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

Abstract

We consider an abstract linear elliptic boundary value problemAu−λu=−f≤0 in a strongly ordered Banach spaceX. The resolvent (λI−A)−1of the closed linear operatorA:X→Xis assumed to be strongly positive and compact for all λ>λ1, where λ1denotes the principal eigenvalue ofA. We prove that there exists a constant δ≡δ(f)>0 depending uponf∈X+\\{0} such that −u=−(λI−A)−1f∈X+holds for all λ∈(λ1−δ,λ1). Here,X+={x∈X:x≥0} denotes the positive cone inXwith the topological interiorX+≠∅. We also present nearly sharp sufficient conditions forAguaranteeing independence of δ>0 fromf, i.e., −(λI−A)−1is strongly positive for all Λ∈(λ1−δ,λ1). 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 λ>λ1) yield an anti-maximum principle of Hopf's type (for λ∈(λ1−δ,λ1) depending uponf): If 0≤f∈Lp(Ω),N<p<∞, andf≢0 in Ω, a boundedC2-domain in RN, thenu<0 in Ω and ∂u/∂ν>0 on ∂Ω whenever λ∈(λ1−δ,λ1).