Efficiency and localisation for the first Dirichlet eigenfunction

Bounds are obtained for the efficiency or mean to peak ratio $E(\Omega)$ for the first Dirichlet eigenfunction (positive) for open, connected sets $\Omega$ with finite measure in Euclidean space $\R^m$. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if $\Omega_n$ is any quadrilateral with perpendicular diagonals of lengths $1$ and $n$ respectively, then the sequence of first Dirichlet eigenfunctions localises, and $E(\Omega_n)=O\big(n^{-2/3}\log n\big)$. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.

The Rayleigh-Ritz variational principle asserts that λ(Ω) = inf The efficiency or mean to max ratio of u Ω is defined by where · p , 1 ≤ p ≤ ∞ denotes the standard L p (Ω) norm.
The study of E(Ω) goes back to the pioneering results of [17,20]. In Theorem 3 of [17], it was shown that if Ω is bounded and convex then with equality in (3) if Ω is a bounded interval in R. A non-linear version has been proved in [10] for the p-Laplacian with 1 < p < ∞. More general results have been obtained in [7]. It follows from inequality (3) and the main theorem in that paper that if Ω is a bounded region in R m , then where B is a ball in R m . Moreover, it was asserted in Table 1 in [17] that 2 π is the limit of the efficiency of a thinning annulus in R m . The proof of this assertion (Theorem 11) will be given in Section 4 below. There we will also compute the efficiency for the equilateral triangle, the square, and the disc. These data support the conjectures that (i) the efficiency of a bounded, convex planar set is maximised by the disc, (ii) if P n ⊂ R 2 is a regular n-gon then n → E(P n ) is increasing. We note that the efficiency for an arbitrarily long rectangle is (2/π) 2 ≈ 0.4053, whereas the efficiency of a disc is approximately 0.4317.
Recently a connection has been established between localisation of eigenfunctions and an effective potential such as the inverse of the torsion function (see [1]). In a similar spirit, it has been shown in certain special cases, such as a bounded interval in R or a square in R 2 , that the eigenfunctions of the Schrödinger operator of Anderson type localise (see [2], and [9]).
The first part of the definition below is very similar to the one in [12], (formula (7.1) for p = 1). Definition 1. Let (Ω n ) be a sequence of non-empty open sets in R m with |Ω n | < ∞.
(i) We say that a sequence f n with f n ∈ L 2 (Ω n ), n ∈ N and f n 2 = 1 localises if there exists a sequence of measurable sets A n ⊂ Ω n such that lim n→∞ |A n | |Ω n | = 0, lim n→∞ An (ii) We say that a sequence f n with f n ∈ L ∞ (Ω n ), f n ≥ 0, f n ∞ > 0, n ∈ N has vanishing efficiency if We have the following elementary observations.

Lemma 2.
If Ω is a non-empty open set with finite Lebesgue measure, and if (ii) The proofs of (5) and (6) are immediate, since by Cauchy-Schwarz, By (6) we have that if f n is localising then the mean to max ratio of f n is vanishing as n → ∞. We were unable to prove that if u Ωn has vanishing efficiency then u Ωn localises.
Denote by ρ(Ω) = sup{min{|x − y| : y ∈ ∂Ω}, x ∈ Ω} the inradius of Ω, by diam(Ω) = sup{|x − y| : x ∈ Ω, y ∈ Ω} the diameter of Ω, and by w(Ω) the width of Ω. For a measurable set A in R k with k < m we denote its k-dimensional Lebesgue measure by |A| k . The indicator function of a set A is denoted by 1 A . We define for ν ≥ 0, j ν to be the first positive zero of the Bessel function J ν .
Below we show that sets with small E(Ω) have small inradius, and large diameter.
If Ω is open, planar, bounded, and convex, then It is straightforward to construct sequences (Ω n ) for which (u Ωn ) is localising and, as a consequence of Lemma 3 and (6), have vanishing efficiency. For example, let Ω n be the disjoint union of one disc B with radius 1, and 4n discs with radii 1/2. All of the L 2 mass of the first eigenfunction of Ω n is supported on B, with |B|/|Ω n | = 1 n+1 , which tends to 0 as n → ∞. Theorem 6 below together with Lemmas 2 and 3, imply localisation for a wide class of sequences (u Ωn ). We first introduce the necessary notation.

