Ground states of two-component attractive Bose–Einstein condensates I: Existence and uniqueness

We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in ${\mathbb{R}}^2$, where the intraspecies interaction $(?a_1,?a_2)$ and the interspecies interaction ?β are both attractive, $i.e$, $a_1$, $a_2$ and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated $L^2$-critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as $\beta \nearrow {\beta }^{?}=a^{?}+\sqrt{(a^{?}?a_1)(a^{?}?a_2)}$, where $0<a_i<a^{?}:={6w6}_2^2$ ($i=1,2$) is fixed and w is the unique positive solution of $\mathrm{\Delta }w?w+w^3=0$ in ${\mathbb{R}}^2$. The semi-trivial limit behavior of ground states is tackled in the companion paper [12].     

Existence and physical uniqueness of the restricted evolution problem of the Einstein relativistic mechanics

The existence and physical uniqueness is proved for the restricted evolution problem in general relativity. Precisely, the Einstein equations for the gravitational field are analyzed in harmonic and generic adapted coordinates in order to prove the existence and uniqueness of solution in both systems of coordinates.     

On uniqueness of tangent cones for Einstein manifolds

We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent cone has a smooth cross-section.     

Existence and uniqueness for Legendre curves

We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.     

Cyclic polygons with given edge lengths: Existence and uniqueness

Let a₁, ..., a_n be positive numbers satisfying the condition that each of the a_i’s is less than the sum of the rest of them; this condition is necessary for the a_i’s to be the edge lengths of a (closed) polygon. It is proved that then there exists a unique (up to an isometry) convex cyclic polygon with edge lengths a₁, ..., a_n. On the other hand, it is shown that, without the convexity condition, there is no uniqueness—even if the signs of all central angles and the winding number are fixed, in addition to the edge lengths.     

Radial Dunkl processes: Existence,uniqueness and hitting time

We give shorter proofs of the following known results: the radial Dunkl process associated with a reduced system and a strictly positive multiplicity function is the unique strong solution for all times t of a stochastic differential equation with a singular drift, the first hitting time of the Weyl chamber by a radial Dunkl process is finite almost surely for small values of the multiplicity function. The proof of the first result allows one to give a positive answer to a conjecture announced by Gallardo–Yor while that of the second shows that the process hits almost surely the wall corresponding to the simple root with a small multiplicity value. To cite this article: N. Demni, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Existence and uniqueness of rectilinear slit maps

We consider a generalization of the parallel slit uniformization in which the angle of inclination of each image slit is assigned independently. Koebe proved that for domains of finite connectivity there is, up to a normalization, a unique rectilinear slit map achieving any given angle assignment. Koebe's theorem is partially extended to domains of infinite connectivity. A uniqueness result is shown for domains of countable connectivity and arbitrary angle assignments, and an existence result is proved for arbitrary domains under the assumption that the angle assignment is continuous and has finite range. In order to prove the existence result a new extremal length tool, called the crossing-module, is introduced. The crossing-module allows greater freedom in the family of admissible arcs than the classical module. Several results known for the module are extended to the crossing-module. A generalization of Jenkins' module condition for the parallel slit problem is given for the rectilinear slit problem in terms of the crossing-module and it is shown that rectilinear slit maps satisfying this crossing-module condition exist.

     

Quasilinear parabolic stochastic partial differential equations: Existence,uniqueness

We present a direct approach to existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone. The proof of uniqueness is very elementary, based on a new method of applying Itô’s formula for the $L^1$-norm. The proof of existence relies on a recent regularity result and is direct in the sense that it does not rely on the stochastic compactness method.     

Existence and uniqueness of algebraic function approximations

The problem of approximating a real-valued, locally analytic functions,f(x), by an algebraic function,Q(x), is considered. Existence and uniqueness theorems are obtained under fairly general conditions, including those of nonnormality.Communicated by Edward B. Saff.     

Existence and uniqueness for the Prandtl equations

Under the hypothesis of analyticity of the data with respect to the tangential variable we prove the existence and uniqueness of the mild solution of Prandtl boundary layer equation. This can be considered an improvement of the results of [8] as we do not require analyticity with respect to the normal variable.     

Existence and uniqueness of physical ground states

It is proved that if A is a bounded Hermitian operator on a probability Hilbert algebra which preserves positivity and is continuous from L² to L^p for some p > 2 then ∥ A ∥ is an eigenvalue of A. A sufficient condition is given for its multiplicity to be one. Applications are given to the proof of existence and nondegeneracy of physical ground states in quantum field theory for physical systems involving Fermions or Bosons.     

Almost contact Einstein–Weyl structures

 Almost contact Weyl manifolds are introduced: in dimension at least 5 they naturally lead to locally conformal cosymplectic spaces. We analyze them from the point of view of Weyl geometry considering in particular the case of compact Einstein–Weyl manifolds. Received: 6 July 2001/Revised version: 5 March 2002     

A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness

We scrutinize both from the physical and the analytical viewpoint the equations ruling the paramagnetic-ferromagnetic phase transition in a rigid three-dimensional body. Starting from the order structure balance, we propose a non-isothermal phase-field model which is thermodynamically consistent and accounts for variations in space and time of all fields (the temperature θ, the magnetic field vector H and the magnetization vector M). In particular, we are able to establish a well-posedness result for the resulting coupled system.