On the L p norm of the torsion function

Bounds are obtained for the Lp norm of the torsion function vΩ, i.e. the solution of −Δv=1,vH10(Ω), in terms of the Lebesgue measure of Ω and the principal eigenvalue 1(Ω) of the Dirichlet Laplacian acting in L2(Ω). We show that these bounds are sharp for 1p2. DOI: https://doi.org/10.1007/s11587-018-0412-x Other titles: On the Lp norm of the torsion function Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-159858 Journal Article Published Version The following work is licensed under a Creative Commons: Attribution 4.0 International (CC BY 4.0) License. Originally published at: Berg, M van den; Kappeler, Thomas (2019). On the L norm of the torsion function. Ricerche di Matematica, 68(2):399-414. DOI: https://doi.org/10.1007/s11587-018-0412-x Ricerche di Matematica https://doi.org/10.1007/s11587-018-0412-x On the Lp norm of the torsion function M. van den Berg1 · T. Kappeler2 Received: 15 February 2018 / Revised: 9 May 2018 © The Author(s) 2018 Abstract Bounds are obtained for the L p norm of the torsion function vΩ , i.e. the solution of −Δv = 1, v ∈ H1 0 (Ω), in terms of the Lebesgue measure of Ω and the principal eigenvalue λ1(Ω) of the Dirichlet Laplacian acting in L2(Ω). We show that these bounds are sharp for 1 ≤ p ≤ 2.


Introduction
Let Ω be a non-empty open set in Euclidean space R m with boundary ∂Ω. It is well-known [2,3] that if the bottom of the Dirichlet Laplacian defined by is bounded away from 0, then has a unique solution denoted by v Ω . The function v Ω is non-negative, pointwise increasing in Ω, and satisfies, The m-dependent constant in the right-hand side of (3) has subsequently been improved [9,16]. We denote the optimal constant in the right-hand side of (3) by suppressing the m-dependence. The torsional rigidity of Ω is defined by It 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 [1,14].
It also arises in the definition of gamma convergence [7] and in the study of minimal submanifolds [12]. Moreover, T 1 (Ω)/|Ω| equals E x (τ Ω ), the expected lifetime τ Ω of Brownian motion in Ω, when averaged with respect to the uniform distribution over all starting points x ∈ Ω.
The torsion function has been studied extensively and numerous works have been written on this subject. We just mention the paper [6], and the references therein. There the Kohler-Jobin rearrangement technique has been applied to the p-torsional rigidity, involving the p-Laplacian, and its first Dirichlet eigenvalue.
A classical inequality, e.g. [14], asserts that the function F 1 defined on the open sets in R m with finite Lebesgue measure Since Ω has finite Lebesgue measure |Ω|, Moreover λ 1 (Ω) is in that case the principal eigenvalue of the Dirichlet Laplacian. Motivated by (5) and (6) we make the following where It follows from the Faber-Krahn inequality that if |Ω| < ∞ then λ 1 (Ω) > 0. The converse does not hold for if Ω is the union of infinitely many disjoint balls of radii 1 then λ 1 (Ω) > 0 but Ω has infinite measure. Note that 2T 2 2 (Ω)/|Ω| equals the second moment of the expected lifetime of Brownian motion in Ω, when averaged with respect to the uniform distribution over all starting points x ∈ Ω.
Note that Ω → T p (Ω) is increasing while Ω → λ 1 (Ω) and Ω → |Ω| −1/ p are decreasing. It is straightforward to verify that Our main results are the following.
(ii) The mapping p → G convex p is non-decreasing, and It follows from (12)  A monotone increasing sequence of cuboids which exhausts the open connected set bounded by two parallel (m − 1)-dimensional hyperplanes is a minimising sequence for G convex ∞ . See also Theorem 2 in [5].
In particular The maximising sequence constructed in [4] for F 1 is also a maximising sequence for In particular if 1 ≤ p ≤ 2, then (vi) and This paper is organised as follows. In Sect. 2 we prove Theorems 1 and 2. The proof of Theorem 3 will be given in Sect. 3.
We note that a general multiplicative inequality involving T p (Ω), λ 1 (Ω) and |Ω| will involve three exponents. However, the requirement that it be invariant under homotheties reduces the number of exponents to two. In Sect. 4 we briefly discuss this two-parameter family of inequalities, and determine which parameter pair yields a finite supremum.
(ii) To prove (10) we observe that since Ω has finite Lebesgue measure the spectrum of the Dirichlet Laplacian acting in L 2 (Ω) is discrete, and consists of an increasing sequence of eigenvalues accumulating at infinity, where we have included multiplicities. We denote a corresponding orthonormal basis of eigenfunctions by {ϕ j,Ω , j = 1, 2, 3, . . .}. The resolvent of the Dirichlet Laplacian acting in L 2 (Ω) is compact, and its kernel H Ω has an L 2 -eigenfunction expansion given by So v Ω , defined by (2), is given by Since v Ω ∈ L 2 (Ω) we have by orthonormality that We conclude that Multiplying both sides of the inequality above with λ 1 (Ω)/|Ω| 1/2 we obtain that This, together with the previous inequality, implies that F 2 (Ω) ≤ 1. We now use Hölder's inequality, and interpolate with 0 < α < 1, ρ > 1 as follows.
Multiplying both sides of the inequality above with λ 1 (Ω) p /|Ω| gives that Proof of Theorem 2 (i) We let Ω n be the disjoint union of one ball of radius 1 and n balls with radii r n , with r n < 1. Then where B 1 = {x ∈ R m : |x| < 1}. Since r n < 1 we have that Since T p p is additive on disjoint open sets we have by scaling that We now choose r n as to minimise the right-hand side of (21), . which implies the assertion.

