Spectral bounds for the torsion function

Let $\Omega$ be an open set in Euclidean space $\R^m,\, m=2,3,...$, and let $v_{\Omega}$ denote the torsion function for $\Omega$. It is known that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $\Leb^2(\Omega)$, denoted by $\lambda(\Omega)$, is bounded away from $0$. It is shown that the previously obtained bound $\|v_{\Omega}\|_{\Leb^{\infty}(\Omega)}\lambda(\Omega)\ge 1$ is sharp: for $m\in\{2,3,...\}$, and any $\epsilon>0$ we construct an open, bounded and connected set $\Omega_{\epsilon}\subset \R^m$ such that $\|v_{\Omega_{\epsilon}}\|_{\Leb^{\infty}(\Omega_{\epsilon})} \lambda(\Omega_{\epsilon})<1+\epsilon$. An upper bound for $v_{\Omega}$ is obtained for planar, convex sets in Euclidean space $M=\R^2$, 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 $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator acting in $\Leb^2(\Omega)$ is bounded away from $0$.


Introduction
Let Ω be an open set in R m , and let ∆ 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 ∆. For x ∈ Ω we denote the first exit time, and expected lifetime of Brownian motion by T Ω = inf{s ≥ 0 : B(s) / ∈ Ω}, 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 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 [11]) to 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 [9]) that for any convex, open Ω ⊂ R m for which λ(Ω) > 0, 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.

Theorem 2.
If Ω is a bounded, planar, open, convex set with width w(Ω), and diameter diam(Ω), then In the Riemannian manifold setting we denote the bottom of the spectrum of the Dirichlet-Laplace-Beltrami operator by (3). We have the following.
Theorem 3. Let M be a complete, non-compact, m-dimensional Riemannian manifold, without boundary, and with non-negative Ricci curvature. There exists K < ∞, depending on M only, such that if Ω ⊂ M is open, and λ(Ω) > 0, then 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 Sections 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 has a unique, minimal, positive fundamental solution p M (x, y; t), where x ∈ M , y ∈ M , t > 0. This solution, the heat kernel for M , is symmetric in x, y, strictly positive, jointly smooth in x, y ∈ M and t > 0, and it satisfies the semigroup property for all x, y ∈ M and t, s > 0, where dz is the Riemannian measure on M . See, for example, [10] for details. If Ω is an open subset of M, then we denote the unique, minimal, positive fundamental solution of the heat equation on Ω by p Ω (x, y; t), where x ∈ Ω, y ∈ Ω, t > 0. This Dirichlet heat kernel satisfies, p Ω (x, y; t) ≤ p M (x, y; t), x ∈ Ω, y ∈ Ω, t > 0. Define Then, and by (1) v It is straightforward to verify that v Ω as in (8)    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 since f restricted to any of the cubes Q i in Ω δ,N,L is normalised. Furthermore Hence where we have used the last hypothesis in the lemma. By (9), (10), the Rayleigh-Ritz variational formula, and the hypothesis N ≥ 10, To obtain an upper bound for v Ω δ,N,L L ∞ (Ω δ,N,L ) , we change the Dirichlet boundary conditions on ∂C L to Neumann boundary conditions. This increases the corresponding heat kernel, torsion function, and L ∞ norm respectively. By periodicity, we have that 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 [3] (with appropriate adjustment in notation) that provided the last term in the round brackets is non-negative. The optimal choice for s gives that 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 (17) Finally, µ 2 ≥ π 2 N 2 L 2 , together with (12), (14), (16), (17), (18), and the choice Proof of Theorem 1. Let 1 < α < 2. By (11) and (19), we have that First consider the planar case m = 2. Recall Lemma 3.1 in [3]: for δ < L/(6N ), Let where 1 < α < 2. Let N 1 ∈ N be such that for all N ≥ N 1 , δ * < L/(6N ). We now use (21) to see that there exists C > 1 such that (In fact C = max{100, 8π/(4 − π)}). We subsequently let N 2 ∈ N be such that for all N ≥ N 2 ,

By (20), (23), and all
where C depends on C and on m only. Finally, we let N 3 ∈ N be such that for all N ≥ N 3 , C N 1−α + N (α−2)/2 < .
Next consider the case m = 3, 4, . . . . We apply Lemma 3.2 in [3] 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 provided We choose This gives inequality (23) by (25). The requirement (26) holds for all N ≥ N 1 , where N 1 is the smallest natural number such that N 2−α 1 ≥ 16κ m . The remainder of the proof follows the lines below (23) with the appropriate adjustment of constants, and the choice of δ * as in (27).
We note that the choice α =

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

It follows that
As w(Ω) ≤ L we obtain by (30) that λ(Ω) ≤ π 2 w(Ω) 2 1 + 7 In order to complete the proof we need the following.

Lemma 5.
If Ω is an open, bounded, convex set in R 2 , and if L is the length of the longest line segment in the closure of Ω, perpendicular to z 1 z 2 , then Proof. Let d 1 , d 2 ∈ ∂Ω such that |d 1 − d 2 | = diam(Ω). We denote the projections of d 1 , d 2 onto the line through z 1 , z 2 by e 1 , e 2 respectively. Let z be the intersection of the lines through z 1 , z 2 and d 1 , d 2 respectively. Then, by the maximality of L, we have that |d 1 − e 1 | ≤ L, |d 2 − e 2 | ≤ L. Furthermore, by convexity, |e 1 − z| + |e 2 − z| ≤ w(Ω). Hence, By (31), we have that . This implies Theorem 2 by (29).

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 [4]) 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 Corollary 3.1 and Theorem 4.1 in [8], 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, (34) Finally, since by (33) the measure of any geodesic ball with radius r is bounded polynomially in r, the theorems of Grigor'yan in [6] imply stochastic completeness. That is, for all x ∈ M, and all t > 0, M dy p M (x, y; t) = 1.
First taking the supremum over all x ∈ Ω(q; R) in the left-hand side of (42), and subsequently taking the supremum over all such x in the right-hand side of (42) gives v Ω(q;R) L ∞ (Ω(q;R)) ≥ λ(Ω(q; R)) −1 .
Observe that the torsion function is monotone increasing in R. Taking the limit R → ∞ in the left-hand side of (43), and subsequently in the right-hand side of (43) completes the proof.