Electron. J. Differential Equations, Vol. 2024 (2024), No. 13, pp. 116.
Failure of the HopfOleinik lemma for a linear elliptic problem with singular convection of nonnegative divergence
Lucio Boccardo, Jesus Ildefonso Diaz, David GomezCastro
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 HopfOleinik 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 GómezCastro
Inst. de Matemática Interdisciplinar
Fac. de Matemáticas, Univ. Complutense de Madrid
Spain
email: dgcastro@ucm.es

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