拟线性次椭圆方程组在Morrey空间上的部分正则性

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))(p2)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.

Partial Regularity in Morrey Spaces for Quasi-linear Subelliptic Systems

This paper is devoted to proving partial regularity in Morrey spaces $L^{p,\lambda}_X(\Omega,\mathbb{R}^{mN})$ with some $ p>2 $ to the $X$-gradient of weak solutions of the following quasilinear subelliptic systems $$ -X^{*}_{\alpha}(a^{\alpha\beta}_{ij}(x,u)X_{\beta}u^{j})=-X^*_{\alpha}f^{\alpha}_{i}+g_{i},\quad i=1,2,\cdots,N,\quad x\in \Omega. $$ Here $X=(X_{1},X_{2},\cdots,X_{m})$ are real smooth vector fields constructed by H\"{o}rmander''s finite rank condition, and $X^{*}_{\alpha}$ is the adjoint vector field of ~$X_{\alpha}$. In addition, the leading coefficients $a^{\alpha\beta}_{ij}(x,u)$ are allowed uniformly vanishing mean oscillation (VMO for short) dependence on the variable $x$ and uniformly continuous dependence on the variable $u$, respectively.