Grâce à une généralisation du théorème de Brauer–Cartan–Hua, nous montrons qu’à l’exact contraire du cas extérieur, quand on dispose d’une extension galoisienne, finie, intérieure et concentrique alors aucune extension galoisienne intermédiaire stricte n’est stable par l’action du groupe de Galois.

In this paper, firstly, we generalize the definition of the bifractional Brownian motion B H,K :=B H,K ; t ≥ 0, with parameters H∈(0,1) and K∈(0,1], to the case where H is no longer a constant, but a function H(·) of the time index t of the process. We denote this new process by B H(·),K . Secondly, we study its time regularities, the local asymptotic self-similarity and the long-range dependence properties.

Dans cet article, on va s’intéresser à la classification de certains entiers naturels reliés à la combinatoire des sous-groupes de congruence du groupe modulaire. Plus précisément, on va s’intéresser ici à la notion de solutions monomiales minimales. Celles-ci sont les solutions d’une équation matricielle (apparaissant également lors de l’étude des frises de Coxeter), modulo un entier N, dont toutes les composantes sont identiques et minimales pour cette propriété. Notre objectif ici est d’étudier les entiers N pour lesquels les solutions monomiales minimales vérifiant certaines conditions fixées possèdent une propriété d’irréductibilité. En particulier, on effectuera la classification des entiers monomialement irréductibles qui sont les entiers pour lesquels toutes les solutions monomiales minimales non nulles sont irréductibles.

We consider the existence problem of conformal metrics with prescribed fractional curvature on the standard sphere S n ,n≥2. It is equivalent to solving a fractional nonlinear variational equation involving a critical nonlinearity. By studying the lack of compactness of the associated variational problem, we extend the existence results of [2] and [3] to any fractional order σ∈(0,n 2) and prove a general existence and multiplicity Theorem under an Euler–Hopf type criterion.

Let G be a minimal split Kac–Moody group over a valued field 𝒦. Motivated by the representation theory of G, we define two topologies of topological group on G, which take into account the topology on 𝒦.

This paper addresses the mathematical analysis of the ocean-atmosphere coupling problem, including Coriolis force, non-local turbulent closure and realistic nonlinear interface conditions. We introduce a 1D vertical model corresponding to a coupled Ekman boundary layer problem with non-local turbulent viscosities. The interest of this model lies in its proximity to realistic ones by considering the numerical strategies employed to take into account the turbulent scale. Well-posedness is first studied in stationary and non-stationary states considering generalized parameterized turbulent viscosities. We establish sufficient criteria on the viscosity profiles for the uniqueness of solution and find that they are not met for parameters in the order of magnitude used in ocean and atmosphere models. To identify precisely the conditions of well-posedness, we therefor establish a necessary and sufficient criterion for the stationary state. We show that there is non-uniqueness of the solution when considering typical viscosity profiles from ocean and atmosphere models. Eventually, we illustrate that non-uniqueness is produced by an inconsistency between the viscosity profile and the boundary layer parametrisation.

In this paper, we consider a nonlinear system of two parabolic equations, with a distributed control in the first equation and an odd coupling term in the second one. We prove that the nonlinear system is locally null-controllable for any arbitrary small time. The main difficulty is that the linearized system is not null-controllable. To overcome this obstacle, we extend in a nonlinear setting the strategy introduced in [18] that consists in constructing odd controls for the linear heat equation. The proof relies on three main steps. First, we obtain from the classical L 2 parabolic Carleman estimate, conjugated with maximal regularity results, a weighted L p observability inequality for the nonhomogeneous heat equation. Secondly, we perform a duality argument, close to the well-known Hilbert Uniqueness Method in a reflexive Banach setting, to prove that the heat equation perturbed by a source term is null-controllable thanks to odd controls. Finally, the nonlinearity is handled with a Schauder fixed-point argument.

The incompressible Micropolar system is given by two coupled equations: the first equation gives the evolution of the velocity field u → while the second equation gives the evolution of the microrotation field ω →. In this article we will consider regularity problems for weak solutions of this system. For this we will introduce the new notion of partial suitable solutions, which imposes a specific behavior for the velocity field u → only, and under some classical hypotheses over the pressure, we will obtain a hölderian gain for both variables u → and ω →.

We introduce the biharmonic Steklov problem on differential forms by considering suitable boundary conditions. We characterize its smallest eigenvalue and prove elementary properties of the spectrum. We obtain various estimates for the first eigenvalue, some of which involve eigenvalues of other problems such as the Dirichlet, Neumann, Robin and Steklov ones. Independently, new inequalities relating the eigenvalues of the latter problems are proved.

We investigate the regularity of the solutions to degenerate and/or singular elliptic equations. We prove the continuity of G(∇u) where u is a locally Lipschitz solution of divG(∇u)=λ∈ℝ in dimension two under some growth assumptions on G. Additionally, we establish a result that holds in any dimension, indicating that the separation between ∇u and the degeneracy set of G is continuous.