Robin Torsion Function

• The Robin torsion function is the unique weak solution of a Poisson problem with Robin boundary conditions and interpolates between Neumann and Dirichlet regimes.
• It admits several variational formulations and explicit expressions on model domains, offering insights into shape optimization and spectral asymptotics.
• Talenti-type comparisons and rigidity results demonstrate that equality in rearrangements forces the domain to be a disk with a radial, decreasing solution.

Searching arXiv for papers on the Robin torsion function and related comparison/rigidity results. The Robin torsion function is the unique weak solution of a Poisson problem with Robin boundary conditions on a bounded domain, typically written as

$$-\Delta u = 1 \quad \text{in }\Omega, \qquad \partial_n u + \beta u = 0 \quad \text{on }\partial\Omega,$$

for a Robin parameter $\beta>0$. Its integral over the domain,

$T(\Omega,\beta)=\int_\Omega u\,dx,$

is the Robin torsional rigidity. In the linear Laplace setting, the function interpolates between the Neumann and Dirichlet regimes as $\beta$ varies, admits both variational and rearrangement characterizations, and serves as a central object in comparison theory, shape optimization, quantitative stability, and spectral asymptotics (Masiello et al., 2022, Amato et al., 14 Nov 2025, Buttazzo et al., 16 Dec 2025).

1. Definition and variational structure

For a bounded Lipschitz domain $\Omega\subset\mathbb R^n$ and $\beta>0$, the Robin torsion function $u_\beta\in H^1(\Omega)$ is the unique weak solution of

$$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$

In weak form this means

$$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$

and existence and uniqueness follow by standard Lax–Milgram arguments; in the planar rigidity setting the solution is also positive in $\Omega$ (Masiello et al., 2022).

The same object admits several equivalent variational formulations. One formulation identifies $\beta>0$0 as the unique minimizer of a quadratic energy such as

$\beta>0$1

or, in equivalent normalization,

$\beta>0$2

over $\beta>0$3 (Li et al., 18 Sep 2025, Sannipoli, 2020, Ognibene, 2024). A dual formulation characterizes the associated torsional rigidity by

$\beta>0$4

a representation that links the torsion problem directly to the first Robin eigenvalue and to geometric shape functionals (Buttazzo et al., 16 Dec 2025, Amato et al., 14 Nov 2025, Barbato et al., 27 Mar 2026).

Some sources use the equivalent sign convention $\beta>0$5 instead of $\beta>0$6, with $\beta>0$7 restoring the standard normalization. This is a change of notation rather than a different problem (Gavitone et al., 29 Sep 2025).

2. Explicit formulas and elementary model cases

On a ball, the Robin torsion function is explicit and radial. If $\beta>0$8, then

$\beta>0$9

with

$T(\Omega,\beta)=\int_\Omega u\,dx,$0

$T(\Omega,\beta)=\int_\Omega u\,dx,$1

(Sannipoli, 2020). In dimension two this becomes

$T(\Omega,\beta)=\int_\Omega u\,dx,$2

which is the radial comparison profile appearing in Talenti-type results (Masiello et al., 2022).

Integrating the ball solution yields exact torsional rigidity formulas. For a ball $T(\Omega,\beta)=\int_\Omega u\,dx,$3,

$T(\Omega,\beta)=\int_\Omega u\,dx,$4

while in the radius-$T(\Omega,\beta)=\int_\Omega u\,dx,$5 ball representation the energy decomposition isolates the Dirichlet torsion contribution and the Robin boundary contribution (Buttazzo et al., 16 Dec 2025, Bandle et al., 2014).

These formulas exhibit the basic limiting regimes. As $T(\Omega,\beta)=\int_\Omega u\,dx,$6, the Robin problem approaches the Dirichlet torsion problem; as $T(\Omega,\beta)=\int_\Omega u\,dx,$7, the solution amplitude blows up, reflecting the incompatibility of the pure Neumann problem with the forcing term $T(\Omega,\beta)=\int_\Omega u\,dx,$8 (Sannipoli, 2020). This suggests regarding the Robin torsion function as an interpolation between clamped and weakly constrained boundary response.

3. Rearrangement, Talenti comparison, and rigidity

A central comparison result is the Talenti-type theorem for Robin boundary conditions. In the planar case, for an open, bounded, Lipschitz set $T(\Omega,\beta)=\int_\Omega u\,dx,$9, let $\beta$0 be the disk with $\beta$1. If $\beta$2 solves the Robin torsion problem on $\beta$3 and $\beta$4 solves the same problem on $\beta$5, then

$\beta$6

where $\beta$7 denotes the Schwarz rearrangement of $\beta$8 (Masiello et al., 2022). In particular, for $\beta$9,

$\Omega\subset\mathbb R^n$0

and the Saint-Venant inequality

$\Omega\subset\mathbb R^n$1

is recovered as a corollary (Masiello et al., 2022).

