• Previous Article
    On differential variational inequalities and projected dynamical systems - equivalence and a stability result
  • PROC Home
  • This Issue
  • Next Article
    Nonexistence of global solutions of nonlinear Schrodinger equations in non star-shaped domains
2007, 2007(Special): 477-486. doi: 10.3934/proc.2007.2007.477

Comparisons of eigenvalues of second order elliptic operators

1. 

Aix-Marseille Université, LATP, Faculté des Sciences et Techniques, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France

2. 

Université Aix-Marseille III, LAPT, Faculté des Sciences et Techniques, Case cour A, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France

3. 

CNRS, LAPT, CMI, 39 rue F. Joliot-Curie, F-13453 Marseille Cedex 13, France

Received  September 2006 Revised  June 2007 Published  September 2007

To any second order elliptic operator $L =$ −div$(A\nabla)$ + $v * \nabla + V$ in a bounded $C^2$ domain $\Omega$ with Dirichlet boundary condition, we associate a second order elliptic operator $L$* in divergence form in the Euclidean ball $\Omega$* centered at 0 and having the same Lebesgue measure as $\Omega$. In $\Omega$, the symmetric matrix field $A$ is in $W^(1,\infty)(\Omega)$, the vector field $v$ is in $L^\infty(\Omega \mathbb{R}^n)$ and $V$ is a continuous function in $bar(\Omega)$. In $\Omega$*, the coefficients of $L$* are radial, they preserve some quantities associated to the coefficients of $L$, and we can construct the operator $L$* in such a way that its principal eigenvalue is not too much larger than that of $L$. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new rearrangement technique, different from the Schwarz symmetrization and interesting by itself.
Citation: François Hamel, Emmanuel Russ, Nikolai Nadirashvili. Comparisons of eigenvalues of second order elliptic operators. Conference Publications, 2007, 2007 (Special) : 477-486. doi: 10.3934/proc.2007.2007.477
[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]

Simona Fornaro, Giorgio Metafune, Diego Pallara, Roland Schnaubelt. Second order elliptic operators in $L^2$ with first order degeneration at the boundary and outward pointing drift. Communications on Pure & Applied Analysis, 2015, 14 (2) : 407-419. doi: 10.3934/cpaa.2015.14.407

[3]

Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747

[4]

Craig Cowan. Optimal Hardy inequalities for general elliptic operators with improvements. Communications on Pure & Applied Analysis, 2010, 9 (1) : 109-140. doi: 10.3934/cpaa.2010.9.109

[5]

Petr Hasil, Petr Zemánek. Critical second order operators on time scales. Conference Publications, 2011, 2011 (Special) : 653-659. doi: 10.3934/proc.2011.2011.653

[6]

Fausto Ferrari. Mean value properties of fractional second order operators. Communications on Pure & Applied Analysis, 2015, 14 (1) : 83-106. doi: 10.3934/cpaa.2015.14.83

[7]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184

[8]

Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023

[9]

M. Euler, N. Euler, M. C. Nucci. On nonlocal symmetries generated by recursion operators: Second-order evolution equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4239-4247. doi: 10.3934/dcds.2017181

[10]

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

[11]

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

[12]

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

[13]

Wenxian Shen, Xiaoxia Xie. On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1665-1696. doi: 10.3934/dcds.2015.35.1665

[14]

Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Cecilia Yarur. Large solutions of elliptic systems of second order and applications to the biharmonic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 411-432. doi: 10.3934/dcds.2012.32.411

[15]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335

[16]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54

[17]

Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581

[18]

He Zhang, Xue Yang, Yong Li. Lyapunov-type inequalities and solvability of second-order ODEs across multi-resonance. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1133-1148. doi: 10.3934/dcdss.2017061

[19]

Jun Cao, Der-Chen Chang, Dachun Yang, Sibei Yang. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435

[20]

Victor Isakov, Nanhee Kim. Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 799-825. doi: 10.3934/dcds.2010.27.799

 Impact Factor: 

Metrics

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

[Back to Top]