Comparisons of eigenvalues of second order elliptic operators

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2007, 2007(Special): 477-486. doi: 10.3934/proc.2007.2007.477

Comparisons of eigenvalues of second order elliptic operators

François Hamel ^1, , Emmanuel Russ ^2, and Nikolai Nadirashvili ^3,

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

Abstract
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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.

Keywords: second order elliptic operators, Principal eigenvalue, rearrangement inequalities..

Mathematics Subject Classification: Primary: 35P15; Secondary: 35J2.

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

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