On capacity and torsional rigidity

We investigate extremality properties of shape functionals which are products of Newtonian capacity $\cp(\overline{\Om})$, and powers of the torsional rigidity $T(\Om)$, for an open set $\Om\subset \R^d$ with compact closure $\overline{\Om}$, and prescribed Lebesgue measure. It is shown that if $\Om$ is convex then $\cp(\overline{\Om})T^q(\Om)$ is (i) bounded from above if and only if $q\ge 1$, and (ii) bounded from below and away from $0$ if and only if $q\le \frac{d-2}{2(d-1)}$. Moreover a convex maximiser for the product exists if either $q>1$, or $d=3$ and $q=1$. A convex minimiser exists for $q<\frac{d-2}{2(d-1)}$. If $q\le 0$, then the product is minimised among all bounded sets by a ball of measure $1$.


Introduction and main results
Several classical inequalities of mathematical physics are of the following the form. Let is invariant under homotheties, and in some cases this quantity is extremal for some open set Ω * ∈ C, G(Ω) ≥ G(Ω * ), Ω ∈ C.
The Faber-Krahn, Krahn-Szegö, and Kohler-Jobin inequalities are of this form, see for example the seminal text [8]. In a recent paper [3] a more general set of inequalities was investigated. These are of the following form: let q ∈ R and consider the shape functional Then, unless q = −β 2 /β 1 , this product is not scaling invariant. However, denoting by |Ω| the Lebesgue measure of Ω, the quantity H(Ω)F (Ω) q |Ω| (β 2 +qβ 1 )/d , is scaling invariant. The case where H is the principal Dirichlet eigenvalue, and F is the torsional rigidity was analysed in [3]. In the present paper we investigate the case where H is the Newtonian capacity and F is the torsional rigidity. Since the Newtonian capacity is most easily defined for compact subsets of R d , d ≥ 3, we restrict ourselves to open sets Ω ⊂ R d , d ≥ 3 which are precompact. In that case the Newtonian capacity scales as a power d − 2 of the homothety. The discussion of the planar case would involve the logarithmic capacity which does not satisfy this scaling property. Throughout this paper we let Ω be a non-empty, open, bounded set in Euclidean space R d , d ≥ 3. For a set A ⊂ R d we denote by A its closure, diam(A) = sup |x − y| : x ∈ A, y ∈ A its diameter, and r(A) = sup r ≥ 0 : (∃x ∈ A), (B r (x) ⊂ A) its inradius, where B r (x) = {y ∈ R d : |x − y| < r} is the ball of radius r centred at x. Before we state the main results we recall some basic facts about the torsion function, torsional rigidity, and Newtonian capacity.
The torsion function for a domain Ω is the solution of −∆u = 1, u ∈ H 1 0 (Ω), and is denoted by u Ω . It is convenient to extend u Ω to all of R d by defining u Ω = 0 on R d \ Ω. It is well known that u Ω is non-negative, bounded ( [1,2,5,9]), and monotone increasing with respect to Ω, that is The torsional rigidity of Ω, or torsion for short, is denoted by where · p , 1 ≤ p ≤ ∞ denotes the usual L p norm. It follows that and that the torsion satisfies the scaling property Moreover T is additive on unions of disjoint families of open sets: It is straightforward to verify that if E(a), with a = (a 1 , a 2 , . . . , a d ) ∈ R d + , is the ellipsoid where is the Lebesgue measure of a ball B 1 with radius 1 in R d . We put .
The Saint-Venant inequality asserts that where Ω * is any ball with |Ω| = |Ω * |. It follows by scaling that Below we recall some basic facts about the Newtonian capacity cap (K) of a compact set K ⊂ R d , d ≥ 3. There are several equivalent definitions of cap (K) of which we choose where ϕ Kε is the restriction of ϕ to K ε = {x ∈ R d : dist(x, K) < ε}. It follows that and that the capacity satisfies the scaling property Moreover if {K i , i ∈ I} is a countable family of compact sets such that ∪ i∈I K i is compact, then cap (∪ i∈I K i ) ≤ i∈I cap (K i ).
It was reported in [6] pp. 260 that the Newtonian capacity of an ellipsoid was computed in volume 8, pp. 103-104 in [4]. The formula is given in terms of an elliptic integral, and reads where We put The isoperimetric inequality for Newtonian capacity (see [8]) asserts that for all compact where K * is a closed ball with |K| = |K * |. It follows by scaling that The shape functional we consider in the present paper is (2) and (7) we obtain that G q is scaling invariant.
All of our main results are for d ≥ 3, and are as follows.
with equality if and only if Ω is (up to sets of capacity 0) a ball in R d .
If q > 1, then the variational problem in the left-hand side of (13) has a maximiser, say Ω + . For any such maximiser, (iii) If q = 1 and d = 3, then the variational problem in the left-hand side of (13) has a maximiser, say Ω + . For any such maximiser, : Ω open, bounded and convex} = 0.
: Ω open, bounded and convex} If 0 < q < (d − 2)/(2(d − 1)), then the variational problem in the left-hand side of (16) has a convex minimiser, say Ω − . For any such minimiser, . (17) We were unable to prove the existence or non-existence of a maximiser for the lefthand side of (13) for q = 1 and d > 3. In these higher-dimensional cases there is a lack of compactness. For example if and if k ≥ 3, then lim ε→0 G 1 (E(a ε )) exists and is strictly positive. Similarly we were unable to prove the existence of a minimiser of the left-hand side of (16) at the critical The proofs of Theorems 1, 2, and 3 are deferred to Sections 2, 3, and 4 respectively. A key ingredient in these proofs is John's ellipsoid theorem [7]. This theorem asserts that for any open, bounded convex set Ω in R d there exists a translation and rotation of Ω, again denoted by Ω, and an ellipsoid E(a) such that Moreover, among all ellipsoids in Ω, E(a/d) has maximal measure.
To prove (iii), we let Q ⊂ R d be a cube with |Q| = 1. Let N ∈ N be arbitrary. The Furthermore, This implies (15) since q > 0, and N ∈ N was arbitrary.

