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We know that the Weak Maximum Principle and Strong Maximum Principle in every PDE book,such as Theorem 3.1 and Theorem 3.5 in David Dilbarg's book. But I never see a author give a nonconstant function as a example can attain it's interior maximum (also on boundary) for the weak principle maximum. Is the example so difficult that they don't provide it ? or the weak maximum principle is the same as the strong maximum principle. In brief, I want an elliptic operator that satisfies the weak maximum principle but not the strong one.

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  • $\begingroup$ Please clarify the question: what kind of functions are considered, harmonic or something else? If it's about harmonic functions, then the strong maximum principle holds, so there aren't any examples like you mentioned. This does not mean that weak m.p. is the same as strong m.p.: they are different statements, one is easier to prove than the other. $\endgroup$ – user147263 Oct 27 '14 at 4:10
  • $\begingroup$ @WeaponofChoice: I think he wants an elliptic operator that satisfies the weak maximum principle but not the strong one. $\endgroup$ – Jose27 Oct 27 '14 at 5:27
  • $\begingroup$ I mean the general elliptic operator, I never find an example which show a nonconstant function satisfies with the condition of the weak principle( Lu≥0 L is elliptic and c≤0)can attain its interior maximum also on boundary of the domain of the functions. Jose27 has pointed out my meaning $\endgroup$ – liaoweichuan Oct 29 '14 at 6:49

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