On Polya's inequality for torsional rigidity and first Dirichlet eigenvalue

Let $\Omega$ be an open set in Euclidean space with finite Lebesgue measure $|\Omega|$. We obtain some properties of the set function $F:\Omega\mapsto \R^+$ defined by $$ F(\Omega)=\frac{T(\Omega)\lambda_1(\Omega)}{|\Omega|} ,$$ where $T(\Omega)$ and $\lambda_1(\Omega)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P\'olya bound $F(\Omega)\le 1,$ and show that $$F(\Omega)\le 1- \nu_m T(\Omega)|\Omega|^{-1-\frac2m},$$ where $\nu_m$ depends only on $m$. For any $m=2,3,\dots$ and $\epsilon\in (0,1)$ we construct an open set $\Omega_{\epsilon}\subset \R^m$ such that $F(\Omega_{\epsilon})\ge 1-\epsilon$.


Introduction
Let Ω be an open set in R m with finite Lebesgue measure |Ω|, and let v Ω : Ω → R + denote the corresponding torsion function, i.e. the unique solution of − ∆v = 1, v ∈ H 1 0 (Ω). (1.1) The torsional rigidity of Ω is defined by T (Ω) = Ω v Ω . As v Ω ≥ 0, the torsional rigidity is the L 1 (Ω) norm of v Ω . The following variational characterisation is well known The torsional rigidity plays a key role in different parts of analysis. For example the torsional rigidity of a cross section of a beam appears in the computation of the angular change when a beam of a given length and a given modulus of rigidity is exposed to a twisting moment [2], [14]. It also arises in the calculation of the heat content of sets with time-dependent boundary conditions [3], in the definition of gamma convergence [6], and in the study of minimal submanifolds [11]. Moreover, T (Ω)/|Ω| equals the expected lifetime of Brownian motion in Ω when averaged with respect to the uniform distribution over all starting points x ∈ Ω.
A classical inequality of Pólya, [14], asserts that the function F defined by We note that F is scale independent i.e. for any homothety αΩ, α > 0, of Ω we have that F (αΩ) = F (Ω).
The main results of this paper are the following. However, for the restriction of F (·) to the class of convex sets in R m , we have the following. (1.7) Theorem 1.2 disproves the conjecture in [4] that F (Ω) ≤ π 2 12 . Theorem 1.5 below goes some way towards proving the π 2 /12 bound for open bounded, planar, convex sets. In order to state our main result for convex sets, we introduce the following notation. For a convex set with finite measure, we denote by w the minimum width of Ω (or simply the width of Ω), which is obtained by minimising among all pairs of parallel supporting hyperplanes of Ω the distance between such hyperplanes. The projection of Ω onto one of the minimising hyperplanes is denoted by E. The first eigenvalue of the (m − 1)-dimensional Dirichlet Laplacian acting in L 2 (E) is denoted by Λ.
If Ω is an open, bounded, convex set in R m with w and Λ as above, then .
The main idea in the proof of Theorem 1.2 is that if Ω is an open, bounded and connected set, then we can find x 0 ∈ Ω and δ > 0 such that punching a hole in Ω centered at x 0 with radius δ increases F . In the proof of Theorem 1.2, we take an m-dimensional cube with side-length L and punch N m holes with the same radius δ in a periodic arrangement. We show that we can find L, N, δ depending on (and m) such that the corresponding value of F for the punched cube exceeds 1 − . As mentioned above, F is invariant under homotheties, and so we could have chosen L = 1. However, it is convenient to keep L undetermined so that we have a homothety or scaling check in the various bounds.
To see that punching a hole increases F , we take Ω open, bounded, connected, and with smooth boundary. Let ϕ 1 ∈ H 1 0 (Ω) be a Dirichlet eigenfunction corresponding to λ 1 (Ω), and let v Ω be the solution of (1.1). We observe that So there exists x 0 ∈ Ω such that Let Ω δ,x0 = Ω \ B(x 0 ; δ), where B(x 0 ; δ) is the closed ball of radius δ > 0 centered at x 0 . We want to show that if δ is small enough, then F (Ω δ,x0 ) > F (Ω). In the planar case m = 2, a classical asymptotic formula (see, for instance, [8,Theorem 1.4.1] and the references therein) gives that Moreover, from [12, Theorem 8.1.6], we have that By (1.11) and (1.12), we have that Hence F (Ω δ,x0 ) > F (Ω) for δ sufficiently small. This paper is organised as follows. In Section 2 we prove Theorem 1.1. In Section 3 we prove Theorem 1.2, and in Section 4 we prove Theorems 1.4 and 1.5 respectively.

