An Abstract Form of Maximum and Anti-maximum Principles of Hopf’s Type

We consider an abstract linear elliptic boundary value problem Auylusyf F0 in a strongly ordered Banach space X. The resolvent l IyA.y1of the closed linear operator A : XªX is assumed to be strongly positive and compact for all l )l1, where l1denotes the principal eigenvalue of A. We prove that there exists a constant d ʹd f.)0 depending upon fgXq_ 04 such that yusy lIy A.y1fgX°q holds for all l g l1yd , l1.. Here, Xqs xgX : xG04 de- notes the positive cone in X with the topological interior X°q/B. We also present nearly sharp sufficient conditions for A guaranteeing independence of d )0 from f, i.e., y l IyA.y1is strongly positive for all Lg l1yd , l1.. In particular, for an elliptic Dirichlet boundary value problem, or for a strictly cooperative system of such problems, the strong maximum and boundary point principles for l )l1. yield an anti-maximum principle of Hopf’s type for l g l1yd , l1.depending upon f .: If 0FfgLp V., N-p-`, and fk0 in V, a bounded C2-domain in RN, then u-0 in V and ­ ur­n )0 on ­V whenever l g l1yd , l1..