This talk begins with a discusion of the classical minimum principle of Hopf and the boundary lemma of Hopf—Oleinik to infer from them the generalized minimum principle of Protter—Weinberger. Then, a technical device of Protter and Weinberger is polished and sharpened to get a fundamental theorem on classification of supersolutions which provides with the theorem of characterization of the strong maximum principle of Amann and Molina-Meyer together with the speaker. Finally, some important applications of this theorem are discussed. The talk adopts the general patterns of Chapters 1, 2, 6 and 7 of the book on Elliptic Operators of the speaker.

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