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

Abstract

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 $\mathrm{div}E\sim \mathrm{dist}(x,\partial \Omega {)}^{-2-\alpha }$. 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. $\partial u/\partial n<0$, fails in the presence of such singular drift terms $E$.