Let X = (X1,.....,Xm) be an infinitely degenerate system of vector fields, we study the existence and regularity of multiple solutions of Dirichelt problem for a class of semi-linear infinitely degenerate elliptic operators associated with the sum of square operator Δx = ∑m(j=1) Xj* Xj.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-23T12:41:56Z No. of bitstreams: 1 arquivototal.pdf: 3952391 bytes, checksum: 515d42b8a346d93bfff74862f9c40c46 (MD5) / Made available in DSpace on 2017-08-23T12:41:56Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 3952391 bytes, checksum: 515d42b8a346d93bfff74862f9c40c46 (MD5) Previous issue date: 2016-02-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we first study some interesting generalizations of the famous inequality Sobolev to limited domains. Then, we will study the existence of positive solution for a nonlinear elliptic equation on a certain condition Neumann, by imposing certain restricitive condition in the nonlinearity. To we consider more general hypotheses we will assume conditions on the boundary of domain. / Neste trabalho, estudaremos primeiramente algumas generaliza¸c˜oes interessantes da famosa desigualdade de Sobolev para dom´ınios limitados. Em seguida, iremos estudar a existˆencia de solu¸c˜oes positivas para uma equa¸c˜ao el´ıptica n˜ao linear, sob uma certa condi¸c˜ao de Neumann e impondo algumas condi¸c˜oes restritivas sobre a n˜ao linearidade. Para considerarmos hip´oteses mais gerais, assumiremos condi¸c˜oes na fronteira do dom˜Anio.

We derive a lower bound for the Wehrl entropy in the setting of SU(1,1). For asymptotically high values of the quantum number k, this bound coincides with the analogue of the Lieb-Wehrl conjecture for SU(1,1) coherent states. The bound on the entropy is proved via a sharp norm bound. The norm bound is deduced by using an interesting identity for Fisher information of SU(1,1) coherent state transforms on the hyperbolic plane and a new family of sharp Sobolev inequalities on the hyperbolic plane. To prove the sharpness of our Sobolev inequality, we need to first prove a uniqueness theorem for solutions of a semi-linear Poisson equation (which is actually the Euler-Lagrange equation for the variational problem associated with our sharp Sobolev inequality) on the hyperbolic plane. Uniqueness theorems proved for similar semi-linear equations in the past do not apply here and the new features of our proof are of independent interest, as are some of the consequences we derive from the new family of Sobolev inequalities. We also prove Fisher information identities for the groups SU(n,1) and SU(n,n).

This thesis is dedicated to the so-called Dirichlet-to-Neumann map associated with the weighted 𝑝-Laplace operator. In Chapter 1, we begin by deriving the Dirichlet-to-Neumann map by using classical modelling and outline why it is interesting to study this boundary operator. In the remaining part of Chapter 1, we dedicate each section an overview about the content of one chapter and summarize the main results. Chapter 2 is dedicated to the Poisson problem and the inverse of the Dirichlet-to-Neumann map. Chapter 3 provides the first main application of the Dirichlet-to-Neumann map, namely, it generates a strongly continuous semigroup of contractions on the Lebesgue space 𝐿2 and this contraction can be extrapolated to a contraction on 𝐿q for all 1 ≤ 𝑞 ≤ ∞. In Chapter 4, we develop an abstract theory to establish global 𝐿𝑞-𝐿∞ regularization estimates satisfied by the semigroup generated by the negative Dirichlet-to-Neumannmap. Chapter 5 is concerned with 𝐿1 and point-wise estimates on the time-derivative of the semigroup generated by the neagtive Dirichlet-to- Neumann map, which are known in the literatur as Aronson-Bénilan type estimates. In Chapter 6, we outline the theory of 𝑗-functional and its application to evolution problems. This theory allows us to study the Dirichlet problem on general open sets Ω, and to realize the Dirichlet-to-Neumann map as an operator in 𝐿2 (𝜕Ω). In Chapter 7, we consider the limit case 𝑝 = 1, which corresponds to the Dirichlet-to-Neumann map associated with the (unweighted) 1-Laplace operator. Each chapter covers parts of the authors papers mentioned in the references.:Chapter 1 Introduction................................................... 