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 $$ -X^{*}_{\alpha}(a^{\alpha\beta}_{ij}(x,u)X_{\beta}u^{j})=-X^*_{\alpha}f^{\alpha}_{i}+g_{i},\quad i=1,2,\cdots,N,\ x\in \Omega $$ 的弱解广义梯度$Xu$在Morrey空间$L^{p,\lambda}_X(\Omega,\mathbb{R}^{mN})\ (p>2)$上的部分正则性, 其中光滑实向量场族$X=(X_{1},X_{2},\cdots,X_{m})$满足H\"ormander 有限秩条件, $X^{\ast}_{\alpha}$是$X_{\alpha}$的共轭; 而且主项系数$a^{\alpha\beta}_{ij}(x,u)$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 $L^{p,\lambda}_X(\Omega,\mathbb{R}^{mN})$ 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.