and multiplicity results in a nonresonant situation via the shooting method and topological arguments.

Cette thèse est divisée en deux parties. Dans la première partie, on considère le système de réaction-diffusion-advection (Pε), qui est un modèle d'haptotaxie, mécanisme lié à la dissémination de tumeurs cancéreuses. Le résultat principal concerne la convergence de la solution du systeme (Pε) vers la solution d'un problème à frontière libre (P0) qui est bien défini. Dans la seconde partie, on considère une classe générale d'équations elliptiques du type Hénon:−∆u = |x|^{α} f(u) dans Ω ⊂ R^N avec α > -2. On examine deux cas classiques : f(u) = e^u, |u|^{p−1} u et deux autres cas : f(u) = u^{p}_{+} puis f(u) nonlinéarité générale. En étudiant les solutions stables en dehors d'un ensemble compact (en particulier, solutions stables et solutions avec indice de Morse fini) avec différentes méthodes, on obtient des résultats de classification.

This thesis is devoted to the study of several problems arising in the field of nonlinear analysis. The work is divided in two parts: the first one concerns existence of oscillating solutions, in a suitable sense, for some nonlinear ODEs and PDEs, while the second one regards the study of qualitative properties, such as monotonicity and symmetry, for solutions to some elliptic problems in unbounded domains. Although the topics faced in this work can appear far away one from the other, the techniques employed in different chapters share several common features. In the firts part, the variational structure of the considered problems plays an essential role, and in particular we obtain existence of oscillating solutions by means of non-standard versions of the Nehari's method and of the Seifert's broken geodesics argument. In the second part, classical tools of geometric analysis, such as the moving planes method and the application of Liouville-type theorems, are used to prove 1-dimensional symmetry of solutions in different situations.

Cette thèse est divisée en six parties. La première partie est consacrée à l'étude de propriétés de Hadamard et à l'obtention de théorèmes de Liouville pour des solutions de viscosité d'équations aux dérivées partielles elliptiques complètement non-linéaires avec des termes de gradient,
This thesis is divided into six parts. The first part is devoted to prove Hadamard properties and Liouville type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations with gradient term

碩士
國立中山大學
應用數學系研究所
95
The Ambarzumyan Theorem states that for the classical Sturm-Liouville problem on $[0,1]$, if the set of Neumann eigenvalue $sigma_N={(npi)^2: nin { f N}cup { 0}}$, then the potential function $q=0$. In this thesis, we study the analogues of Ambarzumyan Theorem for the Sturm-Liouville operators on star-shaped graphs with 3 edges of different lengths. We first solve the direct problem: to find out the set of eigenvalues when $q=0$. Then we use the theory of transformation operators and Raleigh-Ritz inequality to prove the inverse problem. Following Pivovarchik''s work on star-shaped graphs of uniform lengths, we analyze the Kirchoff condition in detail to prove our theorems. In particular, we study the cases when the lengths of the 3 edges satisfy $a_1=a_2=frac{1}{2}a_3$ or $a_1=frac{1}{2}a_2=frac{1}{3}a_3$. Furthermore, we work on Neumann boundary conditions as well as Dirichlet boundary conditions. In the latter case, some assumptions about $q$ have to be made.

碩士
國立臺灣大學
數學研究所
90
We generalize the results for a scalar equation by Modica to the case of a system of equations. It is shown that if a bounded entire solution U(x) of a system of semi-linear equations satisfies the gradient bound |\nabla U|^{2}\leq 2F(U) for all x\in \Bbb R^{n}, then the Liouville theorem holds. Also we show that the inequality above holds if $F(U)$ satisfies some suitable assumptions.