On efficiency and localisation for the torsion function

We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schr\"odinger operator $- \Delta + V$ on an open set $\Omega\subset \R^m$ of finite Lebesgue measure $0<|\Omega|<\infty$ with a real-valued, non-negative, measurable potential $V.$ We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction.


Introduction and main results
Let Ω be an open set in R m , with finite Lebesgue measure, 0 < |Ω| < ∞, and boundary ∂Ω, and let L = −∆ + V, be the Schrödinger operator acting in L 2 (Ω) with the potential V : Ω → R + , R + = [0, ∞) being measurable. The torsion function for Ω is the unique solution of −∆v = 1, v ∈ H 1 0 (Ω). It is denoted by v Ω , and is also referred to as the torsion function for the Dirichlet Laplacian. The function v Ω is non-negative, pointwise increasing in Ω, and satisfies, where λ 1 (Ω) = inf is the first eigenvalue of the Dirichlet Laplacian. Here, and throughout the paper, · p denotes the standard L p norm, 1 ≤ p ≤ ∞. Since |Ω| < ∞ the first eigenvalue is bounded away from 0 by the Faber-Krahn inequality. The mdependent constant in the right-hand side of (1) has subsequently been improved ( [17], [24]). We denote the sharp constant by More generally, the equation Lv = 1 also has a unique solution v Ω,V ∈ H 1 0 (Ω), referred to as the torsion function for L.
In this paper we study the efficiency of the torsion function of Schrödinger operators, and study the phenomenon of localisation. The notion of efficiency, or mean to max ratio, goes back to [21] and [23], where it was introduced for the first Dirichlet eigenfunction. It can be viewed as a (rough) measure of localisation. The mean to max ratio for the torsion function for bounded, open, convex sets in Euclidean space was studied in [15] in the more general context of the p-torsional rigidity. The phenomenon of localisation of eigenfunctions of Schrödinger operators is a prominent and very active research area and has important applications in the applied sciences. The literature is extensive. See for example the review paper [18]. It was discovered in [1] and [2], and the references therein, that under appropriate conditions, v −1 Ω,V can be used for approximating eigenvalues and eigenfunctions of L. It raises the question as to whether under appropriate assumptions, the phenomenon of localisation can also be observed for the torsion function of Schrödinger operators, and suggests to investigate the interplay between the localisation of the torsion function and the one of the first Dirichlet eigenfunction.
The main results of this papers can be described in an informal way as follows. Theorem 1 compares the efficiency of v Ω,V with the one of v Ω under a variety of hypotheses. In addition it shows that for any given Ω, the efficiency for v Ω,V can be arbitrarily close to 1. Theorem 2 asserts that the efficiency for the first eigenfunction of the Dirichlet Laplacian can be arbitrarily close to 1. Among other results, Theorem 3 provides a quantitative estimate, showing that in case the efficiency for the first eigenfunction of the Dirichlet Laplacian is close to one, the corresponding first eigenvalue is large. Finally, Theorem 4 shows that localisation for the torsion function of the Dirichlet Laplacian implies localisation for the first eigenfunction of this operator. (ii) If v Ω,V is the torsion function for L, then its efficiency is its mean to max ratio, If V = 0, then Φ(Ω, 0) is denoted by Φ(Ω), which coincides with the definition in [15].
In fact, where d Ω : Ω → R + is the distance to the boundary function, We denote the spectrum of L with Dirichlet boundary conditions by accumulating at infinity only, and choose a corresponding L 2 -orthonormal basis of eigenfunctions {ϕ 1,Ω,V , ϕ 2,Ω,V , · · · }. If the first Dirichlet eigenvalue λ 1 (Ω, V ) of L has multiplicity 1, then its corresponding eigenspace is one-dimensional, and ϕ 1,Ω,V is uniquely defined up to a sign. Since ϕ 1,Ω,V does not change sign we may choose ϕ 1,Ω,V > 0. In that case we denote the efficiency, or mean to max ratio, of ϕ 1,Ω,V by If V = 0, then E(Ω, 0) is denoted by E(Ω), which coincides with the definition on p.92 in [23]. See also [7]. We note that if Ω is connected, then λ 1 (Ω) is simple.
