Reverse Cheeger inequality for planar convex sets

2015/01/19 by Parini, Enea · 2 citations
#49Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1501.04520

Abstract

We prove the sharp inequality J(Ω) := (λ₁(Ω))/(h₁(Ω)²) lt; (π²)/(4), where Ω is any planar, convex set, λ₁(Ω) is the first eigenvalue of the Laplacian under Dirichlet boundary conditions, and h₁(Ω) is the Cheeger constant of Ω. The value on the right-hand side is optimal, and any sequence of convex sets with fixed volume and diameter tending to infinity is a maximizing sequence. Morever, we discuss the minimization of J in the same class of subsets: we provide a lower bound which improves the generic bound given by Cheeger's inequality, we show the existence of a minimizer, and we give some optimality conditions.

Cited by

• Convex sets with vanishing relative width and the reverse Cheeger inequality
• Optimal domains for the Cheeger inequality

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