Torsion transplantation and a uniform reverse Kohler-Jobin inequality for Robin problems
Henry Zweiman
September 15, 2026
Abstract
We prove that balls maximize the product of the first Robin eigenvalue and a sufficiently large power of the torsional rigidity among bounded planar Lipschitz sets of prescribed area. The exponent threshold is independent of the positive Robin parameter and the area. This answers the planar case of the reverse Kohler-Jobin question posed by Buttazzo, Cito and Solombrino. More generally, the argument applies to the Robin -Laplacian in dimension when , using the integral of the -torsion function. The main step is a composition identity with a nonpositive boundary correction, which transplants the ball eigenfunction through the torsion function and preserves the reference energy. The known pointwise Talenti comparison then yields a linear upper bound for the spectral deficit in terms of the torsional deficit. An elementary global bound completes the optimization argument. For larger exponents, the product deficit is equivalent to the torsional deficit, uniformly in the Robin parameter. The higher-dimensional linear Robin problem is not resolved.
1. Introduction and main results
For a fixed positive Robin parameter, the ball minimizes the first eigenvalue and maximizes torsional rigidity among sets of prescribed volume. These two inequalities act in opposite directions on their product. Increasing the power of torsion suggests that its maximizer should eventually maximize the product, but the separate isoperimetric inequalities do not prove this assertion: one needs quantitative control of the competing increase in the eigenvalue.
Buttazzo, Cito and Solombrino [BCS, Section 5, Q2] ask whether the ball maximizes the first Robin eigenvalue times a sufficiently large power of Robin torsional rigidity. The corresponding Dirichlet reverse Kohler-Jobin inequality is established by Bucur, Lamboley, Nahon and Prunier [BLNP, Corollary 1.6]. Here we answer the Robin question in the plane. Our exponent threshold is uniform in both the positive boundary parameter and the prescribed area. The proof also gives a nonlinear extension in the range of the known pointwise Robin rearrangement theorem.
Let , and . For a nonempty bounded open set with Lipschitz boundary, define
Let be the nonnegative weak solution of
where , and put
Our convention for is the integral of , rather than its -st power. For it is the usual torsional rigidity . Let be a ball with . We omit subscripts when the parameters are fixed and write
Theorem 1.1 (uniform reverse Robin inequality). Fix and . There is a finite number such that, for every , and nonempty bounded open Lipschitz set of volume ,
For every such ,
Equality in (1.5) holds if and only if is a translate of .
In particular, supplies a single exponent threshold for the planar problem [BCS, Q2], simultaneously for all positive Robin parameters. Neither convexity nor connectedness is required. Formula (1.6) says that, for a fixed admissible exponent, the normalized product deficit and torsional deficit are equivalent with constants independent of , and .
The estimate underlying the theorem is the following one-sided spectral control.
Theorem 1.2 (linear spectral control). Under the hypotheses of Theorem 1.1, there is , independent of and , such that
The essential rearrangement input is [AGM, Theorem 1.2(i)]:
where denotes Schwarz decreasing rearrangement. For , this is the comparison of Alvino, Nitsch and Trombetti [ANT]. We use their theorem with constant forcing and exactly the same Robin parameter on both sets. The only dimensional restriction in our proof occurs at (1.8).
We also use the Robin -Faber-Krahn inequality and rigidity [BD, Theorem 1.1]. Its extension from connected domains to the present class is elementary. A bounded Lipschitz set has finitely many components , each Lipschitz, and its first eigenvalue is . The eigenvalue of a ball strictly decreases with its radius at fixed : dilation by changes a trial quotient with gradient part and boundary part into , which is at most . Consequently . Apply [BD] to every component and then enlarge its comparison ball to volume . This proves , with strict inequality if is disconnected; the connected equality case is precisely [BD]. The positivity and simplicity of the ball ground state are standard parts of this Robin eigenvalue theory.
