拟线性次椭圆方程组在Morrey空间上的部分正则性

Partial Regularity in Morrey Spaces for Quasi-linear Subelliptic Systems

证明了拟线性次椭圆方程组-X_α~*(a_(ij)^(αβ)(x,u)X_βu^j)=-X_α~*f_i~α+g_i,i=1,2,…,N,x∈Ω的弱解广义梯度Xu在Morrey空间L_x^(p,λ)(Ω,R^(mN))(p>2)上的部分正则性,其中光滑实向量场族X=(X_1,X_2,…,X_m)满足H(o|¨)rmander有限秩条件,X_α~*是X_α的共轭;而且主项系数a_(ij)^(αβ)(x,u)关于x一致VMO(Vanishing Mean Oscillation的缩写,消失平均震荡)间断,且关于u为一致连续.

This paper is devoted to proving partial regularity in Morrey spaces Lxp,λ （Ω, RmN）with some p 〉 2 to the X-gradient of weak solutions of the following quasilinear subelliptic systems -Xα＊αijαβ（x,u）Xβuj）=-Xα＊fiα＋gi,i=1,2,…,N,x∈Ω Here X = （X1, X2, … , Xm） are real smooth vector fields constructed by HSrmander＇s finite rank condition, and Xα＊ is the adjoint vector field of Xα. In addition, the leading coefficients aijαβ （x, u） are allowed uniformly vanishing mean oscillation （VMO for short） dependence on the variable x and uniformly continuous dependence on the variable u, respectively.