Remarks on the strong maximum principle

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 > N2 . For instance, u(x) = |x| 2 satisfies −∆u+ a(x)u = 0 in B1 with a = 2N |x|2 6∈ L N/2(Ω). If u vanishes on a larger set, one may still hope to conclude, under some weaker condition on a, that u ≡ 0 in Ω. Such a result was obtained by Benilan-Brezis [1, Appendix D] (with a contribution by R. Jensen) in the case where a ∈ L1(Ω) and suppu is a compact subset of Ω. Their maximum principle has been further extended by Ancona [1], who proved Theorem 1 below. We recall that a function v : Ω→ R is quasicontinuous if there exists a sequence of open subsets (ωn) of Ω such that v|Ω\ωn is continuous ∀n ≥ 1 and capωn → 0 as n→∞, where capωn denotes the H1-capacity of ωn.