In this article,we prove that the symmetric function F_n~*(x,r)=i_1+i_ 2_++i_n =r(x_1^(i^1)x_2^(i^2)... x_n^(i^n)1/r is Schur harmonic convex for x∈R^n_+and r∈N={1,2,3,...}.As its applications,some analytic inequalities are established.
In this note, we prove the partial regularity of stationary weak solutions for the Landau- Lifshitz system with general potential in four or three dimensional space. As well known, in general, since the constraint of the methods, in order to get the partial regularity of stationary weak solution of the Landau-Lifshitz system with potential, we need to add some very strongly conditions on the potential. The main difficulty caused by potential is how to find the equation satisfied by the scaling function, which breaks down the blow-up processing. We estimate directly Morrey's energy to avoid the difficulties by blowing up.
In this article, the authors study some basic properties of the so-called quasilinearadditive functions, and some applications to the special functions of quasiconformal analysis are specified.