Proof of Theorem 1.1
Let v Ω be the torsion function of Ω. By choosing v Ω as a test function for the Rayleigh quotient for λ 1 (Ω), we obtain that We have that For every θ ∈ (0, M ), we have that Indeed, since v Ω satisfies the torsion equation (1.1) in Ω, arguing similarly to [16], we have that for θ ∈ (0, M ), and Denote the perimeter of a measurable set A by Per(A). Applying Hölder's inequality to Per({v Ω > θ}) = {v=θ} dH m−1 , we obtain that By the isoperimetric inequality we have that This, together with (2.3), gives the differential inequality Integrating this differential inequality gives (2.1).
Using (2.1) and (2.4), it is straightforward to verify that The inequality (1 + y) α ≥ 1 + αy + y 2 , α ≥ 2, y ≥ −1 then gives that Hölder's inequality then yields that Using the expression for Q and Hölder's inequality gives that This concludes the proof of Theorem 1.1.

Proof of Theorem 1.2
In this section we provide an example of an open connected set Ω in R m which satisfies (1.4). As the technical tools depend heavily on the relation between torsional rigidity and heat equation we recall some of the essential ingredients in Section 3.1 below. The necessary bounds for the first eigenfunction and eigenvalue with Dirichlet boundary conditions on a ball centred in an m-dimensional cube with Neumann boundary conditions will be obtained in Section 3.2. The proof of Theorem 1.2 will be deferred to Section 3.3.

Heat equation and torsional rigidity
We denote the Dirichlet heat kernel for Ω by p Ω (x, y; t), x, y ∈ Ω, t > 0. The integral defined by The interpretation of (3.1), (3.2), and (3.3) is that u Ω (x; t) represents the temperature at point x at time t when the initial temperature in Ω is 1 and the temperature of ∂Ω is 0 for all t > 0. The heat content of Ω at time t is defined as The Dirichlet heat kernel for Ω has the following eigenfunction expansion: It follows from Parseval's formula that The solution of (1.1) is given by It follows that i.e., the torsional rigidity is the integral of the heat content. By the first identity in (3.5), (3.6), and Fubini's theorem we have that where we have used Parseval's identity in the last equality above. This implies Pólya's bound (1.3). The bound also follows by (3.5) and (3.6). By the first identity in (3.7) we obtain that