By examining the proof of Theorem 2 in Section 4 we see that for any m ≥ 2, and ε ∈ (0, 1) there exists an open, bounded and connected set Ω ε ⊂ R m such that (i) E(Ω ε ) ≥ 1 − ε, and (ii) λ 1 (Ω ε )|Ω ε | 2/m is large for ε small. In Theorem 3 below we show that this is a general phenomenon. That is if Ω is any open and connected set in R m , m ≥ 2, with 0 < |Ω| < ∞ such E(Ω) is close to 1, then the eigenfunction is close to its maximum on most of Ω, and λ 1 (Ω)|Ω| 2/m is large. We have a similar phenomenon for the torsion function. Throughout we denote by B(p; R) = {x ∈ R m : |x − p| < R} the open ball with centre p and radius R. We put B R = B(0; R), and ω m = |B 1 |. For Ω open with 0 < |Ω| < ∞, and u ∈ L 1 (Ω), Theorem 3. Let m ≥ 2 and let Ω be a non-empty open set in R m with finite Lebesgue measure, |Ω| < ∞.
Equality occurs if and only if Ω is a ball, and u is a multiple of the torsion function.
(iii) If Ω is connected, then The last result of this paper concerns the localisation of a sequence of torsion functions. We make the following definition.
Let 1 ≤ p < ∞. For n ∈ N, f n ∈ L p (Ω n ), f n ≥ 0, and f n = 0, we define (i) We say that the sequence f n κ-localises in L p if 0 < κ < 1.
(ii) We say that the sequence f n localises in L p if κ = 1.
(iii) We say that the sequence f n does not localise in L p if κ = 0.
Using Cantor's diagonalisation we see the supremum in (13) is in fact a maximum. That is we have a maximising sequence in A((Ω n )). To show that f n localises in L p is equivalent to showing the existence of a sequence of measurable sets A n ⊂ Ω n , n ∈ N such that To show that (f n ) does not localise in L p is equivalent to showing that for any sequence (A n ) of measurable sets A n ⊂ Ω n , n ∈ N the following implication holds: If f n κ-localises in L p then there is a sequence (A n ) ∈ A((Ω n )) which (asymptotically) supports a fraction κ of f n p p . Given such a maximising sequence (A n ) it is possible to construct a sequence (Ã n ) ∈ A((Ω n )) which (asymptotically) supports a fractionκ of f n p p with 0 <κ < κ. Hence the requirement of the supremum in the definition of κ in (13). In Section 2 we analyse two examples in detail where localisation in L 1 and κ-localisation in L 1 occur for a family of torsion functions for Schrödinger operators (Example 1), and for a family of torsion functions for Dirichlet Laplacians (Example 2).
The theorem below asserts that localisation or κ-localisation for the torsion function in L 1 implies localisation for the corresponding first Dirichlet eigenfunction in L 2 .
We see by (16) that for any sequence (Ω n ) of elongating open, bounded and convex sets in R m , (v Ωn ) has non-vanishing efficiency, and so by Theorem 4, (v Ωn ) is not localising or κ-localising in L 1 . This is in contrast with the results of [7], where localisation of a sequence of first Dirichlet eigenfunctions was obtained for a wide class of elongating, open, bounded and convex sets in R m . Further examples demonstrating different behaviour of the torsion function, and the first eigenfunction of the Dirichlet Laplacian around their respective maxima for elongated convex planar domains have been constructed in [3].
This paper is organised as follows. Examples 1 and 2 will be analysed in Section 2. The proofs of Theorems 1, 2, 3 and 4 will be given in Section 3-6 respectively.

Examples
In Example 1 below we analyse localisation and κ-localisation in L 1 for a family of Schrödinger operators in one dimension, parametrised by three real numbers, ν > 1, 0 < α < 1, and c > 0.
To prove (iii) we obtain a lower bound for the supremum in (13) by choosing the sequence A n = (−ε n , ε n ), n ∈ N. Hence by (26) we then get To prove the reverse inequality we let A n ⊂ (−1, 1), n ∈ N be an arbitrary sequence of measurable sets which satisfy lim n→∞ |A n | = 0. In view of (28) we may assume without loss of generality that A n is symmetric, A n = −A n , and that [−ε n , ε n ] ⊂ A n for any n (sufficiently large). It then suffices to show that As in the proof of item (i), we estimatê Altogether we proved that κ = κ c and that the supremum κ is attained by the sequence (A n ) with (−ε n , ε n ).
In Example 2 below we analyse localisation and κ-localisation in L 1 for a family of sequences of open sets in R m , parametrised by three real positive α, β and c.
