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

IF 1 3区数学 Q1 MATHEMATICS

Communications on Pure and Applied Analysis Pub Date : 2022-09-09 DOI:10.3934/cpaa.2023051

Hyunseok Kim, Jisu Oh

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引用次数: 0

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

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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

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来源期刊

Communications on Pure and Applied Analysis 数学-数学

CiteScore

1.90

自引率

10.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： CPAA publishes original research papers of the highest quality in all the major areas of analysis and its applications, with a central theme on theoretical and numeric differential equations. Invited expository articles are also published from time to time. It is edited by a group of energetic leaders to guarantee the journal''s highest standard and closest link to the scientific communities.

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