Sharp fundamental gap estimate on convex domains of sphere

On the Spectral Gap for Convex Domains

Let D be a convex planar domain, symmetric about both the xand y-axes, which is strictly contained in (−a, a) × (−b, b) = Γ. It is proved that, unless D is a certain kind of rectangle, the difference (gap) between the first two eigenvalues of the Dirichlet Laplacian in D is strictly larger than the gap for Γ. We show how to give explicit lower bounds for the difference of the gaps.

The Fundamental Gap Conjecture for Polygonal Domains

In [22], M. van de Berg made a “fundamental gap conjecture” in the context of a free boson gas confined in a container in Rn with hard walls, and in [24], Yau generalized the conjecture to what is now known as the “fundamental gap conjecture.” For a convex domain Ω ⊂ Rn, (0.1) ξ(Ω) := d (λ2(Ω)− λ1(Ω)) ≥ 3π where d is the diameter of the domain, and 0 < λ1(Ω) < λ2(Ω) are the first two eigenvalue...

Sharp bounds for the p-torsion of convex planar domains

We obtain some sharp estimates for the p-torsion of convex planar domains in terms of their area, perimeter, and inradius. The approach we adopt relies on the use of web functions (i.e. functions depending only on the distance from the boundary), and on the behaviour of the inner parallel sets of convex polygons. As an application of our isoperimetric inequalities, we consider the shape optimiz...

Spectral gap for the Cauchy process on convex, symmetric domains

Let D ⊂ R be a bounded convex domain which is symmetric relative to both coordinate axes. Assume that [−a, a] × [−b, b], a ≥ b > 0 is the smallest rectangle (with sides parallel to the coordinate axes) containing D. Let {λn}n=1 be the eigenvalues corresponding to the semigroup of the Cauchy process killed upon exiting D. We obtain the following estimate on the spectral gap: λ2 − λ1 ≥ Cb a2 , wh...

A New Characterization of the Cr Sphere and the Sharp Eigenvalue Estimate for the Kohn Laplacian