Abstract
Let Ω be a bounded C2 domain in ℝn, where n is any positive integer, and let Ω∗ be the Euclidean ball centered at 0 and having the same Lebesgue measure as Ω. Consider the operator L=−div(A∇)+v⋅∇+V on Ω with Dirichlet boundary condition, where the symmetric matrix field A is in W1,∞(Ω), the vector field v is in L∞(Ω,ℝn) and V is a continuous function in Ω⎯⎯⎯. We prove that minimizing the principal eigenvalue of L when the Lebesgue measure of Ω 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 Ω∗ with smooth and radially symmetric coefficients. The constraints which are satisfied by the original coefficients in Ω and the new ones in Ω∗ are expressed in terms of some distribution functions or some integral, pointwise or geometric quantities. Some strict comparisons are also established when Ω is not a ball. To these purposes, we associate to the principal eigenfunction φ of L a new symmetric rearrangement defined on Ω∗, which is different from the classical Schwarz symmetrization and which preserves the integral of div(A∇φ) 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.

KEYWORDS

SHARE & LIKE

COMMENTS

SIMILAR ARTICLES

Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin

Read MoreAbstract For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we

Read MoreAbstract Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let

Read MoreAbstract We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely

Read MoreAbstract We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We

Read MoreAbstract This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal

Read MoreAbstract We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge

Read MoreAbstract We prove that isoparametric hypersurfaces with (g,m)=(6,2) are homogeneous, which answers Dorfmeister-Neher’s conjecture affirmatively and so

Read MoreAbstract We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorificat

Read MoreAbstract If F(x,y)∈ℤ[x,y] is an irreducible binary form of degree k≥3, then a theorem of Darmon and Granville implies that the generalized superellipt

Read More