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

This paper establishes a quantitative isoperimetric inequality for the torsion of multiply connected domains, demonstrating that any domain with torsional rigidity close to the optimal value must be geometrically close to an annulus.

Original authors: Vincenzo Amato, Luca Barbato

Published 2026-06-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Vincenzo Amato, Luca Barbato

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a long, straight beam, like a wooden plank used in construction. If you try to twist this beam (like wringing out a wet towel), how much does it resist? In physics and engineering, this resistance is called torsional rigidity.

The shape of the beam's cross-section (the face you see if you cut the beam) matters a lot. A round pipe is very hard to twist. A square beam is easier. A flat, thin strip is very easy to twist.

For a long time, mathematicians knew a "Golden Rule" for beams with holes (like a pipe or a washer): The shape that resists twisting the most, for a given amount of material and a given size of the hole, is a perfect ring (an annulus). Think of a donut or a washer where the hole is perfectly centered.

This paper is about what happens when your beam is almost a perfect ring, but not quite. Maybe the hole is slightly off-center, or the outer edge is a bit lumpy. The authors ask: If a beam is almost as strong as the perfect ring, does that mean the beam itself must look almost exactly like a perfect ring?

The answer is yes.

Here is a breakdown of their findings using simple analogies:

1. The "Perfect Donut" Benchmark

Imagine you are baking cookies. You have a fixed amount of dough (the outer area) and you must cut out a hole in the middle (the inner area).

The Rule: If you want the cookie to be as strong as possible against twisting, you must make the hole perfectly round and perfectly centered. This is the "Annulus."
The Old Knowledge: Mathematicians already knew that if your cookie is exactly the strongest possible, it must be a perfect ring.

2. The "Stability" Question

Now, imagine you made a cookie that is almost the strongest. It's 99% as strong as the perfect ring.

The Question: Does that mean your cookie is 99% round and centered? Or could it be a weird, lumpy shape that just happens to be strong by accident?
The Paper's Discovery: The authors prove that there are no "accidents." If your cookie is nearly as strong as the perfect ring, it must be nearly a perfect ring. The shape and the strength are tightly locked together. You can't have a weird shape that is almost as strong as the perfect one without the shape itself being almost perfect.

3. Measuring the "Wobble"

The authors created a way to measure exactly how "wobbly" or "off-center" your shape is.

They measure two things separately:
1. How far the outer edge is from being a perfect circle.
2. How far the hole is from being a perfect circle.
They proved that if the "strength gap" (the difference between your cookie's strength and the perfect ring's strength) is small, then the "wobble" of the edges must also be small.

4. The "Concentric" Secret

There is a tricky part. Even if your outer edge is a perfect circle and your hole is a perfect circle, your cookie could still be weak if the hole is off-center (like a donut where the hole is pushed to the side).

The paper shows that if the strength is nearly optimal, not only are the shapes round, but the hole is also almost perfectly centered inside the outer edge.
They call this "almost radiality." It means the whole structure is almost like a target with bullseyes, rather than a lopsided blob.

5. The "Holes" Matter

The paper also deals with beams that might have multiple holes (like a Swiss cheese with many holes, though the math focuses on the total area of the holes).

They found that the "badness" of the holes (how irregular they are) is also controlled by the strength. If the beam is strong, the holes must be small and round.

Summary in a Nutshell

Think of the beam's strength as a "score."

Perfect Score: You have a perfect, centered ring.
High Score (but not perfect): You have a shape that is very close to a perfect ring.
The Paper's Conclusion: You cannot get a "High Score" with a "weird shape." The only way to get a high score is to have a shape that looks almost exactly like the perfect ring. The math proves that the "distance" between your weird shape and the perfect ring is directly tied to how much strength you lost.

The authors didn't just say "it's close"; they gave a precise mathematical formula (a "quantitative version") that tells you exactly how close the shape is based on how close the strength is. It's like saying, "If your cookie is 1% weaker than the perfect one, your hole is definitely less than 1% off-center."

What they did NOT do:
The paper is purely mathematical. It does not talk about building real bridges, designing new materials, or medical applications. It strictly proves a rule about the geometry of shapes and their physical resistance to twisting.

Technical Summary: On the stability of the annulus for the torsion of multiply connected domains

Problem Statement
The paper addresses the stability of the isoperimetric inequality associated with the torsional rigidity of multiply connected domains. While it is a classical result (Pólya and Weinstein, 1950) that among all multiply connected domains with a fixed total area and a fixed total area of holes, the concentric annulus maximizes torsional rigidity, the question of stability remains: if a domain's torsional rigidity is close to this maximum, how close is the domain geometrically to an annulus?

