论文标题
二阶线性椭圆方程的dirichlet问题，带有$ l^{1} $ - 数据
Dirichlet problems for second order linear elliptic equations with $L^{1}$-data
论文作者
论文摘要
We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $Ω$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f \; \; \ text {in $ω$} \ quad \ text {and} \ quad u = 0 \; \; \; \ text {on $ \ partialω$} $} $} $} $} $} $ and $} - {\ rm div} \ left（a d u \ right（a d u \ right） \; \; \ text {in $ω$} \ quad \ text {and} \ quad u = 0 \; \; \; \ text {on $ \ partialω$}，$ $ $ a = [a^a^{a^{ij}] $是对称的，均匀的，均匀的，均匀的，均匀的，均匀的，均匀的eLlistic，and of Farish的均值（vmo）。本文的主要目的是研究两种问题的独特解决性。我们证明，如果$ω$是$ c^{1} $的类$ 1 <s <s <\ frac {3} {2} $和$ c \ ge0 $ in $ω$，然后对于l^1（ω）$中的每个$ f \ in $ f \ in $ w^{1，\ frac {n} {n} {n} {n-1}，\ infty}，\ infty}，\ iffty}，\ frac {n}，\ frac {n}，\ infty}，\ iffty}，\ _0（ω）$第一个问题。此外，在额外的条件下，$ω$是$ c^{1,1} $和l^{n，1}（ω）$的$ c^{1,1} $，我们表明，对于每个$ f \ in l^1（ω; \ m athbb {r}^n）$，第二个问题在$ l^{\ frac {n} {n-1}，\ infty}（ω）$。
We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $Ω$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\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 $L^{\frac{n}{n-1},\infty}(Ω)$.
