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First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains. (English) Zbl 0944.58016
A class of elliptic operators \(L\) with a given representation \[ Lu=\text{div}\bigl(A (\nabla u)\bigr)+D\nabla u+\text{div} (uD')+\gamma u \] where \(A:x\to A_x\in\text{End} T_xM\) is a Borel section of the bundle \(\text{End} TM\) satisfying some supplementary restrictions, \(D\) and \(D'\) are Borel vector fields on \(M\), and \(\gamma\) is a real valued Borel function in \(M\), is considered on a complete Riemannian manifold \(M\) (or on a region \(U\) in \(\mathbb{R}^N)\). One asks, what condition of proximity near infinity (respectively near \(\partial U)\) between two operators \(L_1\) and \(L_2\) insures that their Green’s functions are equivalent in size? The answer is given in the Theorems 1 (general case) and 1’ (for bound metric), in euclidean version in 9.1 and 9.1’, similarly to L. Carleson and J. Serrin but without harmonic analysis techniques [E. B. Fabes, D. S. Jerison, and C. E. Kenig, Ann. Math., II. Ser. 119, 121-141 (1984; Zbl 0551.35024) and V. Y. Eiderman, ‘Measure and capacity of exceptional sets arising in estimations of \(\delta\)-subharmonic functions’ Potential theory, 171-177 (1992)].
Reviewer: M.Rahula (Tartu)

MSC:
58J05 Elliptic equations on manifolds, general theory
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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