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

On the principal eigenvalues of some elliptic problems with large drift

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.
Citation: Tomas Godoy, Jean-Pierre Gossez, Sofia Paczka. On the principal eigenvalues of some elliptic problems with large drift. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1007/s00220-004-1201-9.  Google Scholar

[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.  doi: 10.1002/cpa.3160470105.  Google Scholar

[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.  doi: 10.1512/iumj.2008.57.3204.  Google Scholar

[4]

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

[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.   Google Scholar

[6]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00526-003-0249-2.  Google Scholar

[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, 15 (2007), 137.   Google Scholar

[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.  doi: 10.1007/s10231-009-0120-y.  Google Scholar

[12]

P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity,", Pitman, (1991).   Google Scholar

[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.  doi: 10.1080/03605308008820162.  Google Scholar

[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.  doi: 10.1002/cpa.3160310406.  Google Scholar

[15]

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

[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.  doi: 10.1006/jdeq.1996.0070.  Google Scholar

[17]

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

[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.   Google Scholar

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.  doi: 10.1007/s00220-004-1201-9.  Google Scholar

[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.  doi: 10.1002/cpa.3160470105.  Google Scholar

[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.  doi: 10.1512/iumj.2008.57.3204.  Google Scholar

[4]

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

[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.   Google Scholar

[6]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00526-003-0249-2.  Google Scholar

[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, 15 (2007), 137.   Google Scholar

[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.  doi: 10.1007/s10231-009-0120-y.  Google Scholar

[12]

P. Hess, "Periodic-parabolic Boundary Value Problems and Positivity,", Pitman, (1991).   Google Scholar

[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.  doi: 10.1080/03605308008820162.  Google Scholar

[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.  doi: 10.1002/cpa.3160310406.  Google Scholar

[15]

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

[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.  doi: 10.1006/jdeq.1996.0070.  Google Scholar

[17]

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

[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.   Google Scholar

[1]

Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315

[2]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177

[3]

Fei-Ying Yang, Wan-Tong Li, Jian-Wen Sun. Principal eigenvalues for some nonlocal eigenvalue problems and applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 4027-4049. doi: 10.3934/dcds.2016.36.4027

[4]

Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179

[5]

Wenxiong Chen, Congming Li. Indefinite elliptic problems in a domain. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333

[6]

Haitao Yang. On the existence and asymptotic behavior of large solutions for a semilinear elliptic problem in $R^n$. Communications on Pure & Applied Analysis, 2005, 4 (1) : 187-198. doi: 10.3934/cpaa.2005.4.197

[7]

M. Chuaqui, C. Cortázar, M. Elgueta, J. García-Melián. Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights. Communications on Pure & Applied Analysis, 2004, 3 (4) : 653-662. doi: 10.3934/cpaa.2004.3.653

[8]

Alberto Boscaggin, Maurizio Garrione. Positive solutions to indefinite Neumann problems when the weight has positive average. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5231-5244. doi: 10.3934/dcds.2016028

[9]

M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure & Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411

[10]

Junxiang Li, Yan Gao, Tao Dai, Chunming Ye, Qiang Su, Jiazhen Huo. Substitution secant/finite difference method to large sparse minimax problems. Journal of Industrial & Management Optimization, 2014, 10 (2) : 637-663. doi: 10.3934/jimo.2014.10.637

[11]

Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102

[12]

Jingyu Li. Asymptotic behavior of solutions to elliptic equations in a coated body. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1251-1267. doi: 10.3934/cpaa.2009.8.1251

[13]

Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056

[14]

Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks & Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711

[15]

Tohru Wakasa, Shoji Yotsutani. Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems. Communications on Pure & Applied Analysis, 2010, 9 (2) : 539-561. doi: 10.3934/cpaa.2010.9.539

[16]

Getachew K. Befekadu, Panos J. Antsaklis. On noncooperative $n$-player principal eigenvalue games. Journal of Dynamics & Games, 2015, 2 (1) : 51-63. doi: 10.3934/jdg.2015.2.51

[17]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[18]

Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014

[19]

Zuzana Došlá, Mauro Marini, Serena Matucci. Global Kneser solutions to nonlinear equations with indefinite weight. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3297-3308. doi: 10.3934/dcdsb.2018252

[20]

Bernard Brighi, S. Guesmia. Asymptotic behavior of solution of hyperbolic problems on a cylindrical domain. Conference Publications, 2007, 2007 (Special) : 160-169. doi: 10.3934/proc.2007.2007.160

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]