Atomic Parameters for the 2p53p 2[3/2]2−2p53s 2[3/2]2o2p^53p~^2[3/2]_2 - 2p^53s~^2[3/2]^o_2 Transition of Ne I relevant in nuclear physics

We calculated the magnetic dipole hyperfine interaction constants and the electric field gradients of $2p^53p~^2[3/2]_2$ and $2p^53s~^2[3/2]^o_2$ levels of Ne I by using the multiconfiguration Dirac-Hartree-Fock method. The electronic factors contributing to the isotope shifts were also estimated for the $\lambda = 614.5$ nm transition connecting these two states. Electron correlation and relativistic effects including the Breit interaction were investigated in details. Combining with recent measurements, we extracted the nuclear quadrupole moment values for $^{20}$Ne and $^{23}$Ne with a smaller uncertainty than the current available data. Isotope shifts in the $2p^53p~^2[3/2]_2 - 2p^53s~^2[3/2]^o_2$ transition based on the present calculated field- and mass-shift parameters are in good agreement with the experimental values. However, the field shifts in this transition are two or three orders of magnitude smaller than the mass shifts, making rather difficult to deduce changes in nuclear charge mean square radii. According to our theoretical predictions, we suggest to use instead transitions connecting levels arising from the $2p^53s$ configuration to the ground state, for which the normal mass shift and specific mass shift contributions counteract each other, producing relatively small mass shifts that are only one order of magnitude larger than relatively large field shifts, especially for the $2p^53s~^2[1/2]^o_1 - 2p^6~^1S_0$ transition

Interior derivative estimates and Bernstein theorem for Hessian quotient equations

In this paper, we obtain the interior derivative estimates of solutions for elliptic and parabolic Hessian quotient equations. Then we establish the Bernstein theorem for parabolic Hessian quotient equations, that is, any parabolically convex solution $u=u(x,t)\in C^{4,2}(\mathbb{R}^n\times (-\infty,0])$ for $-u_t\frac{S_n(D^2u)}{S_l(D^2u)}=1$ in $\mathbb{R}^n\times (-\infty,0]$ must be the form of $u=-mt+P(x)$ with $m>0$ being a constant and $P$ being a convex quadratic polynomial

Existence of entire solutions to the Lagrangian mean curvature equations in supercritical phase

In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the main ingredient is to construct appropriate subsolutions and supersolutions as barrier functions. We also prove a nonexistence result to show the convergence rate of the phase functions is optimal