2. A composition identity with a favorable boundary term
Lemma 2.1. Suppose is a bounded nonnegative weak Robin -torsion function on a bounded Lipschitz set, with . Let be , with , and set
Then and
Proof. Extend and as Lipschitz functions beyond their compact intervals. Sobolev composition and the trace identity justify testing (1.2) with . Since , this gives
Adding the Robin energy of the trace gives the identity. Hölder's inequality yields
so . The interior bounds on hold for its Sobolev trace as well: apply the trace operator to the truncations and . Thus all boundary terms above are integrable. This proves (2.2). ∎
If is linear on , then . We will impose precisely this extension below the boundary value of the ball torsion. It makes the boundary inequality an equality on the reference ball, without requiring the trace of the competing torsion function to be constant.
3. Encoding the ball eigenfunction
Let and write . The radial torsion function is
Let be the positive first eigenfunction on , normalized by , and write . Simplicity and rotational invariance make radial. Integrating its radial equation gives
In particular, strictly decreases for . Put and define
For ,
The derivative has the finite positive limit at the center. At , the two Robin boundary conditions give , so the pieces in (3.3) join in . The limiting derivative also verifies differentiability at , for example by the mean value theorem. Thus is increasing and on .
Its derivative is nondecreasing in its argument. Indeed, the radial average in (3.4) decreases with , since decreases, while decreases. Therefore both and take their maxima at , and
For the function in Lemma 2.1, the linear extension makes . Applying the identity on gives
Only first derivatives of are used, including at the center of the ball.
4. From torsional deficit to global product optimality
This section assumes (1.8). Define
The comparison ensures and bounds by . Lemma 2.1, equimeasurability and the monotonicity of imply
The denominator satisfies
The trial function is nonzero, since is strictly increasing and is nonzero and nonnegative. Its Rayleigh quotient yields
In particular, Faber-Krahn and (4.4) imply
Since and has Lipschitz constant , (3.6) gives , hence .
We also need a global bound. Testing the torsion equation with gives . Using in the Rayleigh quotient and then applying Hölder gives
This estimate alone does not identify the maximizing ball, but it controls domains outside the neighborhood covered by (4.5).
Proposition 4.1 (a threshold from ball data). Set
Then for all . For all ,
Proof. By (4.6), , and . If , then (4.4) gives
Thus , proving the product bound for . If , (4.6) gives instead
by the definition of . This covers every .
For , apply the product bound at and use Faber-Krahn to obtain
Bernoulli's inequality proves (4.8). ∎
5. Uniformity in the Robin parameter
The constants in Proposition 4.1 involve only the reference ball. The next step shows that their dependence on its radius and on can be eliminated.
Proposition 5.1. For fixed and , the quantities and in (4.1) and (4.7), computed on a ball with Robin parameter , satisfy
Consequently the supremum of over all radii and all positive Robin parameters is finite.
Proof. Under , the dimensionless boundary parameter is . The eigenvalue scales as , the normalized eigenfunction as , and as . It follows directly from (3.5), (4.1) and (4.7) that and are unchanged by this simultaneous rescaling. We may therefore take and vary in .
Write . Integrating (3.1) gives
Let be the first Dirichlet -eigenvalue on . The Rayleigh principle, using a Dirichlet eigenfunction and a constant function respectively, gives
Equation (3.2) implies
Choose depending only on so that integration of (5.4) yields on . Normalization then bounds uniformly above, while monotonicity gives . The normalized radial eigenfunctions are therefore uniformly bounded and equicontinuous on .