1 1.1 Motivation-physical background ............................. 2 1.2 The Dirichlet-to-Neumann map - an analyst’s perspective . . . . . . . . . 5 1.2.1 Step1. The Dirichlet problem.......................... 5 1.2.2 Step2. The Neumann boundary operator ................ 8 2 1.3 The Dirichlet-to-Neumann map on 𝐿2 ......................... 9 1.4 The Dirichlet-to-Neumann map and Leray-Lions operators . . . . . . . . 11 1.5 The Dirichlet-to-Neumann map is a nonlocal operator . . . . . . . . . . . . 12 1.6 The Dirichlet-to-Neumann map on open sets.................... 13 1.6.1 𝑗-elliptic functionals and their 𝑗-subgradient . . . . . . . . . . . . . 13 1.6.2 The construction of a weak trace on open sets ............ 15 1.6.3 Construction of the Dirichlet-to-Neumann map . . . . . . . . . . . 17 1.7 The Poisson problem and the Neumann-to-Dirichlet map . . . . . . . . . 19 1.8 Evolution problems governed by the Dirichlet-to-Neumann map . . . 21 1.9 𝐿𝑞-𝐿∞ regularization and decay estimates...................... 27 1.10 Aronson-Bénilantypeestimates .............................. 30 1.11 The Dirichlet-to-Neumann map associated with the 1-Laplacian . . . 33 Chapter 2 The Poisson problem and the Neumann-to-Dirichlet map . . . . . . . . . . . 45 2.1 The Poisson problem........................................ 45 2.2 Preliminaries .............................................. 46 2.3 The Dirichlet problem....................................... 48 2.4 The Dirichlet-to-Neumann map............................... 51 2.5 Proof of Theorem 2.1 ....................................... 56 2.5.1 Proof of claim (1) of Theorem 2.1 ...................... 56 2.5.2 Preliminaries for the proof of claim (2) of Theorem 2.1 . . . . 58 2.5.3 Proof of claim( 2) of Theorem 2.1 ...................... 60 Chapter 3 Nonlinear elliptic-parabolic evolution problems.................... 61 3.1 Main result................................................ 61 3.2 Preliminaries .............................................. 64 3.2.1 Some function spaces................................. 64 3.2.2 Nonlinear semigroupt heory - Part I..................... 65 3.2.3 Homogeneous operators - Part I ........................ 75 2 3.3 The Dirichlet-to-Neumann map on 𝐿2 ...................... 77 3.4 The Dirichlet-to-Neumann map on 𝐿1, 𝐿𝜓 and C................ 82 3.5 Proof of Theorem 3.1 ....................................... 84 Chapter 4 𝑳𝒒-𝑳∞ regularization and decay estimates ........................ 89 4.1 Main results............................................... 89 4.2 Preliminaries .............................................. 91 4.3 Sobolev implies 𝐿𝑞 -𝐿𝑟 regularization estimates ................. 92 4.4 Extrapolation towards 𝐿1 .................................... 98 4.5 A nonlinear interpolation theorem.............................100 4.6 Extrapolation towards 𝐿∞ via interpolation of the semigroup . . . . . . 107 4.7 Proof of Theorem 4.1 .......................................115 Chapter 5 Aronson-Bénilan type estimates..................................117 5.1 Main results ...............................................117 5.2 Preliminaries ..............................................119 5.2.1 Nonlinearsemigrouptheory-PartII ....................119 5.2.2 Homogeneousaccretiveoperators ......................130 5.2.3 Homogeneous completely accretive operators . . . . . . . . . . . . 138 5.3 Proof of Theorem 5.1 .......................................141 Chapter 6 The Dirichlet-to-Neumann map on open sets ......................143 6.1 Main results ...............................................143 6.2 The 𝑗-subgradient and basic properties ........................146 6.2.1 Definition and characterisation as a classical gradient . . . . . . 146 6.2.2 Ellipticextensions ...................................151 H 6.2.3 Identification of 𝜑 ..................................152 6.2.4 The case when 𝑗 is a weakly closed operator .............155 6.3 Semigroups and invariance of convex sets ......................156 6.3.1 Positive semigroups ..................................160 6.3.2 Comparison and domination of semigroups ..............161 6.3.3 𝐿∞-contractivity and extrapolation of semigroups . . . . . . . . . 163 6.4 Application:The Dirichlet-to-Neumann map....................168 Chapter 7 The Dirichlet-to-Neumann map associated with the 1-Laplacian . . . . . 