Proof of Theorem 2
Proof. To prove (i) we consider the ellipsoid E(a ε ) with a ε = (1, ε, . . . , ε). We have To obtain a lower bound on cap E(a ε ) we have where c d is a positive constant depending only on d, and Therefore where C d is a positive constant depending only on d. We obtain the desired result by letting ε ↓ 0.
To prove (ii), we require an upper bound for the capacity and we use the inclusion (18), so that and We have by (8), By (8), (9) and (22), By (1) and (3), Putting (20), (23), and (24) together we obtain, This proves (13). To prove the existence of a maximiser, we observe that if the left-hand side of (13) equals G q (B 1 ) then B 1 is a maximiser which satisfies (14). If the left-hand side of (13) is strictly greater than G q (B 1 ), we let Ω bounded and convex be such that By the first two lines in (25) and (26), we have for q > 1, and we obtain by (27)-(29), The inequality above was obtained by putting d i=1 a i = 1 for convenience. As the lefthand side of (30) is also scaling invariant it also holds without the constraint d i=1 a i = 1. Let (Ω n ) be a maximising sequence for the left-hand side of (13). Since this supremum is scaling invariant we fix r(Ω n ) = 1. By (30), diam(Ω n ) ≤ L for some L < ∞, and for all n. By taking translations of (Ω n ) these translates are contained in a closed ball B L of radius L. Since the Hausdorff metric is compact on the space of convex, compact sets in B L , there exists a subsequence of (Ω n ), again denoted by (Ω n ) which converges in the Hausdorff (and in the complementary Hausdorff) metric to an element say Ω + . Since measure, torsion, capacity, diameter, and inradius are all continuous with respect to this metric, G q (Ω + ) = lim n→∞ G q (Ω n ), and Ω + is a maximiser which satisfies (14).
To prove (iii) we let q = 1 and d = 3. Let Ω be an element of a maximising sequence. We may assume that We obtain an upper bound on cap (E(a)) by obtaining a lower bound on e(a). By (9), we have By (8) for d = 3, and (32), Since Ω ⊂ B a 1 , we also have Hence cap (Ω) ≤ κ 3 a 1 min 1, log a −2 2 a −1 3 −1 .
In addition, where we used that a 1 a 2 a 3 = 1. Summarising, from (31) we obtain We first consider the case a 2 ≤ 1. Then using a 3 ≤ a 2 , we obtain that the right-hand side of (33) is bounded from above by 3 7 κ 3 τ 3 ω −2 3 log(1/a 3 ) −1 . We next consider the case a 2 ≥ 1. Then the right-hand side of (33) is bounded from above by 3 7 κ 3 τ 3 ω −2 3 a 3 . Combining these two estimates gives By the first inequality in the right-hand side of (26), and (34), we conclude a 3 ≥ e −3 7 . By (28) and (29), we find that for an element Ω of the maximising sequence satisfying (31), The remaining part of the proof follows similar lines as those in the proof of (ii).
To prove the second part of the assertion under (ii) we let 0 < q < (d − 2)/(2(d − 1)), and note that by the first inequality in (26), (48), We infer that By (28), (29), and (49), . (50) The proof of the existence of a minimiser is similar to the proof of the existence of a maximiser in Theorem 1(ii), and has been omitted. If Ω − is a minimiser then, by continuity of diameter and inradius, Ω − satisfies (50). This proves (17).