Proof. First observe that condition (29) guarantees that the n + 1 open balls do not intersect pairwise. The first inequality in (30) guarantees that the measure of B(p n+1 ; cn −β ) is negligible compared with the measure of Ω n in the limit n → ∞. The second inequality in (30) implies that |Ω n | remains bounded for large n. The torsion function for the open ball B(p; R) is given by To prove (i) we let A n ⊂ Ω n , n ∈ N be an arbitrary sequence of measurable sets which satisfy lim n→∞ |An| |Ωn| = 0. We havê By (32) we havé where we have used (30) to bound the second term in the previous line. By the hypothesis for β under (i) and the hypothesis on (A n ) above we conclude This proves the implication under (15), and concludes the proof of (i).
To prove (ii) let A n = B(p n+1 ; cn −β ), n ∈ N. This gives, By the hypothesis for β under (ii) we have Both requirements under (14) are satisfied. This concludes the proof of (ii).
To prove (iii) we obtain a lower bound for κ c by choosing the sequence A n = B(p n+1 ; cn −β ), n ∈ N. By (34) and the hypothesis for β under (iii) we get To prove the reverse inequality we let A n ⊂ Ω n , n ∈ N be an arbitrary sequence of measurable sets which satisfy lim n→∞ |An| |Ωn| = 0. By (32), (33), and the hypothesis for β under (iii), By taking the lim sup n→∞ in both sides of the inequality in (36), By taking the supremum over all sequences (A n ) ∈ A((Ω n )) we find by (37), This proves by (35) and (38) that (v Ωn ) is κ c -localising with κ c given by (31). This concludes the proof of (iii).

Proof of Theorem 1
To prove Theorem 1 we first need to establish some auxiliary results. It is well known that the torsion function can be expressed in terms of the heat kernel of L.
Let Ω ⊂ R m be open with 0 < |Ω| < ∞, and let V : The torsion function of L then satisfies In case V = 0 we write p Ω for p Ω,V .
Lemma 5 implies that By choosing V = 0 in the expression under the supremum in the right-hand side of (42) we see that c m ≤ d m .
Proof of Lemma 5. To prove the upper bound in (41) we note that Lemma 1 and its proof in [6] hold with λ = λ 1 (Ω, V ), and Lemma 2 and its proof in [6] hold for the semigroup associated with L. Finally Lemma 3 and Theorem 1 and their proofs in [6] hold with λ = λ 1 (Ω, V ). This proves the upper bound in (41). It remains to prove the lower bound. Let with Dirichlet boundary conditions on ∂Ω R . Then L R is self-adjoint, and its spectrum is discrete. Since the first Dirichlet eigenfunction ϕ 1,ΩR,VR is non-negative, and ϕ 1,ΩR,VR 2 = 1, one has, by the Cauchy-Schwarz inequality, By self-adjointnesŝ By (43) -(44) we conclude that Since v ΩR,VR ∞ ≤ v Ω,V ∞ we have The assertion follows since R → λ 1 (Ω R , V R ) is decreasing to λ 1 (Ω, V ) as R → ∞.

Proof of Theorem 2
Proof of Theorem 2. The proof follows a method from [19], which can be summarised as follows: construct a measure µ so that the efficiency of the eigenfunction of the first eigenvalue associated to −∆ + µ almost equals 1 and then approximate µ in the sense of γ-convergence by a sequence of domains. We refer the reader to [14,Definition 4.8] (see also [12,Chapter 4]) for the notions of γ-convergence, relaxed Dirichlet problems and approximations by sequences of domains. Recall that a measure on a domain is said to be capacitary if it is nonnegative, not necessarily finite and Borel, and which is in addition absolutely continuous with respect to the capacity. One can then define a relaxed Dirichlet problem (see [14,Definition 3.1], and [12, Sections 4.3 and 3.6]). The equations −∆v + µv = 1 and −∆v + µv = λv can then be solved in the weak sense. The sequence of sets Ω n ⊂ B 1 is said to γ-convergence to the capacitary measure µ if the sequence of weak solutions v n ∈ H 1 0 (Ω n ) of −∆v n = 1 converges strongly in L 2 (B 1 ) to the weak solution v ∈ H 1 0 (B 1 ) ∩ L 2 (µ) of −∆v + µv = 1. As a consequence of the γ-convergence, the sequence of eigenvalues and corresponding eigenfunctions on the moving domain (Ω n ) converge in a suitable sense The γconvergence is metrisable.