This gives that
(ii) The first part of the assertion follows directly by (9). To prove the second part we recall John's ellipsoid theorem [10,11] which asserts the existence of an ellipsoid Υ with centre c such that x ∈ Υ }. This is the dilation of Υ by the factor m. 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 (2) for Υ is given by By changing to spherical coordinates, we find that where ω m = |B 1 |. Since Ω → v Ω is increasing we have by (8) that Ω → T p (Ω) is increasing, and Since Ω ⊂ mΥ , By the 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 (22), (23), (24), and (8) gives (12).

Proof of Theorem 3
(i) It follows from the second inequality in (9) Taking subsequently the supremum over all Ω with finite measure we obtain the first assertion under (i). As (9) holds for all open sets with finite measure, it also holds for all bounded convex sets. Then, the preceding argument gives the second assertion under (i).

A two-parameter family of inequalities
As mentioned at the end of the Introduction one can define a two-parameter family of products involving T p (Ω), λ 1 (Ω), and |Ω|, which is invariant under homotheties.

Definition 3
For an open set Ω ⊂ R m with finite Lebesgue measure, p ≥ 1, q ∈ R, (ii) It is straightforward to verify that the quantities defined in (39) which tends to infinity as r n tends to 0. Next suppose q ≤ 1. By (40), (4) and Faber-Krahn,

This proves part (ii).
In general it looks very difficult to compute F p,q or even F p = F p,1 , p > 2, with the exception of F p,0 . G. Talenti in [15] obtained a pointwise estimate between the rearrangement of the torsion function of a generic set with finite measure and the torsion function of the ball with the same measure. In particular this estimate implies that the L p norm of the torsion function is maximised by the L p norm of the torsion function for the ball with the same measure. Hence, by (39) and (40) we have Since λ 1 (Ω) = π 2 4a 2 1 , q ≤ 1, and 2a 1 ≤ 1, we have that (2a 1 ) 2−2q ≤ 1. Hence By taking the supremum over all Ω ⊂ R 1 with measure 1 we obtain that F p,q ≤ 2 −(1+2 p)/ p c p π 2q .
To obtain a lower bound for F p,q we make the particular choice of Ω = B 1 . This gives that F p,q ≥ F p,q (B 1 ) = 2 −(1+2 p)/ p π 2q c p .
and (45) follows from (50) and the definition of c p in (47).
To prove (46) we just observe that the maximum of the torsion function and the first Dirichlet eigenvalue are determined by the largest interval in Ω, i.e. a 1 . Since q ≤ 1 we maximise the resulting expression by taking a 1 = 1 2 . Note that as B 1 is convex we also have that and recover the known values F 1 = π 2 12 , F ∞ = π 2 8 [4,5]. Note that F 1 < F 2 = π 2 √ 120 < 1, which is in contrast with the higher dimensional situation m ≥ 2, where F p = 1, 1 ≤ p ≤ 2.
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