Comparisons of eigenvalues of second order elliptic operators

Pages: 477 - 486, Issue Special, September 2007

 Abstract        Full Text (237.9K)              

François Hamel - Aix-Marseille Université, LATP, Faculté des Sciences et Techniques, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France (email)
Emmanuel Russ - Université Aix-Marseille III, LAPT, Faculté des Sciences et Techniques, Case cour A, Avenue Escadrille Normandie-Niemen, F-13397 Marseille Cedex 20, France (email)
Nikolai Nadirashvili - CNRS, LAPT, CMI, 39 rue F. Joliot-Curie, F-13453 Marseille Cedex 13, France (email)

Abstract: 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:  Principal eigenvalue, second order elliptic operators, rearrangement inequalities.
Mathematics Subject Classification:  Primary: 35P15; Secondary: 35J25.

Received: September 2006;      Revised: June 2007;      Published: September 2007.