On the principal eigenvalue of elliptic operators in ℝNand applications

Journal of the European Mathematical Society

Volumn 8, Issue 2, 2006, Pages 195-215

  Berestycki, Henri ^a   Rossi, Luca ^b  

^a ÉCOLE DES HAUTES ÉTUDES EN SCIENCES SOCIALES   (France)

^b SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

Elliptic operators; Generalized principal eigenvalue in n; Limit periodic operators; Principal eigenvalue

Indexed keywords

EID: 33745678495     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/47     Document Type: Article

References (25)

1. N, spectral results and extensions to elliptic operators on Riemannian manifolds. In: Proc. Internat. Conf. on Differential Equations, I. Knowles and R. Lewis (eds.), North-Holland Math. Stud. 92, North-Holland, Amsterdam, 7-17 (1984) Zbl 0564.35033 MR 0799327
2. Avellaneda, M., Lin, F.-H.: Un théorème de Liouville pour des équations elliptiques à coefficients périodiques. C. R. Acad. Sci. Paris Sér. I Math. 309, 245-250 (1989) Zbl 069135022 MR 1010728
3. Berestycki, H.: Le nombre de solutions de certains problèmes semi-linéaires elliptiques. J. Funct. Anal. 40, 1-29 (1981) Zbl 0452.35038 MR 0607588
4. Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: I - Influence of periodic heterogeneous environment on species persistence. J. Math. Biology 51, 75-113 (2005) Zbl 1066.92047
5. Berestycki, H., Hamel, F., Roques, L.: Analysis of the periodically fragmented environment model: II - Biological invasions and pulsating travelling fronts. J. Math. Pures Appl. (9) 84, 1101-1146 (2005) Zbl pre02228673 MR 2155900
6. Berestycki, H., Hamel, F., Rossi, L.: Liouville-type results for semilinear elliptic equations in unbounded domains ,Annali di Matematica Pura ed Applicata,186,3,469-507
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 MR 1258192
8. Bochner, S.: A new approach to almost periodicity. Proc. Nat. Acad. Sci. USA 48, 2039-2043 (1963) Zbl 0112.31401 MR 0145283
9. Engländer, J., Kyprianou, A. E.: Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32, 78-99 (2004) Zbl 1056.60083 MR 2040776
10. dwith spatially dependent branching. Ann. Probab. 27, 684-730 (1999) Zbl 0979.60078 MR 1698955
11. Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983) Zbl 0562.35001 MR 0737190
12. Hansen, L. P., Scheinkman, J. A.: Long term risk: an operator approach. Preprint.
13. Kolmogorov, A. N., Petrovskiǐ, I. G., Piskunov, N. S.: Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. d'Etat à Moscou (Byul. Moskov. Gos. Univ.), Série Internat. A1, 1-26 (1937) Zbl 0018.32106
14. Krasnosel'skij, M. A., Lifshits, Je. A., Sobolev, A. V.: Positive Linear Systems. The Method of Positive Operators, Heldermann, Berlin (1989) Zbl 0674.47036 MR 1038527
15. Krein, M. G., Rutman, M. A.: Linear operators leaving invariant a cone in a Banach space. Uspekhi Mat. Nauk. 3, no. 1, 3-95 (1948) (in Russian);
16. English transl., in Amer. Math. Soc. Transl. 1950, no. 26 MR 0038008
17. Krylov, N. V., Safonov, M. V.: An estimate of the probability that a diffusion process hits a set of positive measure. Soviet Math. Dokl. 20, 253-255 (1979) Zbl 0459.60067
18. Krylov, N. V., Safonov, M. V.: A certain property of solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR 44, 161-175 (1980) Zbl 0464.35035 MR 0563790
19. Kuchment, P., Pinchover, Y.: Integral representations and Liouville theorems for solutions of periodic elliptic equations. J. Funct. Anal. 181, 402-446 (2001) Zbl 0986.35028 MR 1821702
20. Moser, J., Struwe, M.: On a Liouville-type theorem for linear and nonlinear elliptic differential equations on a torus. Bol. Soc. Brasil. Mat. (N.S.) 23, 1-20 (1992) Zbl 0787.35028 MR 1203171
21. Pinsky, R. G.: Positive Harmonic Functions and Diffusion. Cambridge Univ. Press (1995) Zbl 0858.31001 MR 1326606
22. Pinsky, R. G.: Second order elliptic operators with periodic coefficients: criticality theory, perturbations, and positive harmonic functions. J. Funct. Anal. 129, 80-107 (1995) Zbl 0826.35030 MR 1322643
23. Pinsky, R. G.: Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann. Probab. 24, 237-267 (1996) Zbl 0854.60087 MR 1387634
24. Protter, M. H., Weinberger, H. F.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ (1967) Zbl 0153.13602 MR 0219861
25. Safonov, M. V.: Harnack's inequality for elliptic equations and the Holder property of their solutions. J. Soviet Math. 21, 851-863 (1983) Zbl 0511.35029