As , (5.3) and (3.2) show that the oscillation of tends to zero. Normalization implies uniform convergence to . The integrated eigenfunction equation is
Hence . Inserting this limit and (5.2) into the definitions of and gives
For the other endpoint, (5.4) and the Robin condition imply . Given any sequence , take a subsequence along which converges to and converges uniformly to . The limit is nonnegative, normalized, and vanishes at . Equation (3.2) passes to the limit. Its right-hand side is bounded by a constant times , so dominated convergence also gives convergence of the gradient energies. Furthermore,
Therefore and
Thus and is the normalized nonnegative first Dirichlet eigenfunction . Simplicity identifies every subsequential limit, so the whole family converges. Writing , we obtain
For completeness, the ball data are continuous at every . The same radial compactness and gradient convergence apply along . The limiting normalized trial gives . Conversely, the fixed ground state at is an admissible trial for nearby parameters and gives the reverse upper limit. Simplicity again identifies the uniform eigenfunction limit. Thus and , and hence and , are continuous.
The endpoint limits and continuity prove (5.1), since and were proved in Section 4. The denominator in (4.7) is bounded below by the positive constant . Hence has a finite supremum. ∎
Proof of Theorems 1.1 and 1.2. Set
where the ball has radius one and boundary parameter . Proposition 5.1 makes these constants finite. Equation (4.5) proves Theorem 1.2; Proposition 4.1 proves (1.5) and (1.6). If equality holds in (1.5), (1.6) gives . Equation (4.4) then yields ; Faber-Krahn and its rigidity statement give that is a translate of . Such a translate plainly attains equality. ∎
6. Consequences and remaining questions
Theorem 1.1 turns any available lower bound for the normalized torsional deficit into the same lower bound for the product deficit. For example, in the planar linear case the quantitative Robin Saint-Venant inequality gives a positive constant such that
where is Fraenkel asymmetry. This is a direct consequence of (1.6) and the known torsional remainder, as given for example in [ABCMP, Corollary 4.1]. The geometric estimate is a prior input; we do not claim that can be chosen uniformly in by that argument.
The proof of the product theorem uses no compactness of the class of competing domains and no boundary perturbation of the ball. The composition identity preserves the reference energy, and pointwise torsion comparison controls the loss in the trial denominator. The near-optimal estimate (4.4) and the global bound (4.6) are the two distinct ingredients needed to reach every domain.
The same composition construction with zero boundary value is compatible with the Dirichlet problem. The linear Dirichlet reverse inequality is already a theorem of [BLNP]; it is not an additional open-problem solution asserted here. In the Robin setting, the linear case remains beyond the pointwise input (1.8). The known and torsion comparisons do not by themselves control the nonlinear numerator in (4.2). Establishing a suitable substitute, or a different trial construction, remains necessary to extend the present proof.
The exponent (5.8) is sufficient and is not asserted to be optimal. Determining the least admissible exponent, extending the range of , and settling the forward Robin Kohler-Jobin question [BCS, Q1] remain separate problems.
References
[ABCMP] V. Amato, R. Barbato, S. Cito, A. L. Masiello and G. Paoli, A quantitative Talenti-type comparison result with Robin boundary conditions, preprint, arXiv:2511.11316v1.
[AGM] V. Amato, A. Gentile and A. L. Masiello, Comparison results for solutions to p-Laplace equations with Robin boundary conditions, Annali di Matematica Pura ed Applicata 201 (2022), 1189-1212. DOI: 10.1007/s10231-021-01153-y.
[ANT] A. Alvino, C. Nitsch and C. Trombetti, A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions, Communications on Pure and Applied Mathematics 76 (2023), 585-603. DOI: 10.1002/cpa.22090.
[BCS] G. Buttazzo, S. Cito and F. Solombrino, Relations between principal eigenvalue and torsional rigidity with Robin boundary conditions, Milan Journal of Mathematics 94 (2026), 369-386. DOI: 10.1007/s00032-026-00438-2.
[BD] D. Bucur and D. Daners, An alternative approach to the Faber-Krahn inequality for Robin problems, Calculus of Variations and Partial Differential Equations 37 (2010), 75-86. DOI: 10.1007/s00526-009-0252-3.
[BLNP] D. Bucur, J. Lamboley, M. Nahon and R. Prunier, Sharp quantitative stability of the Dirichlet spectrum near the ball, preprint, arXiv:2304.10916v2.