Let B r denote the closure of B r . For any 0 < ε < 1, we consider the Dirichlet-Neumann eigenvalue problem on the annulus A ε = B 1 \ B 1−ε . Denote by λ ε the first eigenvalue and by u ε a corresponding eigenfunction, where ν denotes the inward-pointing normal on the sphere ∂B 1−ε . One can show that λ ε is simple and strictly positive and that u ε is radially symmetric and has a constant sign, say positive. In particular, the restriction of u ε to ∂B 1−ε equals a positive constant, c ε > 0. We continuously extend u ε inside B 1−ε by c ε and denote the resulting function, defined on B 1 , by v ε .
Since the normal derivatives of v ε on both sides of ∂B 1−ε vanish, ∆v ε is an L 2 -function. More precisely, one has where D ′ (B 1 ) denotes the space of distributions on B 1 . By adding on both sides the L 2 function λ ε v ε 1 B1−ε we get We view µ := λ ε 1 B1−ε as a capacitary measure on B 1 . Then formally, we get Combined with the fact that v ε > 0, this means that λ ε is the first Dirichlet eigenvalue of −∆ + µ. We assume from now on that v ε is normalised by v ε L 2 (B1) = 1. Since the measure µ is finite with support in B 1 , the first eigenvalue λ ε is simple.
In view of the Dal Maso-Mosco density result [14,Theorem 4.16], there exists a sequence Ω n ⊆ B 1 , n ≥ 1, such that Ω n γ-converges to µ. We can assume that the boundary ∂Ω n of Ω n is smooth. Indeed, otherwise we can replace each Ω n by an inner approximation with a smooth open set, which is close enough in the sense of the distance associated to the γ-convergence. Furthermore, we can also assume that Ω n is connected. Indeed, since Ω n is smooth, it has a finite number of smooth connected components, which are separated from each other by a positive distance. These components can be joined by a finite number of thin tubes, connecting the set Ω n . As the width of the tubes vanishes, the sequence γ-converges by Sverak's theorem (see [12,Theorem 4.7

.1]).
It is possible to explicitly construct such a sequence (Ω n ) in the spirit of Cioranescu-Murat [13], but such a construction is not needed for the rest of the proof. The only fact we need to keep in mind is that Ω n ⊆ B 1 , so that |Ω n | ≤ |B 1 |. Denote by λ 1 (Ω n ) the first Dirichlet eigenvalue of −∆ on Ω n and by u n the L 2 −normalised, positive eigenfunction corresponding to λ 1 (Ω n ). We extend u n to B 1 by setting it to 0 on B 1 \ Ω n and by a slight abuse of notation, denote this extension again by u n . The γ-convergence of (Ω n ), together with the compact embedding of H 1 0 (B 1 ) in L 1 (B 1 ), imply that (i) λ 1 (Ω n ) → λ ε , and (ii) u n ⇀ v ε weakly in H 1 0 (B 1 ) and strongly in L 1 (B 1 ). Assuming that we get Since the right-hand side is arbitrarily close to 1 when ε ↓ 0, the proof of Theorem 2 is then completed by a diagonal selection procedure. It remains to show (63). This kind of assertion is known to be true in a general setup. In essence it is a consequence of the subharmonicity of the eigenfunctions u n , n ≥ 1. (For a similar result for the torsion function see [19,Theorem 2.2].) For the sake of completeness, we give below a proof. It slightly differs from the one in [19,Theorem 2.2].
First note that since u n → v ε strongly in L 1 , it follows that To prove that lim sup n→+∞ u n ∞ ≤ v ε ∞ we argue as follows. Being convergent, the sequence (λ 1 (Ω n )) is bounded, and so is u n ∞ . Choose M > 0 so that for any n ∈ N Let x n ∈ B 1 be a maximum point for u n . By taking, if necessary, a subsequence we may assume that x n → x * . Furthermore, By the subharmonicity of the function x → u n (x) + M |x−xn| 2 2m around x n , it then follows that for δ > 0 sufficiently small, |B(x n ; δ)| .
Taking the limit n → +∞ we obtain Letting δ ↓ 0, completes the proof.

Remark 1.
If Ω is an open connected subset of R m with m ≥ 2 and 0 < |Ω| < ∞, then where Proof. Putting V = 0 in (39) one obtains by using (51), Integrating both sides of (67) yields, To obtain the stated lower bound for Φ(Ω) it remains to find an upper bound for v Ω ∞ . First consider case (i). By (67), the Cauchy-Schwarz inequality, and the heat semigroup property, one sees that Choosing β = 1 2 in Lemma 1 of [6] gives by domain monotonicity of the Dirichlet heat kernel, and (52) Substitution of (70) into the right-hand side of (69), evaluating the resulting integral with respect to t, and taking the supremum over all (71) Inequality (64) follows from (68), (71) with the values for k 2 and k 3 given in (66).
