We investigate a model corresponding to the experiments for a two-dimensional rotating Bose-Einstein condensate. It consists in minimizing a Gross-Pitaevskii functional defined in R 2 under the unit mass constraint. We estimate the critical rotational speed 1 for vortex existence in the bulk of the condensate and we give some fundamental energy estimates for velocities close to 1 .

We continue the analysis started in [14] on a model describing a two-dimensional rotating Bose-Einstein condensate. This model consists in minimizing under the unit mass constraint, a Gross-Pitaevskii energy defined in R 2 . In this contribution, we estimate the critical rotational speeds Ω d for having exactly d vortices in the bulk of the condensate and we determine their topological charge and their precise location. Our approach relies on asymptotic energy expansion techniques developed by Serfaty [20][21][22] for the Ginzburg-Landau energy of superconductivity in the high κ limit.

We study a class of symmetric critical points in a variational 2D Landau -de Gennes model where the state of nematic liquid crystals is described by symmetric traceless 3×3 matrices. These critical points play the role of topological point defects carrying a degree k 2 for a nonzero integer k. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when k = ±1, 0.

We investigate stability properties of the radially symmetric solution corresponding to the vortex defect (so called "melting hedgehog") in the framework of the Landau -de Gennes model of nematic liquid crystals. We prove local stability of the melting hedgehog under arbitrary Q-tensor valued perturbations in the temperature regime near the critical supercooling temperature. As a consequence of our method, we also rediscover the loss of stability of the vortex defect in the deep nematic regime.

We study a variational model from micromagnetics involving a nonlocal Ginzburg-Landau type energy for S 1 -valued vector fields. These vector fields form domain walls, called Néel walls, that correspond to one-dimensional transitions between two directions within the unit circle S 1 . Due to the nonlocality of the energy, a Néel wall is a two length scale object, comprising a core and two logarithmically decaying tails. Our aim is to determine the energy differences leading to repulsion or attraction between Néel walls. In contrast to the usual Ginzburg-Landau vortices, we obtain a renormalised energy for Néel walls that shows both a tail-tail interaction and a core-tail interaction. This is a novel feature for Ginzburg-Landau type energies that entails attraction between Néel walls of the same sign and repulsion between Néel walls of opposite signs.

The magnetization ripple is a microstructure formed by the magnetization in a thinfilm ferromagnet. It is triggered by the random orientation of the grains in the polycrystalline material. In an approximation of the micromagnetic model, which is sketched in this paper, this leads to a nonlocal (and strongly anisotropic) elliptic equation in two dimensions with white noise as a right hand side. However, like in singular Stochastic PDE, this right hand side is too rough for the non-linearity in the equation. In order to develop a small-date well-posedness theory, we take inspiration from the recent roughpath approach to singular SPDE. To this aim, we develop a Schauder theory for the non-standard symbol |k 1 | 3 + k 2 2 .

Abstract. We study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for S 1 -valued maps m (the magnetization) of two variables x :We are interested in the behavior of minimizers as ε → 0. They are expected to be S 1 -valued maps m of vanishing distributional divergence, ∇ · m = 0, so that appropriate boundary conditions enforce line discontinuities. For finite ε > 0, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel walls have a line energy density of the order 1/|ln ε|. One of the main results is that the boundedness of {|ln ε|E ε (m ε )} implies the compactness of {m ε } ε↓0 , so that indeed the limits m will be S 1 -valued and weakly divergence-free. Moreover, we show the optimality of the 1-d Néel wall under 2-d perturbations as ε ↓ 0.

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