Spectral Bounds for the Torsion Function

Let Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} be an open set in Euclidean space Rm,m=2,3,...\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^m,\, m=2,3,...$$\end{document}, and let vΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\Omega }$$\end{document} denote the torsion function for Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}. It is known that vΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\Omega }$$\end{document} is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in L2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}^2(\Omega )$$\end{document}, denoted by λ(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda (\Omega )$$\end{document}, is bounded away from 0. It is shown that the previously obtained bound ‖vΩ‖L∞(Ω)λ(Ω)≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert v_{\Omega }\Vert _{{\mathcal {L}}^{\infty }(\Omega )}\lambda (\Omega )\ge 1$$\end{document} is sharp: for m∈{2,3,...}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in \{2,3,...\}$$\end{document}, and any ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0$$\end{document} we construct an open, bounded and connected set Ωϵ⊂Rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\epsilon }\subset \mathbb {R}^m$$\end{document} such that ‖vΩϵ‖L∞(Ωϵ)λ(Ωϵ)<1+ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert v_{\Omega _{\epsilon }}\Vert _{{\mathcal {L}}^{\infty }(\Omega _{\epsilon })} \lambda (\Omega _{\epsilon })<1+\epsilon $$\end{document}. An upper bound for vΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\Omega }$$\end{document} is obtained for planar, convex sets in Euclidean space R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}, which is sharp in the limit of elongation. For a complete, non-compact, m-dimensional Riemannian manifold M with non-negative Ricci curvature, and without boundary it is shown that vΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_{\Omega }$$\end{document} is bounded if and only if the bottom of the spectrum of the Dirichlet–Laplace–Beltrami operator acting in L2(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}^2(\Omega )$$\end{document} is bounded away from 0.


Introduction
Let Ω be an open set in R m , and let Δ be the Laplace operator acting in L 2 (R m ). Let (B(s), s ≥ 0; P x , x ∈ R m ) be Brownian motion on R m with generator Δ. For x ∈ Ω we denote the first exit time, and expected lifetime of Brownian motion by respectively, where E x denotes the expectation associated with P x . Then v Ω is the torsion function for Ω, i.e. the unique solution of The bottom of the spectrum of the Dirichlet Laplacian acting in L 2 (Ω) is denoted by 388 M. van den Berg IEOT λ(Ω) = inf It was shown in [1,2] that v Ω L ∞ (Ω) is finite if and only if λ(Ω) > 0. Moreover, if λ(Ω) > 0, then The upper bound in (4) was subsequently improved (see [3] where c = (5(4 + log 2)) 1/2 .
In Theorem 1 below we show that the coefficient 1 of λ(Ω) −1 in the left-hand side of (4) is sharp.
The set Ω is constructed explicitly in the proof of Theorem 1. It has been shown by L. E. Payne (see (3.12) in [4]) that for any convex, with equality if Ω is a slab, i.e. the connected, open set, bounded by two parallel (m − 1)-dimensional hyperplanes. Theorem 2 below shows that for any sufficiently elongated, convex, planar set (not just an elongated rectangle) v Ω L ∞ (Ω) λ(Ω) is approximately equal to π 2 8 . We denote the width and the diameter of a bounded open set Ω by w(Ω) (i.e. the minimal distance of two parallel lines supporting Ω), and diam(Ω) = sup{|x − y| : x ∈ Ω, y ∈ Ω} respectively.
In the Riemannian manifold setting we denote the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator by (3). We have the following.
where K is the constant in the Li-Yau inequality in (35) below.
The proofs of Theorems 1, 2, and 3 will be given in Sects. 2, 3 and 4 respectively.
Below we recall some basic facts on the connection between torsion function and heat kernel. It is well known (see [5][6][7]) that the heat equation , where x ∈ Ω, y ∈ Ω, t > 0. This Dirichlet heat kernel satisfies, Then, and by (1) v It is straightforward to verify that v Ω as in (8) satisfies (2).

