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January  2013, 33(1): 225-237. doi: 10.3934/dcds.2013.33.225

On the principal eigenvalues of some elliptic problems with large drift

Tomas Godoy 1, , Jean-Pierre Gossez 2, and  Sofia Paczka 1,

1. 

FaMAF, Universidad Nacional de Cordoba, Ciudad Universitaria, 5000 Cordoba, Argentina, Argentina
2. 

Departement de Mathematique, C.P. 214, Universite Libre de Bruxelles, 1050 Bruxelles, Belgium

Received  July 2011 Revised  October 2011 Published  September 2012

This paper is concerned with non-selfadjoint elliptic problems having a principal part in divergence form and involving an indefinite weight. We study the asymptotic behavior of the principal eigenvalues when the first order term (drift term) becomes larger and larger. Several of our results also apply to elliptic operators in general form.
Keywords: Nonselfadjoint elliptic problems, principal eigenvalue, indefinite weight, asymptotic behavior, minimax formula, large drift..
Mathematics Subject Classification: 35J25, 35Q9.
Citation: Tomas Godoy, Jean-Pierre Gossez, Sofia Paczka. On the principal eigenvalues of some elliptic problems with large drift. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 225-237. doi: 10.3934/dcds.2013.33.225
References:
[1]

H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480. doi: 10.1007/s00220-004-1201-9.

[2]

H. Berestycki, L. Nirenberg and S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105.

[3]

X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.

[4]

E. N. Dancer, Some remarks on classical problems and fine properties of Sobolev spaces, Diff. Int. Equat., 9 (1996), 437-446.

[5]

A. Devinatz, R. Ellis and A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivative II, Indiana Univ. Math. J., 23 (1974), 991-1011.

[6]

M. Donsker and S. Varadhan, On the principal eigenvalue of second order differential operators, Comm. Pure Appl. Math., 29 (1976), 595-621. doi: 10.1002/cpa.3160290606.

[7]

J. Fleckinger, J. Hernandez and F. de Thelin, Existence of multiple eigenvalues for some indefinite linear eigenvalue problems, Bolletino U. M. I., 7 (2004), 159-188.

[8]

A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivative, Indiana Univ. Math. J., 22 (1973), 1005-1015.

[9]

T. Godoy, J. P. Gossez and S. Paczka, A minimax formula for principal eigenvalues and application to an antimaximum principle, Calculus of Variations and Partial Differential Equations, 21 (2004), 85-111. doi: 10.1007/s00526-003-0249-2.

[10]

T. Godoy, J. P. Gossez and S. Paczka, A minimax formula for the principal eigenvalues of Dirichlet problems and its applications, 2006 International Conference in honor of JacquelineFleckinger, Electronic Journal of Differential Equations, Conference, 15 (2007), 137-154.

[11]

T. Godoy, J. P. Gossez and S. Paczka, On the asymptotic behavior of the principal eigenvalues of some elliptic problems, Annali di Matematica Pura ed Applicata, 189 (2009), 497-521. doi: 10.1007/s10231-009-0120-y.

[12]

P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity," Pitman, 1991.

[13]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Part. Diff. Equat., 5 (1980), 999-1030. doi: 10.1080/03605308008820162.

[14]

C. Holland, A minimum principle for the principal eigenvalue for second order linear elliptic equation with natural boundary condition, Comm. Pure Appl. Math., 31 (1978), 509-519. doi: 10.1002/cpa.3160310406.

[15]

T. Kato, Superconvexity of the spectral radius, and convexity of the spectral bound and the type, Math. Z., 180 (1982), 265-273. doi: 10.1007/BF01318910.

[16]

J. Lopez Gomez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Diff. Equat., 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070.

[17]

S. Paczka, A Neumann periodic parabolic eigenvalue problem with continuous weight, Rendiconti del Seminario Matematico dellUniversita e Politecnico di Torino, 54 (1996), 67-74

[18]

S. Timbo, M. Kimura and H. Notsu, Exponential decay phenomenon of the principal eigenvalue of an elliptic operator with a large drift term of gradient type, Asymptotic Analysis, 65 (2009), 103-123.

show all references

References:
[1]

H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480. doi: 10.1007/s00220-004-1201-9.

[2]

H. Berestycki, L. Nirenberg and S. Varadhan, The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105.

[3]

X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.

[4]

E. N. Dancer, Some remarks on classical problems and fine properties of Sobolev spaces, Diff. Int. Equat., 9 (1996), 437-446.

[5]

A. Devinatz, R. Ellis and A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivative II, Indiana Univ. Math. J., 23 (1974), 991-1011.

[6]

M. Donsker and S. Varadhan, On the principal eigenvalue of second order differential operators, Comm. Pure Appl. Math., 29 (1976), 595-621. doi: 10.1002/cpa.3160290606.

[7]

J. Fleckinger, J. Hernandez and F. de Thelin, Existence of multiple eigenvalues for some indefinite linear eigenvalue problems, Bolletino U. M. I., 7 (2004), 159-188.

[8]

A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivative, Indiana Univ. Math. J., 22 (1973), 1005-1015.

[9]

T. Godoy, J. P. Gossez and S. Paczka, A minimax formula for principal eigenvalues and application to an antimaximum principle, Calculus of Variations and Partial Differential Equations, 21 (2004), 85-111. doi: 10.1007/s00526-003-0249-2.

[10]

T. Godoy, J. P. Gossez and S. Paczka, A minimax formula for the principal eigenvalues of Dirichlet problems and its applications, 2006 International Conference in honor of JacquelineFleckinger, Electronic Journal of Differential Equations, Conference, 15 (2007), 137-154.

[11]

T. Godoy, J. P. Gossez and S. Paczka, On the asymptotic behavior of the principal eigenvalues of some elliptic problems, Annali di Matematica Pura ed Applicata, 189 (2009), 497-521. doi: 10.1007/s10231-009-0120-y.

[12]

P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity," Pitman, 1991.

[13]

P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Part. Diff. Equat., 5 (1980), 999-1030. doi: 10.1080/03605308008820162.

[14]

C. Holland, A minimum principle for the principal eigenvalue for second order linear elliptic equation with natural boundary condition, Comm. Pure Appl. Math., 31 (1978), 509-519. doi: 10.1002/cpa.3160310406.

[15]

T. Kato, Superconvexity of the spectral radius, and convexity of the spectral bound and the type, Math. Z., 180 (1982), 265-273. doi: 10.1007/BF01318910.

[16]

J. Lopez Gomez, The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. Diff. Equat., 127 (1996), 263-294. doi: 10.1006/jdeq.1996.0070.

[17]

S. Paczka, A Neumann periodic parabolic eigenvalue problem with continuous weight, Rendiconti del Seminario Matematico dellUniversita e Politecnico di Torino, 54 (1996), 67-74

[18]

S. Timbo, M. Kimura and H. Notsu, Exponential decay phenomenon of the principal eigenvalue of an elliptic operator with a large drift term of gradient type, Asymptotic Analysis, 65 (2009), 103-123.

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