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 ${\mathbb{R}}^n$, $n\ge 2$: $$-\sum _{i,j=1}^n{a}^{ij}{D}_{ij}u+b\cdot Du+cu=f\text{ in Ω}\text{and}u=0\text{ on ∂Ω}$$ and $$-\mathrm{div}\left(ADu\right)+\mathrm{div}(ub)+cu=\mathrm{div}F\text{ in Ω}\text{and}u=0\text{ on ∂Ω},$$ 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 $\Omega$ is of class $C^1$, $\mathrm{div}A+b\in L^{n,1}(\Omega ;{\mathbb{R}}^n)$, $c\in L^{\frac{n}{2},1}(\Omega )\cap L^s(\Omega )$ for some $1<s<\frac{3}{2}$, and $c\ge 0$ in $\Omega$, then for each $f\in L^1(\Omega )$, there exists a unique weak solution in $W_0^{1,\frac{n}{n-1},\infty }(\Omega )$ 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 $L^{\frac{n}{n-1},\infty }(\Omega )$.

26 pages