171 7.1 Preliminaries ..............................................171 7.1.1 Functions of bounded variation.........................171 7.1.2 Nonlinear semigroup theory - Part III ...................178 7.2 The Dirichlet problem for the 1-Laplace operator................180 7.3 A Robin-type problem for the 1-Laplace operator................187 7.4 Proofs of the main results....................................189 7.4.1 The Dirichlet-to-Neumann operator in 𝐿1 ................189 7.4.2 The Dirichlet-to-Neumann operator in 𝐿2 ................200 7.4.3 The Dirichlet-to-Neumann operator in 𝐿1 (continued)...........204 7.4.4 Long-timestability...................................206 Appendix A Weighted Sobolev Spaces........................................213 A.1 p-admissible weights........................................213 B Mean spaces by Lions and Peetre ................................215 B.1 The connection between mean spaces and 𝐿p spaces.............215 References . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 219 Index .............................................................227

In the modern era of high and infinite dimensional data, classical statistical methodology is often rendered inefficient and ineffective when confronted with such big data problems as arise in genomics, medical imaging, speech analysis, and many other areas of research. Many problems manifest when the practitioner is required to take into account the covariance structure of the data during his or her analysis, which takes on the form of either a high dimensional low rank matrix or a finite dimensional representation of an infinite dimensional operator acting on some underlying function space. Thus, novel methodology is required to estimate, analyze, and make inferences concerning such covariances. In this manuscript, we propose using tools from the concentration of measure literature–a theory that arose in the latter half of the 20th century from connections between geometry, probability, and functional analysis–to construct rigorous descriptive and inferential statistical methodology for covariance matrices and operators. A variety of concentration inequalities are considered, which allow for the construction of nonasymptotic dimension-free confidence sets for the unknown matrices and operators. Given such confidence sets a wide range of estimation and inferential procedures can be and are subsequently developed. For high dimensional data, we propose a method to search a concentration in- equality based confidence set using a binary search algorithm for the estimation of large sparse covariance matrices. Both sub-Gaussian and sub-exponential concentration inequalities are considered and applied to both simulated data and to a set of gene expression data from a study of small round blue-cell tumours. For infinite dimensional data, which is also referred to as functional data, we use a celebrated result, Talagrand’s concentration inequality, in the Banach space setting to construct confidence sets for covariance operators. From these confidence sets, three different inferential techniques emerge: the first is a k-sample test for equality of covariance operator; the second is a functional data classifier, which makes its decisions based on the covariance structure of the data; the third is a functional data clustering algorithm, which incorporates the concentration inequality based confidence sets into the framework of an expectation-maximization algorithm. These techniques are applied to simulated data and to speech samples from a set of spoken phoneme data. Lastly, we take a closer look at a key tool used in the construction of concentration based confidence sets: Rademacher symmetrization. The symmetrization inequality, which arises in the probability in Banach spaces literature, is shown to be connected with optimal transport theory and specifically the Wasserstein distance. This insight is used to improve the symmetrization inequality resulting in tighter concentration bounds to be used in the construction of nonasymptotic confidence sets. A variety of other applications are considered including tests for data symmetry and tightening inequalities in Banach spaces. An R package for inference on covariance operators is briefly discussed in an appendix chapter.