The authors consider a domain $\Omega = G \setminus S$ , where $G$ is a bounded Lipschitz domain and $S = \bigcup_{i=1}^m \Omega_i$ represents $m$ disjoint holes. The torsional rigidity $T(\Omega)$ is defined via a Rayleigh quotient involving the solution to a Poisson equation with Dirichlet boundary conditions on $\partial G$ and constant (but unknown) values on the boundaries of the holes.

Methodology
The authors employ a combination of symmetrization techniques, quantitative geometric inequalities, and the propagation of asymmetry.

Rearrangement and Symmetrization: The analysis relies heavily on the Schwarz rearrangement ( $u^\sharp$ ) and the decreasing rearrangement ( $u^*$ ) of the torsion function. The authors utilize the Pólya-Szegő inequality and its quantitative versions to compare the energy of the solution on the general domain $\Omega$ with that on the optimal annulus $\Omega_O$ .
Quantitative Isoperimetric Inequalities: The proof integrates the quantitative isoperimetric inequality (Theorem 2.9), which relates the perimeter deficit to the Fraenkel asymmetry index $\alpha(\Omega)$ . This allows the authors to translate a deficit in torsional rigidity into a geometric deficit (asymmetry) of the level sets of the torsion function.
Propagation of Asymmetry: A core technique involves "propagating" the asymmetry from the level sets of the solution to the boundaries of the domain. The authors adapt lemmas regarding the stability of sets under small measure perturbations (e.g., Lemma 2.10 and Lemma 5.3) to show that if the global torsional rigidity is near-optimal, the outer boundary $G$ and the union of holes $S$ must individually be close to balls.
Concentricity Analysis: To establish that the domain is close to an annulus (concentric balls) rather than just two separate balls, the authors analyze the "almost radiality" of the torsion function. They define a truncated function $w$ and use the quantitative Pólya-Szegő inequality (Theorem 2.11) to show that the gradient of the solution is close to that of the radial solution, implying the holes are not only spherical but also nearly concentric with the outer boundary.

Key Contributions and Results

Rigidity Characterization (Proposition 1.1 & Theorem 3.1): The authors provide a sharp characterization of the equality case for the Pólya-Weinstein inequality. They prove that if $T(\Omega) = T(\Omega_O)$ , then $\Omega$ must be a concentric annulus (up to translation). This extends previous partial results to a complete characterization for multiply connected domains.
Stability via Fraenkel Asymmetry (Theorem 1.2): The paper establishes a quantitative stability result. Under the assumption that the hole area $|S|$ is not too large relative to the outer domain ( $|S| \leq \frac{2}{3}|G|$ ), the deficit in torsional rigidity controls the sum of the cubes of the Fraenkel asymmetries of the outer domain and the holes:
$T(\Omega_O) - T(\Omega) \geq C \left( |G|^2 \alpha^3(G) + |S|^2 \alpha^3(S) \right).$
This result implies that if the rigidity is near-optimal, both the outer boundary and the holes must be close to balls.
Stability via Annular Asymmetry (Theorem 1.3): The authors introduce an "annular asymmetry index" $\beta(\Omega)$ , measuring the distance of $\Omega$ from the optimal concentric annulus. They prove that the rigidity deficit controls this index:
$T(\Omega_O) - T(\Omega) \geq C \beta^\theta(\Omega).$
This confirms that near-optimality in rigidity forces the domain to be close to a concentric annular configuration, not just a collection of balls.

Significance and Claims
The paper claims to resolve the stability question for the torsional rigidity of multiply connected domains, a problem where sharper characterizations of the equality case were previously unavailable. The significance lies in:

Completing the Rigidity Picture: Moving beyond the existence of the optimal shape to a precise description of the equality case and the behavior of near-optimal shapes.
Handling Multiple Holes: Extending stability analysis to domains with an arbitrary number of holes ( $m \geq 1$ ), proving that the optimal configuration requires not only spherical holes but also their concentricity with the outer boundary.
Methodological Advancement: Demonstrating how the propagation of asymmetry techniques, previously applied to simply connected domains, can be adapted to the more complex geometry of multiply connected domains by separately analyzing the outer boundary and the holes.

The authors remain modest regarding the optimality of the exponents in their stability estimates, explicitly listing the determination of the optimal exponent in Theorems 1.2 and 1.3 as an open problem. They also suggest that a quantified version of the Diaz-Weinstein bounds involving second-order moments could be a future direction.