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On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications

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References

  1. S. Agmon,On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, inMethods of Functional Analysis and Theory of Elliptic Equations, ed. D. Greco, Liguori, Naples, 1982, pp. 19–52.

    Google Scholar 

  2. W. Allegreto and A. B. Mingarelli,On the non-existence of positive solutions for a Schrödinger equation with an indefinite weight-function, C. R. Math. Rep. Acad. Sci. Canada8 (1986), 69–73.

    MathSciNet  Google Scholar 

  3. M. BrelotOn Topologies and Boundaries in Potential Theory, Lecture Notes in Mathematics175, Springer-Verlag, Berlin, 1971.

    MATH  Google Scholar 

  4. R. S. Cantrell and C. Cosner,Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. Roy. Soc. Edinburgh112A (1989), 293–318.

    MathSciNet  Google Scholar 

  5. R. S. Cantrell and C. Cosner,Diffusive logistic equations with indefinite weights: population models in disrupted environments II, Siam J. Math. Anal.22 (1991), 1043–1064.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. D. Donsker and S. R. S. Varadhan,On a variational formula for the principal eigenvalue for operators with maximum principle, Proc. Natl. Acad. Sci. U.S.A.72 (1975), 780–783.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. D. Donsker and S. R. S. Varadhan,On the principal eigenvalue of second-order elliptic differential operators, Comm. Pure Appl. Math.29 (1976), 595–621.

    Article  MATH  MathSciNet  Google Scholar 

  8. C. Holland,A minimum principle for the principal eigenvalue for second order linear elliptic equations with natural boundary conditions. Comm. Pure Appl. Math.31 (1978), 509–519.

    Article  MATH  MathSciNet  Google Scholar 

  9. Y. Kifer,Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Isr. J. Math.70 (1990), 1–47.

    MATH  MathSciNet  Google Scholar 

  10. R. D. Nussbaum,Positive operators and elliptic eigenvalue problems, Math. Z.186 (1984), 247–264.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. D. Nussbaum,Convexity and log convexity for the spectral radius, Linear Algebra Appl.73 (1986), 59–122.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. D. Nussbaum and Y. Pinchover,On variational formulas for the generalized principal eigenvalue of second order elliptic equations with general boundary conditions, in preparation.

  13. Y. Pinchover,On positive solutions of second-order elliptic equations, stability results and classification, Duke Math. J.57 (1988), 955–980.

    Article  MATH  MathSciNet  Google Scholar 

  14. Y. Pinchover,On criticality and ground states of second-order elliptic equations II, J. Differential Equations87 (1990), 353–364.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. G. Pinsky,A generalized Dirichlet principle for second order nonselfadjoint elliptic operators, SIAM J. Math. Anal.19 (1988), 204–213.

    Article  MATH  MathSciNet  Google Scholar 

  16. M. H. Protter and H. F. Weinberger,On the spectrum of general second order operators, Bull. Am. Math. Soc.72 (1966), 251–255.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Reed and B. Simon,Methods of Modern Mathematical Physics. 1. Functional Analysis, revised edition, Academic Press, New York, 1980.

    Google Scholar 

  18. M. Schechter,Hamiltonians for singular potentials, Indiana Univ. Math. J.22 (1972), 483–503.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Schechter,Exact estimates for potentials, SIAM J. Math. Anal.19 (1988), 1324–1328.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Schechter,The spectrum of the Schrödinger operator, Trans. Am. Math., Soc.312 (1989) 115–128.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Schechter,Multiplication operators, Can. J. Math.61 (1989), 234–249.

    MathSciNet  Google Scholar 

  22. M. Sion,On general minimax theorems, Pacific. J. Math.8 (1958), 171–176.

    MATH  MathSciNet  Google Scholar 

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Dedicated to Professor Shmuel Agmon

Partially supported by NSF DMS 89-03018.

Partially supported by Technion VPR-Fund-K. and M. Bank Mathematics R. Fund.

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Nussbaum, R.D., Pinchover, Y. On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications. J. Anal. Math. 59, 161–177 (1992). https://doi.org/10.1007/BF02790223

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