The corresponding rigidity theorem is sharp. If equality holds pointwise in the rearrangement comparison, namely

$\Omega\subset\mathbb R^n$2

then $\Omega\subset\mathbb R^n$3 is, up to translation, a disk and $\Omega\subset\mathbb R^n$4 is radial decreasing. A weaker one-slice criterion also suffices: if

$\Omega\subset\mathbb R^n$5

and there exists one radius $\Omega\subset\mathbb R^n$6 such that $\Omega\subset\mathbb R^n$7, then again $\Omega\subset\mathbb R^n$8 must be a disk and $\Omega\subset\mathbb R^n$9 radial (Masiello et al., 2022).

The proof is driven by differential inequalities for level-set distributions. Writing $\beta>0$0 and $\beta>0$1, one obtains for almost every $\beta>0$2

$\beta>0$3

while equality holds for the radial comparison function $\beta>0$4. Integrating in $\beta>0$5 yields $\beta>0$6 and hence the rearrangement inequality (Masiello et al., 2022). A boundary-integral identity,

$\beta>0$7

is a key ingredient in closing the equality case (Masiello et al., 2022).

A later quantitative extension establishes a stability version of Saint-Venant in terms of the Fraenkel asymmetry

$\beta>0$8

where $\beta>0$9 is the ball with $u_\beta\in H^1(\Omega)$0. There exists a constant $u_\beta\in H^1(\Omega)$1 such that

$u_\beta\in H^1(\Omega)$2

and equality occurs if and only if $u_\beta\in H^1(\Omega)$3 is, up to translation, a ball (Amato et al., 14 Nov 2025). The proof proceeds by a coarea–isoperimetric inequality for level sets, a Gronwall-type comparison, and propagation of asymmetry from $u_\beta\in H^1(\Omega)$4 to the superlevel sets $u_\beta\in H^1(\Omega)$5 (Amato et al., 14 Nov 2025).

4. Shape optimization and geometric inequalities

The Robin torsion function generates several shape functionals of isoperimetric type. Among sets of fixed volume, the disk or ball is the comparison geometry for Talenti- and Saint-Venant-type inequalities, but other extremal problems emphasize different shapes.

For convex domains, one sharp upper bound uses the distance-to-boundary function $u_\beta\in H^1(\Omega)$6: $u_\beta\in H^1(\Omega)$7 This extends the classical Makai bound and reduces to the Dirichlet estimate $u_\beta\in H^1(\Omega)$8 as $u_\beta\in H^1(\Omega)$9 (Barbato et al., 27 Mar 2026). By combining this with elementary bounds on $$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$0 and $$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$1, one obtains the Robin–Makai inequality

$$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$2

where $$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$3 is the inradius (Barbato et al., 27 Mar 2026).

The same work introduces the scale-invariant Robin Makai functional

$$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$4

and states that slab-domains are asymptotically optimal. A slab-domain has the form

$$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$5

with $$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$6 fixed convex and $$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$7; for such domains,

$$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$8

and

$$-\Delta u_\beta=1 \quad \text{in }\Omega,\qquad \partial_n u_\beta+\beta u_\beta=0 \quad \text{on }\partial\Omega.$$9

as $$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$0 (Barbato et al., 27 Mar 2026).

A different shape-optimization perspective concerns products of torsional rigidity and first Robin eigenvalue. Defining

$$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$1

for $$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$2, one finds threshold behavior: $$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$3 where $$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$4 is the infimum over unit-volume Lipschitz domains, and

$$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$5

for the corresponding supremum problem (Buttazzo et al., 16 Dec 2025). The threshold $$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$6 is strictly smaller than the Dirichlet threshold $$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$7 (Buttazzo et al., 16 Dec 2025). This indicates that Robin boundary interaction changes the scaling balance between compliance and vibration in a quantitatively detectable way.

For positive $$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$8, the ball maximizes the Robin torsional rigidity among sets of fixed volume, a fact cited in the context of overdetermined and comparison results (Gavitone et al., 29 Sep 2025). For negative Robin parameters, however, the behavior is more delicate. In the opposite-sign regime, the ball need not provide a minimum, and for nearly spherical domains with elasticity constants close to zero the ball is instead largest for the torsional energy; the transition is tied to Steklov spectral structure and shape derivatives (Bandle et al., 2014).

5. Qualitative properties: concavity, monotonicity, and symmetry

Beyond integral inequalities, the Robin torsion function exhibits qualitative structure under geometric assumptions on the domain.

