X is called to be uniformly local convex in measure if each point in the unit sphere of x is uniformly local convex point in measure ) x 。 x称为依测度一致凸空间( - ur )是指：单位球面上的每一点都是依测度一致凸点。
Second , we introduce some new concepts such as j - viscosity solutions etc and provide some relative properties in order to study the existence and limiting behavior of the solution of the multidimensional landau - lifshitz equations . by virtue of these results we prove that there exists a smooth solution of the multidimensional landau - lifshitz equation with values in unit sphere . we show also that there are two disjoint open subsets such that the solution tends to ( 0 , 1 , 0 ) and ( 0 , - 1 . 0 ) on their arbitrary inner compact sets respectively , and to ( 0 , 0 , 1 ) somewhere in the interface which separates the two open subsets 作为应用，我们利用这些性质证明取值于三维单位球面的n维landau - lifshitz方程存在光滑解，我们还证明存在两个不相交的开子集使得这个光滑解在这两个集合之一内任一紧子集上趋于( 0 . 1 , 0 ) 、在另一集合之内任一紧子集上趋于( 0 , - 1 , 0 ) ，这个光滑解在这两个集合的界面的一些点趋于( 0 , 0 , 1 ) 。
Third , we present some exact global solutions with values in unit sphere for 2 - dimensional landau - lifshitz equations with certain initial - boundary conditions , next , we give a tuft of initial data which generate a group of exact global smooth solutions with values in the unit sphere . these solutions , as we shall prove later , form a continuum 第三、给出具一定初边值条件的二维landau - lifshitz方程的一些精确的取值于单位球面的整体光滑解，同时给出一簇初值，这簇初值使得landau - lifshitz方程有一簇精确的取值于单位球面的整体光滑解，这簇整体光滑解构成一个连续统。
