On the Polygonal Faber-Krahn Inequality

Abstract

It has been conjectured by Pólya and Szegö seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with n sides and fixed area is the regular polygon. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this paper we prove that for each n ≥ 5 the proof of the conjecture can be reduced to a finite number of certified numerical computations. Moreover, the local minimality of the regular polygon can be reduced to a single numerical computation. For n = 5, 6, 7, 8 we perform this computation and certify the numerical approximation by finite elements, up to machine errors.

The nonlocal isoperimetric problem for polygons: Hardy-Littlewood and Riesz inequalities

“…Let us also mention that the same computations above were made also for Q ∈ [2,11]: [5,10] and various choices of t ∈ [1, 100] an oscillatory behavior can be observed, namely the Hessian at the regular N -gon may have positive or negative eigenvalues; however, for Q ≥ 6 the behavior stabilizes and the non-zero eigenvalues become strictly negative.…”

“…Here λ 1 (Ω) is the principal eigenvalue of the Dirichlet Laplacian in Ω, while τ (Ω) is the torsional rigidity of Ω (namely the L 1 (Ω)-norm of the unique solution to the equation −∆u = 1 in H 1 0 (Ω)). The inequalities analogue to (4) have been proved for every N ≥ 3 for the logarithmic capacity [44] and for the Cheeger constant [10] (for related results, see also [5,11,32]). At present, (4) are open for any N ≥ 5, and they can be included among the major open problems in shape optimization.…”

“…An example is shown in Figure 9. Following [33,5], let ϕ i denote the piece-wise affine function on the triangulation T such that ϕ i (A j ) = δ ij .…”

“…The relation between Riesz inequality and the classical isoperimetric inequality is easily individuated. Indeed, the functional J h in (1) differs just by a change of sign and a translation from the nonlocal h-perimeter, defined by (5) P h (Ω) :=…”

“…where the quantity h(x − y) is interpreted as an interaction density between two points x ∈ Ω and y ∈ Ω c := R d \ Ω. A suitable scaling of h in (5) allows to recover the usual notion of perimeter via an asymptotic formula. The concept of nonlocal perimeter has been first introduced in [7], and in recent times it has been widely developed, especially concerning the fractional kernel h(x) = |x| −(d+s) , s ∈ (0, 1), and concerning bounded integrable kernels (see respectively the seminal papers [15,16] and the recent monograph [36]).…”