| 于海燕,郑神州,张志云.拟线性次椭圆方程组在Morrey空间上的部分正则性[J].数学年刊A辑,2017,38(1):0101~116 |
| 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 |
| Received:May 13, 2014 Revised:March 03, 2016 |
| DOI：10.16205/j.cnki.cama.2017.0009 |
| 中文关键词: Subelliptic system, VMO discontinuous coefficients, Morrey spaces $L^{p,lambda}$, Partial regularity |
| 英文关键词:Subelliptic system, VMO discontinuous coefficients, Morrey spaces $L^{p,lambda}$, Partial regularity |
| 基金项目:本文受到国家自然科学基金(No.11371050)的资助. |
| Author Name | Affiliation | E-mail | | YU Haiyan | College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China
Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, China. | 12118381@bjtu.edu.cn | | ZHENG Shenzhou | Corresponding author. Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, China. | shzhzheng@bjtu.edu.cn | | ZHANG Zhiyun | Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, China. | shzhzheng@bjtu.edu.cn |
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| 中文摘要: |
| 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. |
| 英文摘要: |
| 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. |
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