Some inequalities involving perimeter and torsional rigidity

We consider shape functionals of the form $F_q(\Omega)=P(\Omega)T^q(\Omega)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(\Omega)$ denotes the perimeter of $\Omega$ and $T(\Omega)$ is the torsional rigidity of $\Omega$. The minimization and maximization of $F_q(\Omega)$ is considered on various classes of admissible domains $\Omega$: in the class $\mathcal{A}_{all}$ of all domains, in the class $\mathcal{A}_{convex}$ of convex domains, and in the class $\mathcal{A}_{thin}$ of thin domains.


Introduction
In this paper, given an open set Ω ⊂ R d with finite Lebesgue measure, we consider the quantities P (Ω) = perimeter of Ω; T (Ω) = torsional rigidity of Ω.
The perimeter P (Ω) is defined according to the De Giorgi formula The scaling property of the perimeter is P (tΩ) = t d−1 P (Ω) for every t > 0 and the relation between P (Ω) and the Lebesgue measure |Ω| is the well-known isoperimetric inequality: where B is any ball in R d . In addition, the inequality above becomes an equality if and only if Ω is a ball (up to sets of Lebesgue measure zero). Moreover T is increasing with respect to the set inclusion, that is and T is additive on disjoint families of open sets. The scaling property of the torsional rigidity is T (tΩ) = t d+2 T (Ω), for every t > 0, and the relation between T (Ω) and the Lebesgue measure |Ω| is the well-known Saint-Venant inequality (see for instance [16], [17]): Again, the inequality above becomes an equality if and only if Ω is a ball (up to sets of capacity zero). If we denote by B 1 the unitary ball of R d and by ω d its Lebesgue measure, then the solution of (1.2), with Ω = B 1 , is .
We are interested in the problem of minimizing or maximizing quantities of the form P α (Ω)T β (Ω) on some given class of open sets Ω ⊂ R d having a prescribed Lebesgue measure |Ω|, where α, β are two given exponents. Similar problems have been considered for shape functionals involving: -the torsional rigidity and the first eigenvalue of the Laplacian in [2], [3], [6], [8], [11], [19], [20], [21]; -the torsional rigidity and the Newtonian capacity in [1]; -the perimeter and the first eigenvalue of the Laplacian in [14]; -the perimeter and the Newtonian capacity in [10], [13]. The case β = 0 reduces to the isoperimetric inequality, and we have, denoting by Ω * m a ball of measure m, Indeed if we choose Ω n = ∪ n k=1 B n,k where B n,k are disjoint balls of measure m/n each, we get for every n ∈ N inf T (Ω) : The case when α and β have a different sign is also immediate; for instance, if α > 0 and β < 0 we have from (1.1) and (1.3) and similarly, if α < 0 and β > 0 we have . The cases we will investigate are the remaining ones; with no loss of generality we may assume α = 1, so that the optimization problems we consider are for the quantities P (Ω)T q (Ω), with q > 0.
In order to remove the Lebesgue measure constraint |Ω| = m we consider the scaling free functionals In the following sections we study the minimization and the maximization problems for the shape functionals F q on various classes of domains. More precisely we consider the cases below. The class of all domains Ω (nonempty) A all = Ω ⊂ R d : Ω = ∅ will be considered in Section 2; we show that for every q > 0 both the maximization and the minimization problems for F q on A all are ill posed. The class of convex domains Ω A convex = Ω ⊂ R d : Ω = ∅, Ω convex will be considered in Section 3; we show that for 0 < q < 1/2 the maximization problem for F q on A convex is ill posed, whereas the minimization problem is well posed. On the contrary, when q > 1/2 the minimization problem for F q on A convex is ill posed, whereas the maximization problem is well posed. In the threshold case q = 1/2 the precise value of the infimum of F 1/2 is provided; concerning the precise value of the supremum of F 1/2 an interesting conjecture is stated. At present, the conjecture has been shown to be true in the case d = 2, while the question is open in higher dimensions. The class of thin domains A thin , suitably defined, will be considered in Section 4. If h(s) represents the asymptotical local thickness of the thin domain as s varies in a d − 1 dimensional domain A, the maximization of the functional F 1/2 on A thin reduces to the maximization of a functional defined on nonnegative functions h defined on A; this allows us to prove the conjecture for any dimension d on the class of thin convex domains.

Optimization in the class of all domains
In this section we show that the minimization and the maximization problems for the shape functionals F q are both ill posed, for every q > 0.
Theorem 2.1. There exist two sequences Ω 1,n and Ω 2,n of smooth domains such that for every q > 0 we have In particular, we have Proof. In order to show the sup equality it is enough to take as Ω 2,n a perturbation of the unit ball B 1 such that Then we have where we used the monotonicity of the torsional rigidity. Then In order to prove the inf equality we take as Ω ε the unit ball B 1 to which we remove a periodic array of holes; the centers of two adjacent holes are at distance ε and the radii of the holes are It is easy to see that, as ε → 0, we have Concerning the torsion T (Ω ε ), we have (see [9]) Since for every c > 0 we have that Therefore, a diagonal argument allows us to construct a sequence Ω 1,n such that which concludes the proof.