Next consider case (ii). By the first equality in (69) we have by domain monotonicity of the Dirichlet heat kernel
Taking the supremum over all x ∈ Ω and using the formulae for c m and R * gives, This, together with (68), implies the assertion for m ≥ 4.

Proof of Theorem 3
Proof of Theorem 3(i). Since u ∞ > 0 we can re-scale both u and Ω, such that u ∞ = 1, and |Ω| = ω m . Inequality (9) then reads with |Ω| = ω m , u ∞ = 1, and ffl Ω u ≥ 2 m+2 . Note that replacing u by its positive part u + decreases the left-hand side of (72), and furthermore, u + ≤ 1, ffl . So it suffices to prove that for any m ≥ 2, We make some preliminary observations. By Schwarz rearrangement we may consider the infimum in the definition of F over the collection H * 1 0 (B 1 ) of all radially symmetric, decreasing functions u in H 1 0 (B 1 ) since this rearrangement decreases the energy and leaves the other constraints unchanged. So, First note that admits a minimiser. By the Lagrange multiplier theorem, there exists a constant c such that Since u is radially symmetric and decreasing, in the sequel, by a slight abuse of notation, we write u(r) instead of u(x). By a straightforward computation one sees that for any 2/(m + 2) ≤ θ < 1 In particular, u ∞ = u(0) ≥ 1. Note that we could have written 0 ≤ u, u ∞ = 1 instead of 0 ≤ u ≤ 1 in the right-hand side of (73). For any 2/(m + 2) ≤ θ < 1, F * (θ) admits a minimiser. Since the Dirichlet energy is strictly convex, it is unique, and we denote it by u θ . Let Since F (θ) ≥ F * (θ), it suffices to show that If θ = 2 m+2 , then by (74), u 2/(m+2) (r) = 1 − r 2 . Hence We note that 2/(m + 2) < 2/3 < θ * m < 1 for any m ≥ 2. In part (b) we prove that (75) holds for any m ≥ 4 and θ ∈ [θ * m , 1). In part (c) we show that (75) holds for m = 2, 3, using the Euler-Lagrange equation of a variational problem, related to an obstacle problem. See (96) below. Finally, in part (d) we verify that equality in (9) holds if and only if Ω is a ball and u is a multiple of the torsion function for that ball. This completes the proof of Theorem 3 (i).
Finally we consider the case m = 5. One computes that and by the first inequality in (90), Since both θ → θ(1+θ) 1+θ+θ 2 and θ → 1−θ θ −9/5 −1 are non-negative increasing functions on the interval [0, 1), so is the right-hand side of (92) and it remains, by (92), to verify that By (94), the left-hand side of (95) is bounded from below by .206 while the right hand side of (95) is bounded from above by .137. This completes the proof of part (b).
(c) In this part we treat the cases m = 2 and m = 3. We begin with some preliminary considerations. We note that the minimisation problem (73) is related to a volume constraint obstacle problem in B 1 : we claim that there exist c > 0 and 0 ≤ l < 1, depending on θ, so that u θ satisfies the following system of equations, where ν denotes the inward pointing normal on the sphere ∂B l . Indeed, since u θ is radially symmetric, decreasing and since u θ (0) = 1 (see (73)) and u θ (1) = 0, there exists a maximal number 0 ≤ l ≡ l(θ) < 1 so that u θ (r) = 1 for 0 ≤ r ≤ l. By the Lagrange multiplier theorem, there exists a constant c > 0 so that −∆u θ = c on B 1 \ B l in the sense of distributions. It then follows from [16, Theorem 2] that u θ is C 1,α on B 1 which implies that ∂u θ ∂ν = 0 on ∂B l . These observations establish (96). Note that both c and l are uniquely determined by θ.
