论文标题
Hopf-Oleinik引理的失败因线性椭圆问题而与非阴性差异的奇异对流失败
Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence
论文作者
论文摘要
在本文中，我们研究了解决方案对DIRICHLET问题的存在，独特性和集成性$ - \ MATHRM {DIV}（M（X）\ Nabla U）= - \ MathRM {div}（E（e（x）u） + f $，在$ \ Mathbb r r^n $ a $ n \ n \ ge ge ge 3 $ 3 $中的有限域中。我们对$ \ mathrm {div} e \ ge 0 $的单数$ e $特别感兴趣。我们首先回顾$ | e |时已知的存在结果。 \在l^n $中不依赖$ \ mathrm {div} e $的符号。然后，在假设$ \ mathrm {div} e \ ge 0 $分布的假设下，我们将存在理论扩展到$ | e | \ in l^2 $。对于独特性，我们证明了在这种情况下的比较原则。最后，我们以$ ax /| x |^2 $的一点点讨论$ e $ nikular的特定情况，或以$ \ mathrm {div} e \ sim \ sim \ mathrm {dist}（x，x，\partialΩ）^{ - 2-α} $。在这些情况下，$ e $的奇异性导致$ u $ $ $消失在某个订单上。特别是，这表明Hopf-Oleinik引理，即$ \ partial u / \ partial n <0 $，在这种奇异的漂移术语$ e $的情况下失败。
In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem $-\mathrm{div}( M(x) \nabla u ) = -\mathrm{div} (E(x) u) + f$ in a bounded domain of $\mathbb R^N$ with $N \ge 3$. We are particularly interested in singular $E$ with $\mathrm{div} E \ge 0$. We start by recalling known existence results when $|E| \in L^N$ that do not rely on the sign of $\mathrm{div} E $. Then, under the assumption that $\mathrm{div} E \ge 0$ distributionally, we extend the existence theory to $|E| \in L^2$. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of $E$ singular at one point as $Ax /|x|^2$, or towards the boundary as $\mathrm{div} E \sim \mathrm{dist}(x, \partial Ω)^{-2-α}$. In these cases the singularity of $E$ leads to $u$ vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. $\partial u / \partial n < 0$, fails in the presence of such singular drift terms $E$.
