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

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $Ω$ 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 $Ω$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial Ω$}$ and $- {\rm div} \left( A D u \right) + {\rm div}(ub) + cu = {\rm div} F \;\;\text{ in $Ω$} \quad \text{and} \quad u=0 \;\;\text{ on $\partial Ω$} ,$ 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 $Ω$ is of class $C^{1}$, ${\rm div} A + b\in L^{n,1}(Ω;\mathbb{R}^n)$, $c\in L^{\frac{n}{2},1}(Ω) \cap L^s(Ω)$ for some $1<s<\frac{3}{2}$, and $c\ge0$ in $Ω$, then for each $f\in L^1 (Ω)$, there exists a unique weak solution in $W^{1,\frac{n}{n-1},\infty}_0 (Ω)$ of the first problem. Moreover, under the additional condition that $Ω$ is of class $C^{1,1}$ and $c\in L^{n,1}(Ω)$, we show that for each $F \in L^1 (Ω; \mathbb{R}^n)$, the second problem has a unique very weak solution in $L^{\frac{n}{n-1},\infty}(Ω)$.

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