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This paper investigates the link between the Maximum Principle and the signof the (generalized) principal eigenvalue for elliptic operators in unboundeddomains. Our approach covers the cases of Dirichlet, Neumann, and (indefinite)Robin boundary conditions and treat them in a unified way. For a certain classof elliptic operators (including the class of selfadjoint operators), weestablish that the positivity of the principal eigenvalue is a necessary andsufficient condition for the validity of the Maximum Principle. If theprincipal eigenvalue is zero, no general answer holds; instead, under a naturalcondition on the domain's size at infinity, we show that the operator satisfieswhat we call the Critical Maximum Principle. We also address the question ofthe simplicity of the principal eigenvalue, and a series of counterexamples isproposed to disprove some possible misconceptions. Our main results are neweven for the more classical cases of Dirichlet boundary conditions andselfadjoint operators.

Maximum Principle and principal eigenvalue in unbounded domains under general boundary conditions