Concentration behavior of normalized ground states for mass critical Kirchhoff equations in bounded domains

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation $$\{\begin{array}{ll}-(a+b{\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2\mathrm{d}x)\mathrm{\Delta }u+V(x)u=\mu u+{\beta }^{\ast }\mathrm{\mid }u{\mathrm{\mid }}^{\frac{8}{3}}u& in\mathrm{\Omega },\\ u=0& on\mathrm{\partial }\mathrm{\Omega },\\ {\int }_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{d}x=1,\end{array}$$ where $a\ge 0$, $b>0$, the function $V(x)$ is a trapping potential in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$, ${\beta }^{\ast }:=\frac{b}{2}\mathrm{\mid }Q{\mathrm{\mid }}_2^{\frac{8}{3}}$ and $Q$ is the unique positive radially symmetric solution of equation $-2\mathrm{\Delta }u+\frac{1}{3}u-\mathrm{\mid }u{\mathrm{\mid }}^{\frac{8}{3}}u=0.$ We consider the existence of constraint minimizers for the associated energy functional involving the parameter $a$. The minimizer corresponds to the normalized ground state of above problem, and it exists if and only if $a>0$. Moreover, when $V(x)$ attains its flattest global minimum at an inner point or only at the boundary of $\mathrm{\Omega }$, we analyze the fine limit profiles of the minimizers as $a\searrow 0$, including mass concentration at an inner point or near the boundary of $\mathrm{\Omega }$. In particular, we further establish the local uniqueness of the minimizer if it is concentrated at a unique inner point.