We investigate stability for certain geometric and functional inequalities and address the regularity of the free boundary for a problem arising in optimal transport theory. More specifically, stability estimates are obtained for the relative isoperimetric inequality inside convex cones and the Gaussian log-Sobolev inequality for a two parameter family of functions. Thereafter, away from a ``small" singular set, local C^{1,\alpha} regularity of the free boundary is achieved in the optimal partial transport problem. Furthermore, a technique is developed and implemented for estimating the Hausdorff dimension of the singular set. We conclude with a corresponding regularity theory on Riemannian manifolds. / text

Dans ce Manuscrit, nous étudions l'influence de la géométrie sur les équations de Hardy-Sobolev perturbées ou non sur toute variété Riemannienne compacte sans bord de dimension supérieure ou égale à 3. Plus précisément, dans le cas non perturbé nous démontrons que pour toute dimension de la variété strictement supérieure à, l'existence d'une solution (ou plutôt une condition suffisante d'existence) dépendra de la géométrie locale autour de la singularité. En revanche, dans le cas où la dimension est égale à 3, c'est la géométrie globale (particulièrement, la masse de la fonction de Green) de la variété qui comptera. Dans le cas d'une équation à terme perturbatif sous-critique, nous démontrons que l'existence d'une solution dépendra uniquement de la perturbation pour les grandes dimensions et qu'une interaction entre la géométrie globale de la variété et la perturbation apparaîtra en dimension 3. Enfin, nous établissons une inégalité optimale de Hardy-Sobolev Riemannienne, la variété étant avec ou sans bord, où nous démontrons que la première meilleure constante est celle des inégalités Euclidiennes et est atteinte / In this Manuscript, we investigate the influence of geometry on the Hardy-Sobolev equations on the compact Riemannian manifolds without boundary of dimension greateror equal to 3. More precisely, we prove in the non perturbative case that the existence of solutions depends only on the local geometry around the singularity when the dimension is greater or equal to 4 while it is the global geometry of the manifold when the dimension is equal to 3 that matters. In the presence of a perturbative subcritical term, we prove that the existence of solutions depends only on the perturbation when the dimension is greater or equal to 4 while an interaction between the perturbation and the global geometry appears in dimension 3. Finally, we establish an Optimal Hardy-Sobolev inequality for all compact Riemannian manifolds, with or without boundary, where we prove that the Riemannian sharp constant is the one for the Euclidean inequality and is achieved

The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of convergence towards equilibrium in the L²-distance. This result was first obtained by Bovier et. al. in 2004 relying on potential theory. The presented approach in the thesis generalizes to obtain also asymptotic sharp estimates of the constant in the logarithmic Sobolev inequality. The optimal constant in the logarithmic Sobolev inequality characterizes the convergence rate to equilibrium with respect to the relative entropy, which is a stronger distance as the L²-distance and slightly weaker than the L¹-distance. The optimal constant has here no direct spectral representation. The proof makes use of the scale separation present in the dynamics. The Eyring-Kramers formula follows as a simple corollary from the two main results of the work: The first one shows that the associated Gibbs measure restricted to a basin of attraction has a good Poincaré and logarithmic Sobolev constants providing the fast convergence of the diffusion to metastable states. The second main ingredient is a mean-difference estimate. Here a weighted transportation distance is used. It contains the main contribution to the Poincaré and logarithmic Sobolev constant, resulting from exponential long waiting times of jumps between metastable states of the diffusion.

Dans cette thèse, on a d’abord considéré une équation d'onde. On a premièrement montré que l’équation est bien-posée presque sûre par la méthode de décomposition de fréquence de Bourgain sous l’hypothèse de régularité que s > 2(p−3)/(p-1). Ensuite, nous avons réduit de cette exigence de régulation à (p-3)/(p−1) en appelant une estimation probabiliste a priori. Nous considérons également l’approximation des solutions obtenues ci-dessus par des solutions lisses et la stabilité de cette procédure d’approximation. Et nous avons conclu que l’équation est partout mal-posée dans le régime de super-critique. Nous avons considéré ensuite l’équation du faisceau quintique sur le tore 3D. Et nous avons montré que cette équation est presque sûr bien-posée globalement dans certain régimes de super-critique. Enfin, nous avons prouvé que la mesure de l’image de la mesure gaussienne sous l’application de flot de l’équation BBM généralisé satisfait une inégalité de type log-Sobolev avec une petit peu de perte de l’intégrabilité. / In this thesis, we consider a wave equation. We first showed that the equation is almost sure global well-posed via Bourgain’s high-low frequency decomposition under the regularity assumption s > 2(p−3)/(p−1). Then we lowered down this regularity requirement to be (p−3)/(p−1) by invoking a probabilistic a priori estimate. We also consider approximation of the above achieved solutions by smooth solutions and the stability of this approximating procedure. And we concluded that this equation is everywhere ill-posed in the super-critical regime. Next, we considered the quintic beam equation on 3D torus. And we showed that this equation is almost sure global well-posed in certain super-critical regime. Lastly, we proved that the image measure of the Gaussian measure under the generalized BBM flow map satisfies a log-Sobolev type inequality with a little bit loss of integrability.

