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Boccardo, L. and Gómez Castro, David and Díaz Díaz, Jesús Ildefonso
(2022)
*Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence.*
(Unpublished)

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## 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 RN with N≥3. We are particularly interested in singular E with divE≥0. We start by recalling known existence results when |E|∈LN that do not rely on the sign of divE. Then, under the assumption that divE≥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 divE∼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 strong maximum principle fails in the presence of such singular drift terms E.

Item Type: | Article |
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Subjects: | Sciences > Mathematics > Differential equations Sciences > Mathematics > Functions |

ID Code: | 75823 |

Deposited On: | 05 Dec 2022 17:02 |

Last Modified: | 07 Dec 2022 08:15 |

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