On uniformly convex domains $$\int_\Omega \nabla u\cdot \nabla \phi\,dx+\beta\int_{\partial\Omega}u\phi\,d\mathcal H^{n-1}=\int_\Omega \phi\,dx \quad \text{for all }\phi\in H^1(\Omega),$$9 of class $\Omega$0, there exists a threshold $\Omega$1 depending on $\Omega$2, $\Omega$3, and geometric quantities of $\Omega$4 such that for every $\Omega$5, the function

$\Omega$6

is strictly concave on $\Omega$7 (Crasta et al., 2020). Equivalently,

$\Omega$8

for all $\Omega$9 and $\beta>0$00 (Crasta et al., 2020). The proof combines regularity, convergence to the Dirichlet torsion as $\beta>0$01, boundary Hessian estimates, and a constant-rank argument propagated by continuous deformation to a ball (Crasta et al., 2020).

In planar symmetric domains, Li–Wei–Yao establish a one-sided monotonicity theorem. If $\beta>0$02 is smooth, symmetric about the $\beta>0$03-axis, convex in the $\beta>0$04-direction, and satisfies the curvature condition

$\beta>0$05

then the Robin torsion function is even in $\beta>0$06 and satisfies

$\beta>0$07

together with

$\beta>0$08

on the symmetry line (Li et al., 18 Sep 2025). The condition is, in a stated sense, sharp: for any fixed $\beta>0$09 there exists a smooth, symmetric, $\beta>0$10-convex planar domain for which $\beta>0$11 changes sign (Li et al., 18 Sep 2025).

Overdetermined boundary data lead to Serrin-type rigidity. In the formulation using $\beta>0$12 in $\beta>0$13, $\beta>0$14 on $\beta>0$15, and the additional boundary identity

$\beta>0$16

if $\beta>0$17 and

$\beta>0$18

then $\beta>0$19 must be a ball and $\beta>0$20 radially symmetric (Gavitone et al., 29 Sep 2025). The proof is based on the subharmonic $\beta>0$21-function

$\beta>0$22

a fundamental integral identity, and the curvature sign condition (Gavitone et al., 29 Sep 2025).

6. Shape derivatives, spectral relations, and extensions

The Robin torsion function is also a shape-differentiable object. For a perturbation field $\beta>0$23 and perturbed domains $\beta>0$24, define

$\beta>0$25

Then the first variations satisfy

$\beta>0$26

$\beta>0$27

where $\beta>0$28 is a maximum point of $\beta>0$29 and $\beta>0$30 solves

$\beta>0$31

$\beta>0$32

(Sannipoli, 2020). For the ball and first-order volume-preserving perturbations, the first variations vanish: $\beta>0$33 Thus balls are critical shapes for the $\beta>0$34- and $\beta>0$35-norms of the Robin torsion function under a volume constraint (Sannipoli, 2020).

The torsion function also enters directly into Robin spectral asymptotics. For Robin eigenvalues $\beta>0$36 and Dirichlet eigenpairs $\beta>0$37,

$\beta>0$38

The associated torsional rigidity

$\beta>0$39

is identified as the natural geometric quantity governing first-order Dirichlet-limit asymptotics (Ognibene, 2024). A plausible implication is that Robin torsion is not merely a compliance functional but also a boundary-sensitive proxy for the rate at which Robin spectral data collapse to their Dirichlet counterparts.

The notion extends beyond the Euclidean Laplacian. For the Hermite operator in Gaussian space, the Robin torsion function solves

$\beta>0$40

equivalently

$\beta>0$41

and the Gaussian torsional rigidity is

$\beta>0$42

Among sets of prescribed Gaussian measure, half-spaces maximize this quantity, with equality only for half-spaces up to rotation (Chiacchio et al., 2021). This weighted analogue mirrors the Euclidean rearrangement theory while replacing Euclidean isoperimetry with Gaussian isoperimetry (Chiacchio et al., 2021).

Finally, there is a nonlinear $\beta>0$43-Laplacian extension. For $\beta>0$44 and $\beta>0$45, the Robin $\beta>0$46-torsion function is the unique weak solution of

$\beta>0$47

equivalently the minimizer of

$\beta>0$48

(Berg et al., 2013). In the linear case $\beta>0$49, explicit $\beta>0$50 bounds can be expressed in terms of the first Robin eigenvalue (Berg et al., 2013).

The Robin torsion function thus occupies a structurally rich position at the intersection of elliptic PDE, geometric analysis, and shape optimization. Its theory includes exact radial models, Talenti-type comparison and rigidity, quantitative stability in asymmetry, curvature-dependent monotonicity, concavity thresholds, shape derivatives, and direct coupling to Robin spectral invariants (Masiello et al., 2022, Amato et al., 14 Nov 2025, Li et al., 18 Sep 2025, Crasta et al., 2020, Ognibene, 2024).