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In this thesis we study three stochastic partial differential equations (SPDE) that arise as stochastic gradient flows via the fluctuation-dissipation principle. For the first equation we establish a finer regularity statement based on a generalized Taylor expansion which is inspired by the theory of rough paths. The second equation is the thin-film equation with thermal noise which is a singular SPDE. In order to circumvent the issue of dealing with possible renormalization, we discretize the gradient flow structure of the deterministic thin-film equation. Choosing a specific discretization of the metric tensor, we resdiscover a well-known discretization of the thin-film equation introduced by Grün and Rumpf that satisfies a discrete entropy estimate. By proving a stochastic entropy estimate in this discrete setting, we obtain positivity of the scheme in the case of no-slip boundary conditions. Moreover, we analyze the associated rate functional and perform numerical experiments which suggest that the scheme converges. The third equation is the massive $\varphi^4_2$-model on the torus which is also a singular SPDE. In the spirit of Bakry and Émery, we obtain a gradient bound on the Markov semigroup. The proof relies on an $L^2$-estimate for the linearization of the equation. Due to the required renormalization, we use a stopping time argument in order to ensure stochastic integrability of the random constant in the estimate. A postprocessing of this estimate yields an even sharper gradient bound. As a corollary, for large enough mass, we establish a local spectral gap inequality which by ergodicity yields a spectral gap inequality for the $\varphi^4_2$- measure.

[pt] Nesta tese de Doutorado estudamos uma classe de equações diferenciais parciais de segunda ordem, uniformemente elípticas, completamente não-lineares na forma não-divergência, com crescimento superlinear no gradiente e coeficientes mensuráveis. Para equações com crescimento quadrático, provamos que ocorre multiplicidade de soluções quando o operador não é coercivo e investigamos o comportamento qualitativo dos contínuos de soluções obtidos para uma família parametrizada de problemas. Para isso, estendemos a regularidade e as estimativas C1, alfa, de Caffarelli-Swiech-Winter para equações com crescimento, no máximo quadrático, no gradiente, mostrando que as soluções são continuamente diferenciáveis até o bordo. Além disso, mostramos estimativas a priori na norma uniforme via técnicas puramente não-lineares na forma não-divergência, entre elas desigualdades do tipo Harnack e o princípio do máximo forte de Vázquez para equações de nosso tipo. / [en] In this Ph.D. thesis we study a class of uniformly elliptic partial differential equations of second order in fully nonlinear nondivergence form with superlinear growth in the gradient and measurable coefficients. For equations with quadratic growth, we prove that multiplicity of solutions occurs when the operator is not coercive. We investigate the qualitative behavior of the continuums of solutions obtained for a parameterized family of problems. For this, we extend the Caffarelli-Swiech-Winter C1, alpha, regularity estimates to equations with at most quadratic gradient growth, showing that the solutions are continuously differentiable up to the boundary. Furthermore, we show a priori estimates in the uniform norm using purely nonlinear techniques in the nondivergence form, such as Harnack type inequalities and a Vázquez’s strong maximum principle for equations of our type.

Dans cette thèse, on s'intéresse à deux problèmes provenant de l'étude mathématique des fluides incompressibles visqueux : la propagation de la régularité tangentielle et le mouvement d'une surface libre.La première question concerne plus particulièrement l'étude qualitative de l'évolution de quantités thermodynamiques telles que la température dans l'équation de Boussinesq sans diffusion et la densité dans le système de Navier-Stokes non homogène. Typiquement, on suppose que ces deux quantités sont, à l'instant initial, discontinues le long d'une interface à régularité h"oldérienne. Comme conséquence de résultats de propagation de régularité tangentielle pour le champ de vitesses, on établit que la régularité des interfaces persiste pour tout temps aussi bien en dimension deux d'espace, qu'en dimension supérieure (avec condition de petitesse). Notre approche suit celle du travail de J.-Y. Chemin dans les années 90 pour le problème des poches de tourbillon dans les fluides incompressiblesparfaits.Dans le cas présent, outre cette hypothèse de régularité tangentielle, nous n'avons besoin que d'une régularité critique sur le champ de vitesses.La démonstration repose sur le calcul para-différentiel et les espaces de multiplicateurs.Dans la dernière partie de la thèse, on considère le problème à frontière libre pour le système de Navier-Stokes incompressible à deux phases. Ce système permet de décrire l'évolution d'un mélange de deux fluides non miscibles tels que l'huile et l'eau par exemple. Différents cas de figure sont étudiés : le cas d'un réservoir borné, d'une goutte ou d'une rivière à profondeur finie.On établit l'existence et l'unicité à temps petit pour ce problème. Notre démonstration repose fortement sur des propriétés de régularité maximale parabolique de type $L_p$-$L_q / This thesis is dedicated to two different problems in the mathematical study of the viscous incompressible fluids: the persistence of tangential regularity and the motion of a free surface.The first problem concerns the study of the qualitative properties of some thermodynamical quantities in incompressible fluid models, such as the temperature for Boussinesq system with no diffusion and the density for the non-homogeneous Navier-Stokes system. Typically, we assume those two quantities to be initially piecewise constant along an interface with H"older regularity.As a consequence of stability of certain directional smoothness of the velocity field, we establish that the regularity of the interfaces persist globally with respect to time both in the two dimensional and higher dimensional cases (under some smallness condition). Our strategy is borrowed from the pioneering works by J.-Y.Chemin in 1990s on the vortex patch problem for ideal fluids.Let us emphasize that, apart from the directional regularity, we only impose rough (critical) regularity on the velocity field. The proof requires tools from para-differential calculus and multiplier space theory.In the last part of this thesis, we are concerned with the free boundary value problem for two-phase density-dependent Navier-Stokes system.This model is used to describe the motion of two immiscible liquids, like the oil and the water. Such mixture may occur in different situations, such as in a fixed bounded container, in a moving bounded droplet or in a river with finite depth. We establish the short time well-posedness for this problem. Our result strongly relies on the $L_p$-$L_q$ maximal regularity theoryfor parabolic equations

