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

TL;DR

This paper investigates the existence, uniqueness, and boundary behavior of solutions to a linear elliptic PDE with singular convection terms, revealing conditions under which the Hopf-Oleinik lemma fails due to singular drift effects.

Contribution

It extends the existence theory for elliptic equations with singular divergence-free convection fields and demonstrates the failure of the Hopf-Oleinik lemma in these cases.

Findings

Existence results for $|E| \, \in L^2$ when $\mathrm{div} E \ge 0$

Comparison principle established for uniqueness

Failure of the Hopf-Oleinik lemma with singular drift terms

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 $E$.