Eigenfunction and eigenvalue bounds
We introduce the following notation. Let Ω L = (− L 2 , L 2 ) m be an open cube in R m with measure L m , and let K be a compact subset of Ω L . We denote the first eigenvalue of the Laplacian acting in L 2 (Ω L − K) with Neumann boundary conditions on ∂Ω L and Dirichlet boundary conditions on ∂K by µ 1,K,L . We denote the corresponding normalised eigenfunction by ϕ 1,K,L .
The following shows that the L 1 norm of the first eigenfunction converges to L m/2 as µ 1,K,L ↓ 0.
Proof. To prove (3.8), we note that by Cauchy-Schwarz, This proves the right-hand side of (3.8). To prove the left-hand side of (3.8), we denote the heat kernel with Neumann boundary conditions on ∂Ω L and Dirichlet boundary conditions on ∂K by π K,L (x, y; t).
By the eigenfunction expansion of π K,L (x, y; t), we have for t > 0 that where π Ω L (x, y; t) is the Neumann heat kernel for the cube Ω L , and where we have used the eigenfunction expansion of the latter together with separation of variables. Taking the supremum over all x ∈ Ω L − K gives that Furthermore, since ϕ 1,K,L 2 L 2 (Ω L −K) = 1, we have by the positivity of ϕ 1,K,L that We choose t > 0 as to maximise the right-hand side above. This proves the left-hand side of (3.8).
In the sequel we need upper and lower bounds for the first Dirichlet eigenvalue µ 1,K,L where K = B(0; δ) ⊂ Ω L . These were obtained for general compact sets K ⊂ Ω L ⊂ R m , m = 3, 4, ... in [17] and [18] in terms of the Newtonian capacity cap (K) of K in R m . The various m-dependent constants in [17, Propositions 2.2, 2.3, 2.4] and in [18,Theorem A] have not been evaluated. We supply these in the Lemmas 3.2 and 3.3 below. We consider general compact subsets as the proofs (for m = 3, ...) are hardly more involved than the special case of a ball.
Proof. By the L 2 -eigenfunction expansion of π K,L (x, y; t) we have that As in [17] and [18], we introduce some Brownian motion tools. Let (B(s), s ≥ 0;P x , x ∈ Ω L ) be Brownian motion with reflection on ∂Ω L . For a compact subset K ⊂ Ω L we let Integrating both sides of (3.13) with respect to x over Ω L − K gives, with (3.15), that It follows that Following [18, p.449], we defineK as the subset of R m by the method of images, so that in each tiling L-cube of R m we have a reflected image of K. Let (B(s), s ≥ 0; P x , x ∈ R m ) be Brownian motion on R m , and define the first hitting time of a closed set A by . For a compact set K ⊂ R m , we define the last exit time by where we put L K = +∞ if the supremum is over the empty set. Then P By [13], we have that where p(x, y; s) = (4πs) −m/2 e −|x−y| 2 /(4s) , (3.18) and where µ K (dy) is the equilibrium measure of the compact K. Next we choose t = L 2 . By the above, we have that For y ∈ K and x ∈ Ω L − K, we have that |x − y| ≤ diam(Ω L ) = mL 2 . So, by (3.19), we conclude that where k m is given by (3.11). This proves part (i) of the lemma. To prove part (ii) of the lemma, we follow the Remark on p.451 in [18], and define the trial function where is the Newtonian capacity of the ball with radius 1 in R m . Then Hence In order to compute the integral with respect to x over R m , we write Changing variables s = σ 2 , s = σ 2 gives that the right-hand side above equals (3.24) By (3.17) and (3.18), we have that for y ∈ K, Putting (3.21)-(3.25) together gives that (3.26) The last inequality in (3.26) follows from uniform bounds on the Γ function. See for example [1, 6.1.38].
We obtain a lower bound for ψ L 2 (Ω L −K) as follows. By (3.20), By rearrangement, we have that where Ω * L is the ball centered at 0 with the same measure as Ω L . Hence (3.30) Proof. We define 1, x ∈ Ω L ∩ {|x| > L 2 }. Then This proves the upper bound in (3.30).
To prove the lower bound we use the method of descent as in [18, p.451], and observe that for m = 2, µ 1,B(0;δ),L equals the bottom of the spectrum of the Laplacian with Neumann boundary conditions on the boundary of the cube Ω L = (− L 2 , L 2 ) 3 , and Dirichlet boundary conditions on the cylinder C L,δ = {(x 1 , x 2 , x 3 ) : − L 2 < x 1 < L 2 , x 2 2 + x 2 3 < δ 2 } of height L and radius δ through the centre of that cube. By the lower bound in Lemma 3.2 for m = 3, we obtain that : µ is a probability measure supported on K .
By monotonicity, we have that cap (C L,δ ) ≥ cap (∪ N j=1 B j ). To bound the latter, we let σ j be the surface measure on the boundary of the jth ball, and let We wish to find an upper bound for the energy (3.32) If N = 1 then the expression above equals the inverse of cap (B(0; δ)). The contribution from the N terms with j = k in (3.32) equals 1 4πN δ . Furthermore, the contribution of the terms with |j − k| = 1 in (3.32) is bounded by N −1 (4πN δ 2 ) 2 σ1(dx)σ2(dy) 4π|x−y| . As δ −1 dσ j is the equilibrium measure for the jth ball, we have that δ −1 σ2(dy) 4π|x−y| ≤ 1. We conclude that Similarly the contribution of the terms with |j − k| = 2 in (3.32) is bounded by It remains to find an upper bound for the terms in (3.32) for |j − k| ≥ 3. For x, y on the surface of the j, kth balls we have that |x − y| ≥ 2|k − j − 1|δ. Hence the contribution from the terms with |j − k| ≥ 3 in (3.32) is bounded from above by Collecting all terms, we see that the expression under (3.32) is bounded from above by 3+log N 4πN δ . Hence where we have used that N ≥ 3, δ ≤ L/6. Numerical evaluation gives that k 3 ≥ 0.0101... ≥ 1 100 . The lower bound in Lemma 3.3 follows by (3.31) and (3.33).