Ma thèse de doctorat se concentre principalement sur le comportement en temps long des processus de Markov, les inégalités fonctionnelles et les techniques relatives. Plus spécifiquement, Je vais présenter les taux de convergence sous-exponentielle explicites des processus de Markov dans deux approches : la méthode Meyn-Tweedie et l’hypocoercivité (faible). Le document se divise en trois parties. Dans la première partie, Je vais présenter quelques résultats importants et des connaissances connexes. D’abord, un aperçu de mon domaine de recherche sera donné. La convergence exponentielle (ou sous-exponentielle) des chaînes de Markov et des processus de Markov (à temps continu) est un sujet d’actualité dans la théorie des probabilité. La méthode traditionnelle développée et popularisée par Meyn-Tweedie est largement utilisée pour ce problème. Dans la plupart des résultats, le taux de convergence n’est pas explicite, et certains d’entre eux seront brièvement présentés. De plus, la fonction de Lyapunov est cruciale dans l’approche Meyn-Tweedie, et elle est aussi liée à certaines inégalités fonctionnelles (par exemple, inégalité de Poincaré). Cette relation entre fonction de Lyapounov et inégalités fonctionnelles sera donnée avec les résultats au sens L2. En outre, pour l’exemple de l’équation cinétique de Fokker-Planck, un résultat de convergence exponentielle explicite de la solution sera introduite à la manière de Villani : l’hypocoercivité. Ces contenus sont les fondements de mon travail, et mon but est d’étudier la décroissance sous-exponentielle. La deuxième partie, fait l’objet d’un article écrit en coopération avec d’autres sur les taux de convergence sous-exponentielle explicites des processus de Markov à temps continu. Comme nous le savons, les résultats sur les taux de convergence explicites ont été donnés pour le cas exponentiel. Nous les étendons au cas sous-exponentielle par l’approche Meyn-Tweedie. La clé de la preuve est l’estimation du temps de passage dans un ensemble ”petite”, obtenue par Douc, Fort et Guillin, mais pour laquelle nous donnons une preuve plus simple. Nous utilisons aussi la construction du couplage et donnons une ergodicité sous exponentielle explicite. Enfin, nous donnons quelques applications numériques. Dans la dernière partie, mon second article traite de l’équation cinétique de Fokker-Planck. Je prolonge l’hypocoercivité à l’hypocoercivité faible qui correspond à inégalité de Poincaré faible. Grâce à cette extension, on peut obtenir le taux de convergence explicite de la solution, dans des cas sous-exponentiels. La convergence est au sens H1 et au sens L2. A la fin de ce document, j’étudie le cas de l’entropie relative comme Villani, et j’obtiens la convergence au sens de l’entropie. Enfin, Je donne deux exemples pour les potentiels qui impliquent l’inégalité de Poincaré faible ou l’inégalité de Sobolev logarithmique faible pour la mesure invariante. / My Ph.D dissertation mainly focuses on long time behavior of Markov processes, functional inequalities and related techniques. More specifically, I will present the computable sub-exponential convergence rate of the Markov process in two approaches : Meyn-Tweedie’s method and (weak) hypocoercivity. The paper consists of three parts. In the first part, I will introduce some important results and related knowledge. Firstly, overviews of my research field are given. Exponential (or subexponential) convergence of Markov chains and (continuous time) Markov processes is a hot issue in probability. The traditional method - Meyn-Tweedie’s approach is widely applied for this problem. Most of the results about convergence rate is not explicit, and some of them will be introduced briefly. In addition,Lyapunov function is crucial in Meyn-Tweendie’s aproach, and it is also related to some functional inequalities (for example, Poincar´e inequality). The relationship of them will be given with results in L2 sense. Furthermore, as a example of kinetic Fokker-Planck equation, a computable result of exponential convergence of the solution of it will be introduced in Villani’ way - hypocoercivity. These contents are foundations of my work, and my destination is to study the sub-exponential decay. In the second part, it is my article cooperated with others about subexponential convergence rate of continuous time Markov processes. As we all know, the explicit results of convergence rate is about the exponential case. We extend them to sub-exponential case in Meyn-Tweedie’s approach. The key of the proof is the estimation of the hitting time to small set which was got by Douc, Fort and Guillin, for which we also propose an alternative simpler proof. We also use coupling construction as others and give a quantitative sub-exponential ergodicity. At last, we give some calculations for examples. In the last part, my second article deal with the kinetic Fokker-Planck equation. I extend the hypocoercivity to weak hypocoercivity which correspond to weak Poincar´e inequality. Through the extension, one can get the computable rate of convergence of the solution, which is also sub-exponential case. The convergence is in H1 sense and in L2 sense. In the end of this paper, I study the relative entropy case as C.Villani, and get convergence in entropy. Finally, I give two examples for potentials that implies weak Poincar´e inequality or weak logarithmic Sobolve inequality for invarient measure.