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 $1<s<\frac{3}{2}$, and $c\ge0$ in $\Omega$, then for each $f\in L^1 (\Omega )$, there exists a unique weak solution in $W^{1,\frac{n}{n-1},\infty}_0 (\Omega)$ of the first problem. Moreover, under the additional condition that $\Omega$ is of class $C^{1,1}$ and $c\in L^{n,1}(\Omega)$, we show that for each $F \in L^1 (\Omega ; \mathbb{R}^n)$, the second problem has a unique very weak solution in $L^{\frac{n}{n-1},\infty}(\Omega)$.