Optimization in the class of convex domains
In this section we consider only domains Ω which are convex. A first remark is in the proposition below and shows that in some cases the optimization problems for the shape functional F q is still ill posed.
Proof. Let A be a smooth convex d − 1 dimensional set and for every ε > 0 consider the domain Ω ε ∈ A convex given by We have (for the torsion asymptotics see for instance [2]) Letting ε → 0 achieves the proof.
We show now that in some other cases the optimization problems for the shape functional F q is well posed. Let us begin to consider the case q = 1/2.
and the infimum is asymptotically reached by domains of the form Proof. Thanks to a classical result by Polya ( [23], see also Theorem 5.1 of [11]) it holds for any bounded open convex set. Taking into account (3.1), we get (3.2).
Concerning the supremum of F 1/2 (Ω) in the class A convex we can only show that it is finite. Proposition 3.3. For every Ω ∈ A convex we have .
Proof. By the John's ellipsoid Theorem [18], there exists an ellipsoid, that without loss of generality we may assume centered at the origin, Since the solution of (1.2) for E a is given by Combining these formulas we have from (3.4) and finally, by Jensen inequality, as required.
On the precise value of sup F 1/2 (Ω) : Ω ∈ A convex we make the following conjecture.

Conjecture 3.4. We have
and it is asymptotically reached by taking for instance Remark 3.5. We recall that Conjecture 3.4 has been shown to be true in the case d = 2 (see [23], [22], and the more recent paper [12]). In Section 4 we prove the conjecture above for every d ≥ 2 in the class of convex thin domains.
We show now that for F q in the class A convex the minimization problem is well posed when q < 1/2 and the maximization problem is well posed when q > 1/2. From the bounds obtained in Propositions 3.2 and 3.3 we can prove the following results.
Proposition 3.6. We have

Hence it is enough to apply the bounds (3.2) and (3.3), together with the Saint Venant inequality (1.3) to get that for every Ω
By the expression (1.4) for T (B) we conclude the proof.
We now prove the existence of a convex minimizer when q < 1/2 and of a convex maximizer when q > 1/2. Theorem 3.7. There exists a solution for the following optimization problems: min F q (Ω) : Ω ∈ A convex for every q < 1/2; max F q (Ω) : Ω ∈ A convex for every q > 1/2.
Proof. Suppose q < 1/2 and consider Ω n a minimizing sequence for F q (Ω). By the John's ellipsoid Theorem we can assume that there exists a sequence of ellipsoids E an such that E an ⊂ Ω n ⊂ dE an .
Observe that this implies that the diameter of Ω n is uniformly bounded in n. We claim that a 1n ≥ c for every n ∈ N where c is a positive constant. Then the proof is achieved by extracting a subsequence Ω n k which converges both in the sense of characteristic functions and in the Hausdorff metric to some open, non empty, convex, bounded set Ω − and by using the continuity properties of torsional rigidity, perimeter and volume (see for instance, [7], [17]).
To prove the claim we use a strategy similar to the one already used in the proof of Proposition 3.3. Let Q an be the cuboid d i=1 ] − a in , a in [. Since d −1/2 Q an ⊂ E an we have, for n large enough, An explicit computation shows Observe that, by Cauchy-Schwarz inequality, while for the last term it holds Therefore, putting together (3.5)-(3.7) and using the fact that q < 1/2 we obtain that, if n is large enough, the sequence a 1n must be greater than some positive constant c, which proves the claim.
The case q > 1/2 can be proved in a similar way. If Ω n is a maximizing sequence for F q (Ω) and E an are ellipsoids such that E an ⊂ Ω n ⊂ dE an , we have Hence (3.8) implies, for a suitable constant C q,d depending only on q and on d, where in the last inequality we used the Cauchy-Schwarz inequality (3.6). Finally, since a in ≤ a dn = 1, we obtain in ) (2q−1)/d and, since q > 1/2, the conclusion follows as in the previous case.

Optimization in the class of thin domains
In this section we consider the class of thin domains where ε is a small positive parameter, A is a (smooth) domain of R d−1 , and h − , h + are two given (smooth) functions. We denote by h(s) the local thickness and we assume that h(s) ≥ 0. The following asymptotics hold for the quantities we are interested to (for the torsional rigidity we refer to [5]): which together give the asymptotic formula when q = 1/2 where we use the notation By Hölder inequality we have lim ε→0 F 1/2 (Ω ε ) ≥ 3 −1/2 and the value 3 −1/2 is actually reached by taking the local thickness function h constant, which corresponds to Ω ε a thin slab. A sharp inequality from above is also possible for F 1/2 (Ω ε ), if we restrict the analysis to convex domains, that is to local thickness functions h which are concave.
The following result will be used, for which we refer to [4], [15]. where the constant C p,q is given by In addition, the inequality above becomes an equality when A is a ball of radius 1 and f (x) = 1 − |x|.
We are now in a position to prove the Conjecture 3.4 for convex thin domains. In addition, the inequality above becomes an equality taking for instance as A the unit ball of R d−1 and as the local thickness h(s) the function 1 − |s|.
Proof. By