On the stability of the annulus for the torsion of multiply connected domains

Calc. Var. (2026) 65:199 https://doi.org/10.1007/s00526-026-03362-w Calculus of Variations On the stability of the annulus for the torsion of multiply connected domains Vincenzo Amato1 · Luca Barbato1 Received: 9 June 2025 / Accepted: 21 May 2026 © The Author(s) 2026 Abstract We establish a quantitative version of the isoperimetric inequality for the torsional rigidity of multiply connected domains, among sets with given area and with given joint area of the holes. Since the optimal shape is the annulus, we study how a domain approaches an annular configuration when its torsional rigidity is close to optimal. Our result shows that when the torsional rigidity is nearly optimal, the domain  must be close to an annulus. Mathematics Subject Classification 35J05 · 35B35 · 35B45 1 Introduction The aim of this paper is to establish a stability result for the isoperimetric inequality associated with the torsional rigidity of multiply connected domains. Let us consider a homogeneous beam whose cross-section is a multiply connected domain . Definition 1.1 A set  is a multiply connected domain if  = G \ S and the following properties hold: (i) there exist G, 1 , . . . , m bounded Lipschitz domain of Rn ; (ii) i ⊂ G for i = 1, . . . , m; (iii) i ∩  j = ∅ if i  = j; m  (iv) S = i . i=1 It is well known that the torsional rigidity of , namely that of a beam with cross-section , can be expressed as the maximum of a Rayleigh quotient (see, for instance, [12, 41]), that Communicated by A. Mondino. B Vincenzo Amato Luca Barbato 1 Mathematical and Physical Sciences for Advanced Materials and Technologies, Scuola Superiore Meridionale, Largo San Marcellino 10, 80138 Napoli, Italy 0123456789().: V,-vol 123 199 is Page 2 of 32 V. Amato, L. Barbato ⎧  ⎫ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ dx ⎨ ⎬ G 1 : ϕ ∈ H0 (G), ϕ ≡ constant on i ∀i = 1, . . . m . T () = max  ⎪ ⎪ ⎪ ⎪ ⎪ |∇ϕ|2 d x ⎪ ⎩ ⎭ (1) G In the case of simply connected domains (m = 0), de Saint-Venant conjectured in 1856 that among all beams with cross-section of prescribed area, the one with circular cross-section has the largest torsional rigidity. A complete proof of this conjecture was given by Pólya in 1948, see [39]. The de Saint-Venant inequality was later generalized in several directions. For instance, in [22], Díaz and Weinstein derived upper and lower bounds for the torsional rigidity of a beam, also in the case of multiply connected domains, in terms of its second-order moment. In 1950, Pólya and Weinstein proved in [41] that among all beams with multiply connected cross-sections with given area and given total area of the holes, the annulus maximizes the torsional rigidity. Namely, using definition (1), one has T () ≤ T (O ), (2) where O = G  \ S  , and G  and S  denote concentric balls having the same measure as G and S, respectively. This result has also been extended to any dimension and to certain degenerate elliptic operators in [12, 13]. Moreover, this problem can be viewed as a limiting case of a double-phase problem, which has been studied by several authors, see for example [20, 21, 36]. Once an isoperimetric inequality is established, a natural question concerns rigidity, namely, under which conditions the equality holds. In the simply connected case, this was proved by Pólya in [39] and later revisited in [8]. For multiply connected domains, Pólya and Weinstein provided a partial answer: they showed that if equality holds in (2), then the domain must have exactly one hole. To the best of our knowledge, no sharper characterization of the equality case is available for multiply connected domains (m ≥ 1). We prove the following result. Proposition 1.1 Let  ⊂ R2 satisfy Definition 1.1. If equality holds in (2), that is T () = T (O ), then, up to translation,  = O and therefore there exists x0 ∈ R2 such that G = G  + x0 and S = S  + x0 . In Section 3, we prove Proposition 1.1 as a particular case of a more general theorem valid in any dimension and for a broader class of linear elliptic operators. Annuli are the only sets for which equality holds in (2). This naturally leads to the question of stability. More precisely, one would like to improve (2) by adding a remainder term on the left-hand side that quantifies how far a set is from the optimal annulus, that is the one with the same measure and the same hole measure. To this end, we first introduce a way to measure the deviation from the corresponding annulus. We separately measure the distance of the outer boundary from a sphere and the distance of the holes from a ball. One possible tool is the Fraenkel asymmetry index; see, for instance, [26] for its properties. This index is defined by  | Br (x)| α() := minn : |Br (x)| = || , x∈R |Br (x)| 123 On the stability of the annulus Page 3 of 32 199 where denotes the symmetric difference (see Subsection 2.3). We can now state our first main theorem. Theorem 1.2 Let  ⊂ R2 satisfy Definition 1.1 such that |S| ≤ T (O ) − T () ≥ 1 32 29 πγ 2 2 |G|. Then 3   2 3 |G| α (G) + |S|2 α 3 (S) . where γ2 denotes the constant appearing in the quantitative isoperimetric inequality in dimension two (see [23]). The proof of this theorem relies on the propagation of asymmetry discussed in [30], a technique that has been further developed and applied in several works, including [3–5, 9, 17, 35]. We also point out that inequalities where annuli are optimal sets already appear in the literature, see [18, 19, 28, 37, 38]. For instance in [19], the authors studied a mixed eigenvalue problem on a domain with holes, providing an interpretation of the distance between  and the corresponding annulus. In that case as well, the proof is divided into two parts, dealing separately with the outer and inner boundaries. Motivated by the classical Fraenkel asymmetry index, we introduce an annular asymmetry index defined by β() = inf{| A| : A = B1 \ B2 , |B1 | = |G|, |B2 | = |S|}, which vanishes if and only if  is an annulus. Using this notion, we state our second main result. Theorem 1.3 Let  ⊂ R2 satisfy Definition 1.1. Then, there exists C > 0 and θ > 0, depending on |G| and |S|, such that T (O ) − T () ≥ Cβ θ (). The proof of this theorem relies on the closeness of the torsion function to the one of the optimal annulus. Since the holes correspond to regions where the gradient of the torsion function vanishes, this closeness implies not only that the outer domain and the holes are close to balls (as shown in Theorem 1.2), but also that these balls are almost concentric. The paper is organized as follows. In Section 2, we introduce notation and preliminary results. Section 3 is devoted to the proof of rigidity. In Section 4, we show that the outer domain is close to a ball; in Section 5, we prove that the asymmetry of the holes is small; and in Section 6, we study the almost radiality of the torsion function. Finally, in Section 7, we present a list of open problems. 2 Notation and preli (...truncated)