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Let $\Omega$ be a bounded $C^{2}$ domain in $\R^n$, and let $\Omega^{\ast}$be the Euclidean ball centered at 0 and having the same Lebesgue measure as$\Omega$. Consider the operator $L=-\div(A\nabla)+v\cdot \nabla +V$ on $\Omega$with Dirichlet boundary condition. We prove that minimizing the principaleigenvalue of $L$ when the Lebesgue measure of $\Omega$ is fixed and when $A$,$v$ and $V$ vary under some constraints is the same as minimizing the principaleigenvalue of some operators $L^*$ in the ball $\Omega^*$ with smooth andradially symmetric coefficients. The constraints which are satisfied by theoriginal coefficients in $\Omega$ and the new ones in $\Omega^*$ are expressedin terms of some distribution functions or some integral, pointwise orgeometric quantities. Some strict comparisons are also established when$\Omega$ is not a ball.

Rearrangement inequalities and applications to isoperimetric problems for eigenvalues