## 偏微分方程式セミナー(2015/12/7): On the Gagliardo-Nirenberg inequality with magnetic field and its application to Bose-Einstein condensation, 倉田 和浩 氏

2015年 　 12月 7日 16時 30分 ～ 2015年 　 12月 7日 17時 30分

In this talk, I will consider the minimization problem associated with the Gagliardo-Nirenberg inequality with magnetic field in two space dimension:
$0<\sigma(A):=\inf \biggl\{ \frac{\|(\nabla-iA)\phi \|_2^2\|\phi\|_2^2} {\|\phi\|^4_4}; \phi\in H^1_A({\bf R^2}; {\bf C}), \phi\neq 0\biggr\}.$
Here $A(x)=(A_1(x), A_2(x))\in C^1({\bf R}^2; {\bf R}^2)$ is a magnetic vector potential and $H^1_A({\bf R}^2; {\bf C})=\{\phi\in H^1_{loc}({\bf R}^2; {\bf C}) ; (\nabla -i A) \phi \in L^2({\bf R}^2)\}$. We also use the notation $\|\phi\|_p =(\int_{{\bf R}^2} |\phi|^p\, dx)^{1/p}$ for $p\ge 1$.

I will show that $\sigma(A)$ always coincides with $\sigma(0)$. Moreover, I prove that the corresponding minimization problem does not have a minimizer if and only if the magnetic field $B(x)(=\partial_2 A_1(x)-\partial_1 A_2(x) )\not\equiv 0$. Inspired by the work of Guo and Seiringer(2014), as an application, I study an existence of the ground state of a Bose-Einstein model with an attractive interaction and its concentration phenomenon when the particle number tends to the critical number.