Rearrangement inequalities and applications to isoperimetric problems for eigenvalues

• 成果类型: Article
• 署名作者: Hamel, Francois; Nadirashvili, Nikolai; Russ, Emmanuel
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.2.1
• 发表日期: 2011
• 页码: 647-755
• 关键词: 2nd-order elliptic-operators; faber-krahn inequality; principal eigenvalue; rayleighs conjecture; boundary-conditions; membrane problem; robin problems; clamped plate; laplacian; EQUATIONS

摘要

Let Omega be a bounded C-2 domain in R-n, where n is any positive integer, and let Omega* be the Euclidean ball centered at 0 and having the same Lebesgue measure as Omega. Consider the operator L = -div(A del) v . del + V on Omega with Dirichlet boundary condition, where the symmetric matrix field A is in W-1,W-infinity(Omega), the vector field v is in L-infinity(Omega, R-n) and V is a continuous function in (Omega) over bar. 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 phi of L a new symmetric rearrangement defined on Omega*, which is different from the classical Schwarz symmetrization and which preserves the integral of div(A del phi) 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.