Proof of Theorem 1
We introduce the following notation.
. The set Ω δ,N,L also features in [9], where the sharpness of an inequality due to Pólya has been established. Below we will show that for any > 0 we can choose δ, N such that In Lemma 4 below we show that λ(Ω δ,N,L ) is approximately equal to the first eigenvalue, μ 1,B(0;δ),L/N , of the Laplacian with Neumann boundary conditions on ∂C L/N , and with Dirichlet boundary conditions on ∂B(0; δ). The requirement μ 1,B(0;δ),L/N not being too small stems from the fact that the approximation of replacing the Neumann boundary conditions on C L is a surface effect which should not dominate the leading term μ 1,B(0;δ),L/N .
Proof. Let ϕ 1,B(0;δ),L/N be the first eigenfunction (positive) corresponding to μ 1,B(0;δ),L/N , and normalised in L 2 (C L/N − B(0; δ)). In order to prove the lemma we construct a test function by periodically extending ϕ 1,B(0;δ),L/N to all cubes Q 1 , . . . Q N m of Ω δ,N,L . We denote this periodic extension by f . We define . So C L,N is the sub-cube of C L with the outer layer of cubes of size L/N removed. Letf Vol. 88 (2017) Spectral Bounds for the Torsion Function 391 Thenf ∈ H 1 0 (Ω δ,N,L ), and since f restricted to any of the cubes Q i in Ω δ,N,L is normalised. Furthermore where we have used the last hypothesis in the lemma. By (9), (10), the Rayleigh-Ritz variational formula, and the hypothesis N ≥ 10, where ϕ 1 = ϕ 1 L ∞ (C L/N −B(0;δ)) . By (13), we have that the third term in the right-hand side of (14) equals The term with j = 1 in (15) is bounded from above by where we used the fact that 1 = C L/N −B(0;δ) ϕ 2 1 ≤ ϕ 1 C L/N −B(0;δ) ϕ 1 . It was shown on p.586, lines -3,-4, in [9] (with appropriate adjustment in notation) that provided the last term in the round brackets is non-negative. The optimal choice for s gives that Vol. 88 (2017) Spectral Bounds for the Torsion Function 393 By further restricting the range for μ 1 , we have that the first term with j = 1 in (15) is then bounded from above by The terms with j ≥ 2 in (15) give, by Cauchy-Schwarz for both the series in j, and the integral over To bound the first series in the right-hand side of (17), we note that the μ j 's are bounded from below by the Neumann eigenvalues of the cube C L/N . So Similarly to the proof of Lemma 3.1 in [9], we have that Finally, μ 2 ≥ π 2 N 2 L 2 , together with (12), (14), (16), (17), (18), and the choice
Next consider the case m = 3, 4, . . . . We apply Lemma 3.2 in [9] to the case K = B(0; δ), and denote the Newtonian capacity of K by cap(K). Then cap(B(0; δ)) = κ m δ m−2 , where κ m is the Newtonian capacity of the ball with radius 1 in R m . Then Lemma 3.2 gives that there exists C ≥ 1 such that We note that the choice α = 4 3 in either (22) or in (27) gives, by (24), the decay rate

Proof of Theorem 2
In view of Payne's inequality (6) it suffices to obtain an upper bound for v Ω L ∞ (Ω) λ(Ω). We first observe, that by domain monotonicity of the torsion function, v Ω is bounded by the torsion function for the (connected) set bounded by the two parallel lines tangent to Ω at distance w(Ω). Hence In order to obtain an upper bound for λ(Ω), we introduce the following notation. For a planar, open, convex set, with finite measure, we let z 1 , z 2 be two points on the boundary of Ω which realise the width. That is there are two parallel lines tangent to ∂Ω, at z 1 and z 2 respectively, and at distance w(Ω).
Let the x-axis be perpendicular to the vector z 1 z 2 , containing the point 1 2 (z 1 + z 2 ). We consider the family of line segments parallel to the x-axis, obtained by intersection with Ω, and let l 1 , l 2 be two points on the boundary of Ω which realise the maximum length L of this family. The quadrilateral with vertices, z 1 , z 2 , l 1 , l 2 is contained in Ω. This quadrilateral in turn contains a rectangle with side-lengths h, and 1 − h w(Ω) L respectively, where h ∈ [0, w(Ω)) is arbitrary. Hence, by domain monotonicity of the Dirichlet eigenvalues, we have that Minimising the right-hand side above with respect to h gives that

396
M. van den Berg IEOT As w(Ω) ≤ L we obtain by (30) that In order to complete the proof we need the following.

Proof of Theorem 3
We denote by d : M × M → R + the geodesic distance associated to (M, g). For x ∈ M, R > 0, B(x; R) = {y ∈ M : d(x, y) < R}. For a measurable set A ⊂ M we denote by |A| its Lebesgue measure. The Bishop-Gromov Theorem (see [10]) states that if M is a complete, non-compact, m-dimensional, Riemannian manifold with non-negative Ricci curvature, then for p ∈ M , the map r → |B(p;r)| r m is monotone decreasing. In particular |B(p; r 2 )| |B(p; r 1 )| ≤ r 2 r 1 m , 0 < r 1 ≤ r 2 .
Corollary 3.1 and Theorem 4.1 in [11], imply that if M is complete with nonnegative Ricci curvature, then for any D 2 > 2 and 0 < D 1 < 2 there exist constants 0 < C 1 ≤ C 2 < ∞ such that for all x ∈ M, y ∈ M, t > 0,