Electron. J. Differential Equations, Vol. 2024 (2024), No. 13, pp. 1-16.

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

Lucio Boccardo, Jesus Ildefonso Diaz, David Gomez-Castro

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 October 31, 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.
DOI: 10.58997/ejde.2024.13

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Lucio Boccardo
Istituto Lombardo & Sapienza Università di Roma
Italy
email: boccardo@mat.uniroma1.it
Jesús Ildefonso Díaz
Inst. de Matemática Interdisciplinar
Fac. de Matemáticas, Univ. Complutense de Madrid
Spain email: jidiaz@ucm.es
David Gómez-Castro
Inst. de Matemática Interdisciplinar
Fac. de Matemáticas, Univ. Complutense de Madrid
Spain
email: dgcastro@ucm.es

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