Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-19T16:50:04.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators

Published online by Cambridge University Press:  22 January 2016

Alano Ancona
Alano Ancona*
Affiliation:
Département de Mathématiques, Bâtiment 425, Université Paris-Sud, 91405 Orsay, France, ancona@matups.math.u-psud.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M be a manifold and let L be a sufficiently smooth second order elliptic operator in M such that (M, L) is a transient pair. It is first shown that if L is symmetric with respect to some density in M, there exists a positive L-harmonic function in M which dominates L-Green’s function at infinity. Other classes of elliptic operators are investigated and examples are constructed showing that this property may fail if the symmetry assumption is removed. Another part of the paper deals with the existence of critical points for certain L-harmonic functions with periodicity properties. A class of small perturbations of second order elliptic operators is also described.

Keywords

31C1231C3560J1060J60
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 165 , March 2002 , pp. 123 - 158
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

References

[An1] Ancona, A., Une propriété de la compactification de Martin d’un domaine euclidien, Ann. Inst. Fourier (Grenoble), 29, 4 (1979), 71–90.Google Scholar
[An2] Ancona, A., Negatively curved manifolds, elliptic operators and the Martin boundary, Ann. of Math., 125 (1987), 495–536.Google Scholar
[An3] Ancona, A., Théorie du Potentiel sur les graphes et les variétés, Ecole d’été de Probabilités de Saint-Flour XVIII 1988, Lecture Notes in Math. 1427, Springer-Verlag (1990), pp. 5–112.Google Scholar
[An4] Ancona, A., First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains, J. d’Analyse Math., 72 (1997), 45–92.Google Scholar
[An5] Ancona, A., Green’s functions, generalized first eigenvalues and perturbations of diffusions or Markov chains, Random walks and discrete Potential Theory (M. Picardello and W. Woess, eds.), Symposia Mathematica XXXIX, Cambridge Univ. Press (1999), pp. 1–25.Google Scholar
[Aro] Aronszajn, N., A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., (9) 36 (1957), 235–249.Google Scholar
[Bre] Brelot, M., Axiomatique des fonctions harmoniques, Les Presses de l’Universitè de Montréal, 1969.Google Scholar
[Cho] Choquet, G., Diamètre tranfini et comparaison de diverses capacités, Séminaire de Théorie du Potentiel, faculté des Sciences de Paris, année 1958–59 (1960), exposé n°4.Google Scholar
[Duf] Duffin, R. J., Discrete potential theory, Duke Math. J., 20 (1953), 233–251.Google Scholar
[Ev] Evans, G. C., Potentials and positively infinite singularities of harmonic functions, Monatsch. Math. Phys., 43 (1936), 419–424.Google Scholar
[Fek] Fekete, M., Uber die Verteilung des Wurzeln bei geweissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Zeitschr., 17 (1923), 228–249.Google Scholar
[Her] Hervé, R.-M., Recherche sur la théorie axiomatique des fonctions surharmoniques et du Potentiel, Ann. Inst. Fourier, XII (1962), 415–471.Google Scholar
[H-H] Hervé, M. and Hervé, R.-M., Les fonctions surharmoniques associées à un opérateur elliptique du second ordre à coefficients discontinus, Ann. Inst. Fourier, XIX (1969), 305–359.Google Scholar
[Hör] Hörmander, L., The analysis of linear partial differential operators III, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1985.Google Scholar
[I-M] Ito, K. and McKean, H. P., Potentials and the random walk, Illinois J. Math., 4 (1960), 119–132.Google Scholar
[K-V] Kalton, N. J., Verbitsky, I. E., Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc., 351 (9) (1999), 3441–3497.Google Scholar
[Mok] Mokobodzki, G., Cônes de potentiels et noyaux subordonnés, Potential Theory (C.I.M.E., I Ciclo, Stresa, 1969), Edizioni Cremonese, Roma (1970), pp. 207–248.Google Scholar
[Mu] Murata, M., Structure of positive solutions to (-Δ + V)u = 0 in Rn , Duke Math. J., 53 (1986), 869–943.Google Scholar
[Naï] Naïm, L., Sur le rôle de la frontière de R. S. Martin dans la théorie du Potentiel, Annales de l’institut Fourier, 7 (1957), 183–281.Google Scholar
[Ni1] Ninomiya, N., Sur le principe de continuité dans la théorie du Potentiel, J. Inst. Polytech. Osaka City Univ. Ser. A. 8 (1957), 51–56.Google Scholar
[Ni2] Ninomiya, N., A note on transfinite diameter, Kōdai Math. Sem. rep., 27, 3 (March, 1976), 300–307.Google Scholar
[Owh] Owhadi, H., Private communications (1999).Google Scholar
[Pi1] Pinchover, Y., Criticality and ground states for second-order elliptic equations, J. Differential Equations, 80 (1989), 237–250.Google Scholar
[Pi2] Owhadi, H., Private communications (September, 1998 and March, 1999).Google Scholar
[Pi3] Owhadi, H., Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations, Math. Ann., 314 (1999), 555–590.Google Scholar
[Pi4] Owhadi, H., On the maximum and anti-maximum principles, Differential Equations and Mathematical Physics (R. Weikard and G. Weinstein, eds.), Symposia Mathematica XXXIX, International Press, Cambridge, MA (to appear).Google Scholar
[Sta] Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1) (1965), 189–258.Google Scholar