论文标题
$ w^{1,2+ \ varepsilon} $的弱解决方案的存在和唯一性，用于椭圆方程，在弱$ l^{n} $ space中漂移
Existence and uniqueness of weak solution in $W^{1,2+\varepsilon}$ for elliptic equation with drifts in weak-$L^{n}$ spaces
论文作者
论文摘要
我们考虑具有以下dirichlet问题的椭圆形方程，具有单数漂移$ \ mathbf {b} $：\ [\ [\ text {（a）} - \ propatatorName {div}（a \ nabla u）+\ operatatOrNAME+\ operatatorName {div}（div}（div}） - \ operatorName {div}（a^t \ nabla v） - \ Mathbf {b} \ cdot \ nabla v = g \ quad \ quad \ text {in}ω，\ \]，其中$ω$是$ \ \ mathbb {r} r}^n $ n $，$ n $，$ n $，$ n $，$ n $，$ n $，假设$ \ mathbf {b} \ in l^{n，\ infty}（ω）^n $在$ω$中具有非负弱差异，我们确定了$ w^{1,2+ \ varepsilon} _0（b）$ w^{1,2+ \ varepsilon} _0（ω）的弱解决方案的存在和独特性。作为一个应用程序，我们证明了$ \ bigCap_ {q <2} w^{1，q} _0（ω）$ in \ bigCap_ in \ bigCap_ {q <2} w^^^{ - 1，q}（ω）
We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$: \[ \text{(a) } -\operatorname{div}(A \nabla u)+\operatorname{div}(u\mathbf{b})=f,\quad \text{(b) } -\operatorname{div}(A^T \nabla v)-\mathbf{b} \cdot \nabla v =g \quad \text{in } Ω, \] where $Ω$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$. Assuming that $\mathbf{b}\in L^{n,\infty}(Ω)^n$ has non-negative weak divergence in $Ω$, we establish existence and uniqueness of weak solution in $W^{1,2+\varepsilon}_0(Ω)$ of the problem (b) when $A$ is bounded and uniformly elliptic. As an application, we prove unique solvability of weak solution $u$ in $\bigcap_{q<2} W^{1,q}_0(Ω)$ for the problem (a) for every $f\in \bigcap_{q<2} W^{-1,q}(Ω)$.
