, Volume 72, Issue 1, pp 45-92

First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains

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Abstract

Given a complete Riemannian manifoldM (or a regionU inR N ) and two second-order elliptic operators L1, L2 in M (resp.U, conditions, mainly in terms of proximity near infinity (resp. near ∂U) between these operators, are found which imply that their Green’s functions are equivalent in size. For the case of a complete manifold with a given reference pointO the conditions are as follows:L 1 andL 2 are weakly coercive and locally well-behaved, there is an integrable and nonincreasing positive function Ф on [0, ∞[ such that the “distance” (to be defined) betweenL 1 andL 2 in each ballB(x, 1 ) ⊂M is less than Ф(d(x, O)). At the same time a continuity property of the bottom of the spectrum of such elliptic operators is proved. Generalizations are discussed. Applications to the domain case lead to Dini-type criteria for Lipschitz domains (or, more generally, Hölder-type domains).