Proof of Theorem 1.2
We partition Ω L into N m disjoint open cubes C 1 , . . . , C N m each with measure (L/N ) m . We denote the centres of these cubes by c 1 , . . . , c N m respectively. Let 0 < δ < L 2N , and put We denote the Dirichlet heat kernel for Ω δ,N,L and Ω L by p Ω δ,N,L (x, y; t) and p Ω L (x, y; t) respectively. The heat kernel with Neumann boundary conditions on ∂Ω L and Dirichlet boundary conditions on ∂Ω δ,N,L − ∂Ω L will be denoted by π Ω δ,N,L (x, y; t). Let T > 0, and let > 0 be arbitrary. We bound the torsional rigidity for Ω δ,N,L from below as follows.
Hence the second term in the right-hand side of (3.35) is bounded in absolute value by  (C1−B(c1;δ)) .
We conclude that the first term in the left-hand side of (3.35) is bounded from below by gives that For m = 2 we obtain by (3.41) and Lemma 3.3 that x ⊂ Υ}. This is the dilation of Υ by a factor of m with centre c. Υ is the ellipsoid of maximal volume in Ω. By translating both Ω and Υ we may assume that It is easily verified that the unique solution of (1.1) for Υ is given by By changing to spherical coordinates, we find that Since Ω ⊂ mΥ, By monotonicity of Dirichlet eigenvalues, we have that λ 1 (Ω) ≥ λ 1 (mΥ). The ellipsoid mΥ is contained in a cuboid with lengths 2ma 1 , . . . , 2ma m . So we have that Combining (4.1), (4.2) and (4.3) gives the lower bound in (1.6). To prove part (ii) we note (see [10]) that for bounded, convex Ω in R 2 , Furthermore, by [7, Theorem 5.1], we have that for Ω convex in R m , (4.5) The assertion under (1.7) follows by (4.4) and (4.5).
Proof of Theorem 1.5. We claim that it is always possible to choose z 1 , z 2 ∈ ∂Ω such that |z 1 − z 2 | = w, and therefore the vector z 1 − z 2 is orthogonal at z 1 and z 2 to two parallel supporting hyperplanes achieving the minimal distance w.
To show this, the first step is to prove that for any direction ν, there exist two pointsz 1 ,z 2 ∈ ∂Ω such that the supporting hyperplanes tangent to ∂Ω at these points are parallel to each other. Indeed, assuming that the set is smooth and strictly convex (the general case follows at once from an approximation argument), for every η ∈ S m−1 such that η ·ν > 0, there exists a unique pointx(η) ∈ ∂Ω where the outer unit normal is η. Moreover, there exists a unique pointx(x) ∈ ∂Ω such thatx −x is parallel to ν. We denote by ξ(x) the inner unit normal to Ω atx and observe that ξ · ν > 0. Therefore, denoting by S ν = {η ∈ S, η · ν ≥ 0}, the map ξ(x(x(η))) (possibly extended so that ξ = −η when η · ν = 0) is a continuous map from S ν into itself. Brouwer's fixed point theorem provides the existence ofη such that ξ(η) =η and this completes the first step. Now, in view of the above result, assuming that T 1 and T 2 are two supporting hyperplanes at distance w, there exist two points z 1 , z 2 ∈ ∂Ω such that z 1 − z 2 is orthogonal to T 1 and T 2 , and the supporting hyperplanes tangent to ∂Ω at z 1 and z 2 are parallel to each other. On one hand we have w ≤ |z 1 − z 2 |, and on the other hand, by construction, |z 1 − z 2 |, is not greater than the distance between T 1 and T 2 . This forces z 1 and z 2 to belong to T 1 and T 2 and hence w = |z 1 − z 2 |, which proves our claim.
We introduce a reference frame in R m , (x, y) ∈ R × R m−1 where x points in the direction z 1 − z 2 and (0, 0) = z 1 + z 2 2 . Denoting by E the projection of Ω onto the hyperplane x = 0, we have Ω = {(x, y) ∈ R m : l(y) < x < L(y), y ∈ E}, (4.6) where L : E → R is concave, l : E → R is convex, l ≤ L and max{L(y) − l(y) : y ∈ E} = w. This maximum is achieved at y = 0. We note that {(x, y) ∈ R m : x = 0, (x, y) ∈ Ω} ⊃ 1 2 E, where 1 2 E is the homothety of E by 1 2 with respect to y = 0. We consider the two-sided cone with base 1 2 E and vertices ( w 2 , 0) and ( −w . This two-sided cone contains a cylinder C h with height 2h and base 1 − 2h w 1 2 E. By monotonicity of Dirichlet eigenvalues, we have that λ 1 (Ω) ≤ λ 1 (C h ). By separation of variables, we have that (4.7) Minimising the right-hand side of (4.7) with respect to h gives that To prove part (ii), we note that for m = 2 Theorem 1.1 gives that for any Ω with finite Lebesgue measure, F (Ω) ≤ 1 − π λ 1 (Ω)|Ω| + π .

(4.13)
For w |E| small we use (4.13) as an upper bound, while for w |E| large we use (4.11) as an upper bound. The cross-over point value of w |E| where the right-hand side of (4.11) equals the right-hand side of (4.13) is bounded from below by 0.0015197. This, together with the bound under (4.11), gives the assertion under (1.10).
Below we list some known numerical values of F for some convex planar shapes. To obtain the third line in the table we note that for a half-disc of area πa 2 /2, the torsional rigidity is given by π 8 − 1 π a 4 (see [19, pp. 265-267]). The first Dirichlet eigenvalue of the half-disc is the second Dirichlet eigenvalue of the full disc and equals j 2 1,1 a 2 . The first Dirichlet eigenvalue of an equilateral triangle with side lengths a is given by 16π 2 3a 2 . So we obtain the last line in the table above.