The first contribution of this thesis is a new regularity theorem for time harmonic Maxwell's equations with less than Lipschitz complex anisotropic coefficients. By using the L^p theory for elliptic equations, it is possible to prove H¹ and Hölder regularity results, provided that the coefficients are W^1,p for some p = 3. This improves previous regularity results, where the assumption W^1,∞ for the coefficients was believed to be optimal. The method can be easily extended to the case of bi-anisotropic materials, for which a separate approach turns out to be unnecessary. The second focus of this work is the boundary control of the Helmholtz and Maxwell equations to enforce local constraints inside the domain. More precisely, we look for suitable boundary conditions such that the corresponding solutions and their derivatives satisfy certain local non-zero constraints. Complex geometric optics solutions can be used to construct such illuminations, but are impractical for several reasons. We propose a constructive approach to this problem based on the use of multiple frequencies. The suitable boundary conditions are explicitly constructed and give the desired constraints, provided that a finite number of frequencies, given a priori, are chosen in a fixed range. This method is based on the holomorphicity of the solutions with respect to the frequency and on the regularity theory for the PDE under consideration. This theory finds applications to several hybrid imaging inverse problems, where the unknown coefficients have to be imaged from internal measurements. In order to perform the reconstruction, we often need to find suitable boundary conditions such that the corresponding solutions satisfy certain non-zero constraints, depending on the particular problem under consideration. The multiple frequency approach introduced in this thesis represents a valid alternative to the use of complex geometric optics solutions to construct such boundary conditions. Several examples are discussed.

L'objectif de cette thèse est d'étudier le comportement local d'une mesure de probabilité, notamment au travers d'un indice de régularité locale. Dans la première partie, nous établissons la normalité asymptotique de l'estimateur des kn plus proches voisins de la densité et de l'histogramme. Dans la deuxième, nous définissons un estimateur du mode sous des hypothèses affaiblies. Nous montrons que l'indice de régularité intervient dans ces deux problèmes. Enfin, nous construisons dans une troisième partie différents estimateurs pour l'indice de régularité à partir d'estimateurs de la fonction de répartition, dont nous réalisons une revue bibliographique.

Cette thèse a deux objectifs principaux. En premier lieu, on voudrait offrir un aperçu des connaissances académiques actuelles, tant théoriques qu'empiriques, des processus d’accommodation linguistique entre interlocuteurs, au sens général, et des caractéristiques rythmiques de la langue espagnole, en particulier. En second lieu, on présente deux études empiriques conçues pour analyser l’influence de la régularité rythmique au niveau des phrases et de l’arrangement phonologique sur les processus d’accommodation linguistique. Dans l'ensemble, les données rassemblées dans cette thèse indiquent que les phrases avec un rythme régulier, disposées en groupes accentuels, produisent une plus grande ressemblance entre les hispanophones en matière de rythme et de l’étendue de la F0, par rapport aux phrases avec un rythme irrégulier et aux phrases disposées en pieds accentuels. De plus, certains faits connus concernant les femmes ayant une moyenne de F0 supérieure, une étendue de F0 plus large, et un débit de parole plus lent quant aux hommes ont également été observés au cours de la première expérience. En outre, une valeur inférieure de la moyenne de F0 et une étendue de F0 plus étroite ont été observées lors de l’utilisation de phrases avec un rythme régulier et de phrases disposées en groupes accentuels, par rapport aux conditions expérimentales opposées. En ce qui concerne la tâche de perception, les phrases des dyades mixtes ont été notées de manière plus similaire les unes aux autres par rapport aux phrases des dyades de femmes et des dyades d’hommes (parmi d'autres résultats trouvés). / This thesis has two principal aims. In the first place, we would like to offer an overview of the current academic knowledge, both theoretical and empirical, of the processes of linguistic accommodation between interlocutors, in a general sense, and of the rhythmic characteristics of the Spanish language, in particular. In the second place, we present two empirical studies designed to analyze the influence of sentence-level rhythmic regularity and phonological phrasing on the processes of linguistic accommodation. Taken together, the data gathered in this thesis indicate that regular rhythmic sentences, arranged in accentual groups, generate a greater amount of resemblance between Spanish speakers in terms of rhythm and F0 range, with respect to irregular rhythmic sentences and sentences arranged in accentual feet. Moreover, a lower value of F0 mean and a narrower F0 range were observed during the use of both regular rhythmic sentences and sentences arranged in accentual groups compared to the opposite conditions. In addition, some known facts related to women having a higher F0 mean, a wider F0 range, and speaking slower regarding men were also found during the first experiment. As for the perceptual task, sentences of mixed dyads were rated more similar to each other with respect to sentences of female only and male only dyads (among other patterns found).