Theorem 6.
Let Ω ⊂ R m be horn-shaped with |Ω| < ∞ and |Ω ′ | m−1 < ∞. If λ ≥ λ(Ω), If Ω ⊂ R 2 is open, bounded and convex, then it is always possible to find an isometry of Ω such that this isometric set is horn-shaped: let p and q be points on ∂Ω such that |p − q| = w(Ω), and p − q is perpendicular to the pair of straight parallel lines tangent to ∂Ω at both p and q which define the width w(Ω). That such a pair p, q exists was shown for example in Theorem 1.5 in [5]. Let T p,q (Ω) = {x − 1 2 (p + q) : x ∈ Ω} be the translation of Ω which translates the midpoint of p and q to the origin. Let ϕ be the angle between the positive x 1 axis and the unit vector (p − q)/|p − q|, and let R ϕ be rotation over an angle π 2 − ϕ. Then R ϕ T p,q (Ω) is isometric with Ω, horn-shaped, The points p and q need not be unique, and so this isometry need not be unique. However, the construction above always gives (13). If Υ is an ellipse with semi axes a 1 and a 2 with a 1 > a 2 then
It follows by scaling properties of both u Ω and |Ω| that if Ω is open and connected with |Ω| < ∞, and if α > 0, then where αΩ is a homothety of Ω by a factor α. Similarly, Example 9 then implies that a sequence of suitable translations, rotations and hometheties of sectors (S n (r)), with S n (r) := {(ρ, θ) : 0 < ρ < r, 0 < θ < π/n} and (u Sn(r) ) localises as n → ∞. This could have been obtained directly using separation of variables, Kapteyn's inequality, and extensive computations involving Bessel functions. See [15] for similar computations. and and (u Ωn,α ) is localising. and and lim If △ ⊂ R 2 is an equilateral triangle, then If ⊂ R 2 is a rectangle, then If B ⊂ R 2 is a disc, then Inequalities (6.9) in [12], and (4.7) in [16] state that for Ω open, bounded, planar, and convex, and both papers refer to [17] for details. However, no such inequality can be found in [17]. Inequality (22) would, by first maximising its right-hand side over all x ∈ Ω, and subsequently its left-hand side over all x ∈ Ω, imply that Since the Dirichlet eigenvalues are monotone in the domain, and Ω contains a disc of radius ρ(Ω), This, by (2) and (23), implies that for a bounded, planar convex set Ω, Inequality (23) was also quoted in formula (2.24) in [11]. However, (23) and (24) cannot hold true. Example 8 above implies that lim n→∞ E(Ω n ) = 0 for a collection of sequences of convex quadrilaterals (Ω n ). This collection includes a sequence of rhombi with vertices (n/2, 0), (−n/2, 0), (0, 1/2), (0, −1/2). This contradicts (24). This paper is organised as follows. The proofs of Lemma 3, and Theorem 4 are deferred to Section 2 below. The proofs of Theorem 6, Corollary 7, and Examples 8, 9, and 10 will be given in Section 3. The proof of Theorem 11 will be given in Section 4.

Proofs of Lemma 3 and Theorem 4
Proof of Lemma 3. First suppose (7) holds. That is if then lim n→∞ a n = 0.
3 Proofs of Theorem 6, Corollary 7, and Examples 8,9,10 To prove Theorem 6 we proceed via a number of lemmas.
We have by Cauchy-Schwarz that For a non-empty open set Ω ⊂ R m , we denote by p Ω (x, y; t), x ∈ Ω, y ∈ Ω, t > 0 its Dirichlet heat kernel. .
P. Kröger observed that one can get upper bounds for the first Dirichlet eigenvalue of the circular sector S n (r) with radius r and opening angle π/n, which have the correct leading term by choosing an optimal rectangle inside the sector [15]. Similar observations were used in the proof of Theorem 1.5 in [5], and also in the proof of Theorem 1.3 in [13].
Proof of Example 8. Theorem 1.5 in [5] implies the existence of a constant c 1 < ∞ such that We note that Ω n is horn-shaped with respect to the coordinate system which defines it in Example 8. Note that |Ω ′ n | 1 = 1. Straightforward computations show, and By (47) we see that (10) holds for all n ≥ N Ω := min{n ∈ N : n 2/3 ≥ π −2 c 1 }.