Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 13, pp. 1-16.

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

Authors: Lucio Boccardo (Univ. di Roma, Italy)
         Jesus Ildefonso Diaz (Univ. Complutense de Madrid, Spain)
         David Gomez-Castro (Univ. Complutense de Madrid, Spain)

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 $E$.

Submitted November 29, 2023. Published January 31, 2024.
Math Subject Classifications: 35J25, 35J75, 35B50, 35B60.
Key Words: Linear elliptic equation; convection with singularity on the boundary; strong maximum principle; flat solutions.