Dirichlet problems for second order linear elliptic equations with $ L^{1} $-data

Abstract

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $\Omega$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f \;\;\text{ in $\Omega$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial \Omega$} $$ and $$ - {\rm div} \left( A D u \right) + {\rm div}(ub) + cu = {\rm div} F \;\;\text{ in $\Omega$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial \Omega$} , $$ where $A=[a^{ij}]$ is symmetric, uniformly elliptic, and of vanishing mean oscillation (VMO). The main purposes of this paper is to study unique solvability for both problems with $L^1$-data. We prove that if $\Omega$ is of class $C^{1}$, $ {\rm div} A + b\in L^{n,1}(\Omega;\mathbb{R}^n)$, $c\in L^{\frac{n}{2},1}(\Omega) \cap L^s(\Omega)$ for some $1

查看原文

分享

微信好友 朋友圈 QQ好友 复制链接

本刊更多论文

具有$L^{1}$数据的二阶线性椭圆型方程的Dirichlet问题

在$\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f \;\;\text{ in $\Omega$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial \Omega$} $$和$$ - {\rm div} \left( A D u \right) + {\rm div}(ub) + cu = {\rm div} F \;\;\text{ in $\Omega$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial \Omega$} , $$的有界区域$\Omega$上，考虑二阶线性椭圆方程的非散度和散度形式的Dirichlet问题，其中$A=[a^{ij}]$是对称的，均匀椭圆的，并且具有消失的平均振荡(VMO)。本文的主要目的是研究$L^1$ -数据下这两个问题的唯一可解性。我们证明了如果$\Omega$是$C^{1}$, $ {\rm div} A + b\in L^{n,1}(\Omega;\mathbb{R}^n)$, $c\in L^{\frac{n}{2},1}(\Omega) \cap L^s(\Omega)$对于$1

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文 去求助

相关文献