PDE Seminar (2015/12/7): On the Gagliardo-Nirenberg inequality with magnetic field and its application to Bose-Einstein condensation

2015-12-7 16:30 - 2015-12-7 17:30
Faculty of Science Building #3, Room 309
Kazuhiro Kurata (Tokyo Metropolitan University)
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.