具有二阶逼近效应场多维Landau-Lifshitz方程解的极限行为

Limiting behavior of solutions to multidimensional LandauLifshitz equations with second approximation of effective field

本文给出在Rn+1到S2整体光滑映射控制下的极大值原理,通过这个原理建立具有二阶逼近效应场多维Landau-Lifshitz方程的δ-黏性上解和δ-黏性下解.利用这些δ-黏性上下解,我们建立具有二阶逼近效应场多维Landau-Lifshitz方程的黏性解并获得解的极限行为,即存在两个不相交的开子集M和N使得δ-黏性上解和δ-黏性下解分别在M内任一紧子集上趋于(0,1,0),在N内任一紧子集上趋于(0,1,0).

In this paper, by a maximum principle under global control map from R^n＋1 to S2, 5-viscosity supersolution and subsolution to multidimensional Landau-Lifshitz equations with the second approximation of effective field are constructed. Utilizing the δ-viscosity supersolution and subsolution, the viscosity solution to the equations is showed and the limiting behavior of the solution to the equations is obtained. Exactly, there exist two disjoint open subsets such that the solutions tend to the point （0 one of them and tend to the point （0,-1, 0） on arbitrary compact subsets 1, O） on arbitrary compact subsets in in the other of them, respectively.