PDE Seminar (2015/12/7): On the GagliardoNirenberg inequality with magnetic field and its application to BoseEinstein condensation
 Date

2015127 16:30 
2015127 17:30
 Place
 Faculty of Science Building #3, Room 309

Speaker/Organizer
 Kazuhiro Kurata (Tokyo Metropolitan University)

 In this talk, I will consider the minimization problem associated with the GagliardoNirenberg inequality with magnetic field in two space dimension:
\[ 0<\sigma(A):=\inf \biggl\{ \frac{\(\nablaiA)\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 BoseEinstein model with an attractive interaction and its concentration phenomenon when the particle number tends to the critical number.