Journal d’Analyse Mathematique

, Volume 72, Issue 1, pp 45–92

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

Article

Abstract

Given a complete Riemannian manifoldM (or a regionU inRN) and two second-order elliptic operators L1, L2in 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:L1 andL2 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) betweenL1 andL2 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).

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© Hebrew University 1997

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité Paris SudOrsayFrance

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