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

Abstract

In this article we study the existence, uniqueness, and integrability of solutions to the Dirichlet problem \(-\hbox{div}( M(x) \nabla u ) = -\hbox{div} (E(x) u) + f\) in a bounded domain of \(\mathbb{R}^N\) with \(N \ge 3\). We are particularly interested in singular \(E\) with \(\hbox{div} E \ge 0\). We start by recalling known existence results when \(|E| \in L^N\) that do not rely on the sign of \(\hbox{div} E \). Then, under the assumption that \(\hbox{div} E \ge 0\) 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 \(\hbox{div} E \sim \hbox{dist}(x, \partial \Omega)^{-2-\alpha}\). 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 \(E\). For more information see https://ejde.math.txstate.edu/Volumes/2024/13/abstr.html