Comparisons of eigenvalues of second order elliptic operators

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.