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

2022/11/18 by Lucio Boccardo, Boccardo, Lucio, David Gómez‐Castro +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2211.10122

openalex publication_date 2022/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem -div( M(x) ∇ u ) = -div (E(x) u) + f in a bounded domain of \mathbb R^N with N ≥ 3. We are particularly interested in singular E with div E ≥ 0. We start by recalling known existence results when |E| ∈ L^N that do not rely on the sign of div E . Then, under the assumption that div E ≥ 0 distributionally, we extend the existence theory to |E| ∈ L². 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 div E ∼ dist(x, ∂ Ω)^-2-α. 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. ∂ u / ∂ n < 0, fails in the presence of such singular drift terms E.

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