Rearrangement inequalities and applications to isoperimetric problems for eigenvalues

Abstract

Let $\Omega$ be a bounded $C^{2}$ domain in $\Bbb{R}^n$, where $n$ is any positive integer, and let $\Omega^{\ast}$ be the Euclidean ball centered at $0$ and having the same Lebesgue measure as $\Omega$. Consider the operator $L=-\mathrm{div}(A\nabla)+v\cdot \nabla +V$ on $\Omega$ with Dirichlet boundary condition, where the symmetric matrix field $A$ is in $W^{1,\infty}(\Omega)$, the vector field $v$ is in $L^{\infty}(\Omega,\Bbb{R}^n)$ and $V$ is a continuous function in $\overline{\Omega}$. We prove that minimizing the principal eigenvalue 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 principal eigenvalue of some operators $L^*$ in the ball $\Omega^*$ with smooth and radially symmetric coefficients. The constraints which are satisfied by the original coefficients in $\Omega$ and the new ones in $\Omega^*$ are expressed in terms of some distribution functions or some integral, pointwise or geometric quantities. Some strict comparisons are also established when $\Omega$ is not a ball. To these purposes, we associate to the principal eigenfunction $\varphi$ of $L$ a new symmetric rearrangement defined on $\Omega^*$, which is different from the classical Schwarz symmetrization and which preserves the integral of $\mathrm{div}(A\nabla\varphi)$ on suitable equi-measurable sets. A substantial part of the paper is devoted to the proofs of pointwise and integral inequalities of independent interest which are satisfied by this rearrangement. The comparisons for the eigenvalues hold for general operators of the type $L$ and they are new even for symmetric operators. Furthermore they generalize, in particular, and provide an alternative proof of the well-known Rayleigh-Faber-Krahn isoperimetric inequality about the principal eigenvalue of the Laplacian under Dirichlet boundary condition on a domain with fixed Lebesgue measure.