Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence

In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem $-\mathrm{d}\mathrm{i}\mathrm{v}(M(x)\mathrm{\nabla }u)=-\mathrm{d}\mathrm{i}\mathrm{v}(E(x)u)+f$ in a bounded domain of ${\mathbb{R}}^N$ with $N\ge 3$. We are particularly interested in singular $E$ with $\mathrm{d}\mathrm{i}\mathrm{v}E\ge 0$. We start by recalling known existence results when $\mathrm{\mid }E\mathrm{\mid }\in L^N$ that do not rely on the sign of $\mathrm{d}\mathrm{i}\mathrm{v}E$. Then, under the assumption that $\mathrm{d}\mathrm{i}\mathrm{v}E\ge 0$ distributionally, we extend the existence theory to $\mathrm{\mid }E\mathrm{\mid }\in L^2$. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of $E$ singular at one point as $Ax\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$, or towards the boundary as $\mathrm{d}\mathrm{i}\mathrm{v}E\sim \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{-2-\alpha }$. 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. $\mathrm{\partial }u\mathrm{/}\mathrm{\partial }n<0$, fails in the presence of such singular drift terms $E$.