The P-Landau-Lifshitz Type Equation With Stereograph Projection In Complex Plane

In this paper, we mainly focused on discussing p-Landau-Lifshitz type equation and p-Ginsburg-Landau type energy. Firstly by saturation constraint and stereograph projection, p-Landau-Lifshitz type equation is transformed into a complex differential equation.Secondly inspired by the computation of Vm and field energy calculus, we construct a p-Ginsburg-Landau type energy. Then we show that the previous complex differential equation is exactly the Euler-Lagrange equation of this p-Ginsburg-Landau type energy. This implies that when the external magnetic field disappears in ferromagnetic, the magnetic moment will stop deflecting in magnets, magnetic energy reaches the minimum value. This phenomenon conforms to the view that material structure is relatively more stable when the energy is relatively smaller.Moreover, given p> 2, we prove the existence of global minimizer for the p-Ginsburg-Landau type energy over the category of radially symmetric maps on R2 with degree d=1 at infinity. Assuming that the minimizer is radially symmetric, the p-Ginsburg-Landau type energy can be transformed into the integral of a real Lagrange function. By computation we show the convexity of the real Lagrange function with respect to the derivative of the minimizer and show the Lagrange function is bounded below. According to the Theorem 1 in session 8.2 of the book "Partial differential equation" written by Evans L.C, the energy is weakly lower semi continuous. Because of the arbitrariness of the weak convergence interval, we conclude that the limit of weak convergent function sequence is a minimizer of energy functional.Finally, we prove some corresponding property of the minimizer.