×

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

Citations:

Zbl 0551.35024
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ancona, A., Comparaison des fonctions de Green et des mesures harmoniques pour des opérateurs elliptiques, C. R. Acad. Sci. Paris, 294, 505-508 (1982) · Zbl 0504.35037
[2] Ancona, A., On strong barriers and an inequality of Hardy for domains in ℝ^N+, J. London Math. Soc., 34, 2, 274-290 (1986) · Zbl 0629.31002 · doi:10.1112/jlms/s2-34.2.274
[3] Ancona, A., Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math., 125, 495-536 (1987) · Zbl 0652.31008 · doi:10.2307/1971409
[4] Ancona, A., Théorie du potentiel sur les graphes et les variétés, Ecole d’été de Probabilités de Saint-Flour XVIII — 1988, 5-112 (1990), Berlin: Springer-Verlag, Berlin · Zbl 0719.60074
[5] Ancona, A., Positive harmonic functions and hyperbolicity, Potential Theory, Surveys and Problems, 1-23 (1988), Berlin: Springer-Verlag, Berlin · Zbl 0677.31006 · doi:10.1007/BFb0103341
[6] Aronson, D. G., Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., 73, 890-896 (1967) · Zbl 0153.42002 · doi:10.1090/S0002-9904-1967-11830-5
[7] Berestycki, H.; Nirenberg, L.; Varadhan, S. R. S., The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math., 47, 47-92 (1994) · Zbl 0806.35129 · doi:10.1002/cpa.3160470105
[8] M. Brelot,Axiomatique des fonctions harmoniques, Les Presses de l’Université de Montréal, 1969. · Zbl 0148.10401
[9] Carleson, L., On mappings conformai at the boundary, J. Analyse Math., 19, 1-13 (1967) · Zbl 0186.13701 · doi:10.1016/0022-247X(67)90017-0
[10] Cranston, M.; Zhao, Z., Conditional transformation of drift formula and Potential Theory for 1/2δ +b(.).∇, Comm. Math. Phys., 112, 613-627 (1987) · Zbl 0647.60071 · doi:10.1007/BF01225375
[11] Dahlberg, B., On the absolute continuity of elliptic measures, Amer. J. Math., 108, 1119-1138 (1986) · Zbl 0644.35032 · doi:10.2307/2374598
[12] Fabes, E.; Kenigand, C.; Jerison, D., Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure, Ann. of Math., 119, 121-141 (1984) · Zbl 0551.35024 · doi:10.2307/2006966
[13] Fefferman, R. A.; Kenig, C.; Pipher, J., The theory of weight and the Dirichlet problem for elliptic equations, Ann. of Math., 134, 65-124 (1991) · Zbl 0770.35014 · doi:10.2307/2944333
[14] Gardiner, S. J.; Klimek, M., Convexity and subsolutions of partial differential equations, Bull. London Math. Soc., 18, 41-43 (1986) · Zbl 0562.31014 · doi:10.1112/blms/18.1.41
[15] Giaquinta, M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Math. Studies105 (1983), Princeton, N.J.: Princeton University Press, Princeton, N.J. · Zbl 0516.49003
[16] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Berlin: Springer-Verlag, Berlin · Zbl 0361.35003
[17] U. HamenstÄdt,Harmonic measures for compact negatively curved manifolds, Preprint, Bonn UniversitÄt (1995). · Zbl 0899.58031
[18] Hervé, R.-M., Recherche sur la théorie axiomatique des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, XII, 415-471 (1962)
[19] Hervé, M.; Hervé, R.-M., Les fonctions surharmoniques associées à un opérateur elliptique du second ordre à coefficients discontinus, Ann. Inst. Fourier, XIX, 305-359 (1969) · Zbl 0176.09801
[20] Hueber, H.; Sieveking, M., On the quotients of Green functions, Ann. Inst. Fourier, 32, 1, 105-118 (1982) · Zbl 0465.35028
[21] Ladyzhenskaia, O. A.; Uraltseva, N. N., Linear and Quasi-Linear Equations (1968), New York and London: Academic Press, New York and London · Zbl 0164.13002
[22] Meyers, N. G., An Lpestimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa, 17, 189-206 (1963) · Zbl 0127.31904
[23] G. Mokobodzki,Algèbre des multiplicateurs d’un espace de Dirichlet, to appear.
[24] Murata, M., Structure of positive solutions to (-δ+V)u = 0in R^n, Duke Math. J., 53, 869-943 (1986) · Zbl 0624.35023 · doi:10.1215/S0012-7094-86-05347-0
[25] Nečas, J., Les méthodes directes en théorie des opérateurs elliptiques (1967), Masson (Paris): Academia, Masson (Paris) · Zbl 1225.35003
[26] NÄkki, R.; VÄisÄlÄ, J., John disks, Exposition. Math., 9, 3-43 (1991) · Zbl 0757.30028
[27] Pinchover, Y., Sur les solutions positives d’équations elliptiques et paraboliques, C. R. Acad. Sci. Paris, 302, 447-450 (1986) · Zbl 0598.35036
[28] Pinchover, Y., On positive solutions of second order elliptic equations, stability results and classification, Duke Math. J., 57, 955-980 (1988) · Zbl 0685.35035 · doi:10.1215/S0012-7094-88-05743-2
[29] Pinchover, Y., Criticality and ground states of second order elliptic equations, J. Differential Equations, 80, 237-250 (1989) · Zbl 0697.35036 · doi:10.1016/0022-0396(89)90083-1
[30] Pinchover, Y., On the equivalence of Green functions of second order elliptic equations in R^n, Differential Integral Equations, 5, 481-490 (1992) · Zbl 0772.35015
[31] Saloff-Coste, L., Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom., 36, 417-450 (1992) · Zbl 0735.58032
[32] Saloff-Coste, L., Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, 4, 4, 429-467 (1995) · Zbl 0840.31006 · doi:10.1007/BF01053457
[33] Serrin, J., On the Harnack inequality for linear elliptic equations, J. Analyse Math., 4, 292-308 (1956) · Zbl 0070.32302 · doi:10.1007/BF02787725
[34] Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15, 1, 189-258 (1965) · Zbl 0151.15401
[35] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton, N.J.: Princeton University Press, Princeton, N.J. · Zbl 0207.13501
[36] Taylor, J. C., The Martin boundaries of equivalent sheaves, Ann. Inst. Fourier, 20, 1, 433-456 (1970) · Zbl 0185.19801
[37] Widman, K.-O., On the boundary behavior of solutions to a class of elliptic partial differential equations, Ark. Mat., 6, 485-533 (1967) · Zbl 0166.37702 · doi:10.1007/BF02591926
[38] Widman, K.-O., Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations, Math. Scand., 21, 17-37 (1967) · Zbl 0164.13101
[39] Zhao, Z., Subcriticality. positivity and gauge ability of the Schrödinger operator, Bull. Amer. Math. Soc., 23, 513-517 (1990) · Zbl 0744.35042 · doi:10.1090/S0273-0979-1990-15965-8
[40] Zhao, Z., Subcriticality and gaugeability of the Schrödinger operator, Trans. Amer. Math. Soc., 334, 75-96 (1992) · Zbl 0765.60063 · doi:10.2307/2153973
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.