Remarks on the strong maximum principle

@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

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