On the stability of 2D dipolar Bose-Einstein condensates

We study the existence of energy minimizers for a Bose-Einstein condensate with dipole-dipole interactions, tightly confined to a plane. The problem is critical in that the kinetic energy and the (partially attractive) interaction energy behave the same under mass-preserving scalings of the wave-function. We obtain a sharp criterion for the existence of ground states, involving the optimal constant of a certain generalized Gagliardo-Nirenberg inequality

Mean field equations on tori: existence and uniqueness of evenly symmetric blow-up solutions

We are concerned with the blow-up analysis of mean field equations. It has been proven in [6] that solutions blowing-up at the same non-degenerate blow-up set are unique. On the other hand, the authors in [18] show that solutions with a degenerate blow-up set are in general non-unique. In this paper we first prove that evenly symmetric solutions on a flat torus with a degenerate two-point blow-up set are unique. In the second part of the paper we complete the analysis by proving the existence of such blow-up solutions by using a Lyapunov-Schmidt reduction method. Moreover, we deduce that all evenly symmetric blow-up solutions come from one-point blow-up solutions of the mean field equation on a "half" torus

The Nonexistence of Vortices for Rotating Bose-Einstein Condensates in Non-Radially Symmetric Traps

We consider ground states of rotating Bose-Einstein condensates with attractive interactions in non-radially harmonic traps $V(x)=x_1^2+\Lambda ^2x_2^2$, where $0<\Lambda \not =1$ and $x=(x_1, x_2)\in R^2$. For any fixed rotational velocity $0\le \Omega <\Omega ^*:=2\min \{1, \Lambda\}$, it is known that ground states exist if and only if $a<a^*$ for some critical constant $00$ denotes the product for the number of particles times the absolute value of the scattering length. We analyze the asymptotic expansions of ground states as $a\nearrow a^*$, which display the visible effect of $\Omega$ on ground states. As a byproduct, we further prove that ground states do not have any vortex in the region $R(a):=\{x\in R^2:\,|x|\le C (a^*-a)^{-\frac{1}{12}}\}$ as $a\nearrow a^*$ for some constant $C>0$, which is independent of $0<a<a^*$.Comment: 29 pages, this is the revised versio