Nesta tese abordaremos dois tipos de problemas relacionados aos célebres Teorema da Fibração de Milnor e Teorema da Fibração de Milnor-Lê para o caso real com valores críticos não isolados. Primeiramente, asseguramos fibrações do tipo Milnor-Lê para F : (Xm, 0) &rarr; (Yn, 0), germe de aplicação subanalítico com X e Y espaços subanalíticos sobre C \\ {0} uma curva subanalítica conexa em Y e sobre um subespaço analítico suave W &sub; Y de dimensão p, n &ge; p &ge; 2, sob algumas condições. Em particular, mostramos a existência das fibrações sobre o discriminantes de germe de aplicações subanalíticos, caso esse ainda não estudado na literatura, normalmente o conjunto dos valores críticos são desconsiderados. Finalizando nossa análise da categoria subanalítica, certificamos que existe a fibração de Milnor-Lê para f : (X, 0) &rarr;(Rp, 0), com dimensão de X maior que p &ge; 2, subanalítica e X subanalítico com valores críticos não isolados, definindo d-regularidade. Abordamos estes problemas utilizando resultados de campos de vetores rugosos. Em uma segunda etapa apresentamos um novo critério necessário e suficiente para verificar a importante propriedade de transversalidade de um germe de aplicação real f de classe Cl, l &ge; 1. Fazendo uso também de uma recente ferramenta desenvolvida, a D-regularidade, verificamos condições para a existência das fibrações do germe de aplicação &Psi; F, X : (Cn, 0) &rarr; (C, 0) não holomorfo, dado por &Psi; (z, z&#772;) = &Sigma;nj=1 kjtjzj a<sub<jzj bj, aj, bj &ge; 0 com aj = bj para pelo menos um j e aj &ne; bj para ao menos um j, com j = 1, ... , n. Observamos que &Psi;F, X são polinômios homogêneos pesados mistos com R+ -ação. Consideramos &Psi;F, X : (R2n ,0) &rarr; (R2, 0) germe de aplicação analítico real. Estudamos a topologia dessas fibrações nos reais, constatando que o discriminante tem dimensão 1 e por isso tem ambas as fibrações conhecidas. Por fim exibimos um homeomorfismo entre as fibras dos valores regulares e dos valores críticos. / In this thesis two types of problems related to the famous Milnor Fibration Theorem and Milnor-Lê Fibration Theorem for the real case with non-isolated critical values will be addressed. Primarily, we assure the fibrations of type Milnor-Lê for the germ F : (X, 0) &rarr; (Y, 0) subanalytic with X and Y subanalytic spaces on C \\ {0} a subanalytic connected curve in Y and over a smooth analytical subspace W &sub; Y of dimension p, n &ge p &ge; 2, under some conditions. In particular, we show the existence of the fibrations about the discriminants of subanalytical map-germ, if this not been studied in the literature, usually the set of critical values are disregarded. Finalizing our analysis of this subanalytic category, we certify that there exist the fibrations of type Milnor-Lê to f : (X, 0) &rarr; (Rp, 0), with dimension of X greater than p &ge; 2, subanalytic and X subanalytic with non-isolated critical values, setting d -regularity. We address these problems using results of the rugose vector fields. In a second part, we present a new necessary and sufficient criterion to verify the important transversality property of a real map-germ f of class Cl, l &ge; 1. Using a recent developed tool, D-regularity, we verify conditions for the existence of the fibrations of map-germ &Psi; F, X : (Cn, 0) &rarr; (C, 0) non holomorphic, given by &Psi; (z, z&#772;) = &Sigma;nj=1 kjtjzj ajzb<sup<j, aj, bj &ge; 0 with aj = bj for at least one j and aj &ne; bj for at leeast one j = 1, ..., n. We note that &Psi; F, X are mixed weighted homogeneous polynomials with R+-action. We consider &Psi;F, X : (R2n, 0) &rarr; (R2, 0) real analytic map-germ. We studied the topology of these fibrations, noting that the discriminant has dimension 1 and therefore has both the fibrations known. Lastly we show a homeomorphism between the fibers of the regular values and the critical values for a case special this family.

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