We claim that the map is an increasing bijection. To prove the latter assertion, we construct for any given 0 ≤ l < 1 a unique radially symmetric, decreasing solution u(·; l) of (96) and show that θ ≡ θ(l) := ffl B1 u(·; l), satisfies 2/(m + 2) ≤ θ < 1 with θ(0) = 2/(m + 2) and lim l→1 θ(l) = 1. First we note that c is uniquely determined by l since for any given 0 ≤ l < 1, the solution of (96) is given by a formula. To obtain it, note that the general radially symmetric solution of −∆u = c on the annulus B 1 \ B l is of the form for some real constants a, b, c. The condition ∂u ∂ν (l) = 0 implies that a = l m c m so that the boundary condition u(1) = 0 leads to The value of c is now obtained by the requirement u(l) = 1, One verifies in a straightforward way that the resulting function u ≡ u(·; l) is decreasing for l ≤ r ≤ 1, that θ(0) = 2/(m + 2), and that l → c ≡ c(l) is a continuous, strictly increasing function of 0 ≤ l < 1. We claim that l → θ(l) is also strictly increasing. To verify that this is indeed the case, one could explicitly compute θ in terms of l, but the formula is rather complicated. Instead we prove the claim by using the maximum principle. By contradiction, suppose there exist 0 ≤ l 2 < l 1 < 1 with θ 2 := θ(l 2 ) > θ 1 := θ(l 1 ). By the considerations above, c 2 := c(l 2 ) < c 1 := c(l 1 ). Hence −∆(u 1 − u 2 ) = c 1 − c 2 > 0 on B 1 \ B l1 where u j := u(·, l j ) for j = 1, 2. Since θ 1 < θ 2 , there exist l 1 < r 1 < r 2 < 1 so that (u 1 − u 2 )(r) < 0 for any r 1 < r < r 2 , contradicting the maximum principle. From the formula (97) of u(·; l) one infers that θ(l) is a continuous function of l and that lim l↑1 θ(l) = 1. Hence for any 0 ≤ l < 1, u(·; l) coincides with u θ where θ ≡ θ(l) = ffl B1 u(·; l). Altogether we have shown that b is a continuous, increasing bijection.
Without any additional effort we may consider the general case m ≥ 3, and follow the line of arguments above. Integrating by parts, one haŝ In the case m = 3, one gets in this way and a lengthy computation leads to the formula g ′ (l) = 24π 25 2l(20l 4 + 67l 3 + 84l 2 + 46l + 8) (2l + 1) 4 .
Clearly, g ′ (l) > 0 on (0, 1) for m = 3. 1 1 For general m ≥ 4, the formula for g ′ can be computed to be a quotient of two polynomials with degrees depending on m. We believe that g ′ (l) is strictly positive for every l on (0, 1), but a direct proof, covering all dimensions m ≥ 5, based on the formula of g ′ seems out of reach. For m = 4 the quotient of the polynomials simplifies, and gives g(l) = ω 4 16 3 l 2 + 16 9 . We see that for m = 4, g(1) = 64ω 4 9 agrees with the value f (1) given in Remark 2. We also have that g(0) = 16ω 4 3 agrees with the value f (1/3) from Theorem 3(i). Indeed for m = 4 and θ = 1 3 we have equality in (9). Note that for m = 4, g(l) is increasing on (0, 1).
(d) In this last part we prove that equality in (9) holds if and only if Ω is a ball and u is a multiple of the torsion function for that ball. Clearly, if Ω is a ball and u is a multiple of the torsion function for that ball, then (9) holds (see (76)). Conversely, assume that equality holds in (9). We re-scale the measure of Ω and the L ∞ norm of u as in the proof of Theorem 3(i). Equality in (9) implies that u has the same Dirichlet integral as its Schwarz rearrangement u * , and its Schwarz rearrangement is the solution of the obstacle problem on the ball B 1 -see (96). In view of the strict monotonicity of f on [ 2 m+2 , 2 3 ) (see part (a)) and the (strict) inequalities obtained above 2 3 (see parts (a)-(c)), this implies that θ = 2 m+2 , which corresponds to l = 0 and to c = 2m (see (76), (98)). It means that u * is a multiple of the torsion function on B 1 (see (96)).
In order to justify that u has to be equal to u * , recall that (101) holds and that u * , being a multiple of the torsion function on B 1 , has a critical set of zero measure. Equality between u and u * , up to a translation, comes from the classical result of Brothers and Ziemer [11, Theorem 1.1].
Proof of Theorem 3(ii). The key ingredient into the proof is inequality (9). First note that since for any t > 0, 1 t 2 v Ω (tx) is the torsion function of 1 t Ω. Choosing t = (|Ω|/ω m ) 1/m , one infers that it suffices to prove estimate (10) in the case |Ω| = ω m .