Optimal convergence rate to the nonrelativistic limit of Chandrasekhar variational model for Neutron stars

In this paper, we consider the nonrelativistic limit of Chandrasekhar variational model for neutron stars. We show that the minimizer ${\rho }_c$ of Chandrasekhar energy $E_c(N)$ converges strongly to the minimizer ${\rho }_{\mathrm{\infty }}$ of limit energy $E_{\mathrm{\infty }}(N)$ in $L^1\cap L^{\frac{5}{3}}({\mathbb{R}}^3)$ as the speed of light $c\to \mathrm{\infty }$, this is a limit between two free boundary problems. Moreover, we develop a novel approach to obtain the convergence rates, we show that the above nonrelativistic limit has the optimal convergence rate $\frac{1}{c^2}$. For the radius $R_c$ of the compact support of ${\rho }_c(x)$ and the radius $R_{\mathrm{\infty }}$ of the compact support of ${\rho }_{\mathrm{\infty }}(x)$, we also get the optimal convergence rate $\frac{1}{c^2}$, this means that $R_{\mathrm{\infty }}-R_c=O(\frac{1}{c^2})$ as $c\to \mathrm{\infty }$. Moreover, we also obtain the optimal uniform bounds of $R_c$ and $L^{\mathrm{\infty }}$-norm of ${\rho }_c$ with respect to $N$ as $c\to \mathrm{\infty }$.