@inproceedings{Brezis2006RemarksOT,
title={Remarks on the strong maximum principle},
author={H{\"a}ım Brezis and Augusto C. Ponce},
year={2006}
}

Häım Brezis, Augusto C. Ponce

Published 2006

The strong maximum principle asserts that if u is smooth, u ≥ 0 and −∆u ≥ 0 in a connected domain Ω ⊂ RN , then either u ≡ 0 or u > 0 in Ω. The same conclusion holds when −∆ is replaced by −∆ + a(x) with a ∈ Lp(Ω), p > N2 (this is a consequence of Harnack’s inequality; see e.g. Stampacchia [1], and also Trudinger [1], Corollary 5.3). Another formulation of the same fact says that if u(x0) = 0 for some point x0 ∈ Ω, then u ≡ 0 in Ω. A similar conclusion fails, however, when a 6∈ Lp(Ω), for any p… CONTINUE READING