The principal eigenvalue and maximum principle for second‐order elliptic operators in general domains

Communications on Pure and Applied MathematicsVolume 47, Issue 1 p. 47-92 Article The principal eigenvalue and maximum principle for second-order elliptic operators in general domains H. Berestycki, H. Berestycki Université de Paris VISearch for more papers by this authorL. Nirenberg, L. Nirenberg Courant InstituteSearch for more papers by this authorS. R. S. Varadhan, S. R. S. Varadhan Courant InstituteSearch for more papers by this author H. Berestycki, H. Berestycki Université de Paris VISearch for more papers by this authorL. Nirenberg, L. Nirenberg Courant InstituteSearch for more papers by this authorS. R. S. Varadhan, S. R. S. Varadhan Courant InstituteSearch for more papers by this author First published: January 1994 https://doi.org/10.1002/cpa.3160470105Citations: 365AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat Bibliography 1 Agmon, S., On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, pp. 1952 in: Methods of Functional Analysis and Theory of Elliptic Equations, D. Greco, ed., Liguori Ed. Napoli, 1983. 2 Bakelman, I. Ya., Convex Functions and Elliptic Equations, manuscript of book. 3 Barta, J., Sur la vibration fondamentale d'une membrane, C. R. Acad. Sci. Paris 204, 1937, pp. 472–473. 4 Berestycki, H., and Nirenberg, L., Monotonicity, symmetry and antisymmetry of solutions of semi-linear elliptic equations, J. Geom. Phys. 5, 1988, pp. 237–275. 5 Berestycki, H., and Nirenberg, L., On the method of moving planes and the sliding method, Boll. Soc. Brasil Mat. Nova Ser. 22, 1991, pp. 1–37. 6 Birindelli, I., Second-Order Elliptic Equations in General Domains: Hopf's Lemma and Anti-Maximum Principle, Doctoral Dissertation, New York University, 1992. 7 Bony, J. M., Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris Série A 265, 1967, pp. 333–336. 8 Donsker, M. D., and Varadhan, S. R. S., On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Nat. Acad. Sci. USA 72, 1975, pp. 780–783. 9 Donsker, M. D., and Varadhan, S. R. S., On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math. 29, 1976, pp. 595–621. 10 Gilbarg, D., and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order; 2nd ed., Grundlehren der mathematischen Wissenschaften, No. 224, Springer-Verlag, Berlin-New York, 1983. 11 Krylov, N. V., Nonlinear elliptic and parabolic equations of the second order, Math. and its Applications, Reidel, Norwell, Massachusetts, 1987. 12 Lieb, E. H., On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74, 1983, pp. 441–448. 13 Lions, P. L., A remark on Bony's maximum principle, Proc. AMS 88, 1983, pp. 503–508. 14 Miller, K., Barriers on cones for uniformly elliptic operators, Ann. Mat. Pura Appl. 76, 1967, pp. 93–106. 15 Nussbaum, R. D., and Pinchover, Y., On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications, J. Analyse Math. 59, 1992, pp. 161–177. 16 Protter, M. H., and Weinberger, H. F., On the spectrum of general second order operators, Bull. AMS 72, 1966, pp. 251–255. 17 Protter, M. H., and Weinberger, H. F., Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1967. 18 Pucci, C., Maximum and minimum first eigenvalues for a class of elliptic operators, Proc. AMS 17, 1966, pp. 788–795. 19 Rabinowitz, P. H., Théorie du degré topologique et applications à des problèmes aux limites non linéaires, Lecture Notes Lab. Analyse Numerique, Université Paris VI, 1975. 20 Serrin, J., A remark on the preceding paper of Amann, Arch. Rat. Mech. Anal. 44, 1972, pp. 183–186. 21 Stroock, D., and Varadhan, S. R. S., On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. Pure Appl. Math. 25, 1972, pp. 651–713. 22 Venturino, M., Primo antovalore di operatori lineari ellittici in forma non variazionale, Boll. Un. Math. It. 5, 1978, pp. 576–591. 23 Walter, W., A theorem on elliptic differential inequalities with an application to gradient bounds, Math. Z. 200, 1989, pp. 293–299. 24 Zhao, Z., Subcriticality, positivity, and gaugeability of the Schrödinger operator; Bull. AMS, 23, 1990, pp. 513–517. 25 Brezis, H., and Lions, P. L., An estimate related to the strong maximum principle, Boll. Un. Math. Ital. A 17, 1980, pp. 503–508. Citing Literature Volume47, Issue1January 1994Pages 47-92 ReferencesRelatedInformation