Euclidean persistence and sharp stability for anisotropic eigenvalue-torsion products
Henry Zweiman
September 15, 2026
Abstract
We optimize the product of the first Dirichlet eigenvalue and a power of torsional rigidity on a fixed Euclidean ball, allowing the anisotropic energy to vary over all normalized symmetric seminorms. For every dimension and every energy exponent , we prove that the Euclidean norm uniquely maximizes the product for all sufficiently small positive torsion exponents and uniquely minimizes it for all sufficiently large exponents. In either regime, the product deficit is equivalent to the spherical average of . Consequently it controls the supremum norm distance from the Euclidean norm with the optimal power . The proof combines radial variational tests, convex symmetrization, and an integral comparison between support and radial deficits of convex bodies inside a ball. For quadratic growth, this answers the ball cases of two Euclidean-optimality questions of Buttazzo and Fernandes Horta, beyond the quadratic-seminorm class. No regularity or strict convexity of the competing anisotropies is assumed.
Fix the Euclidean unit ball , where , and an exponent . Denote its volume by , write , and let be probability surface measure on . We consider the class of symmetric seminorms normalized by
Thus is convex, satisfies , and may vanish on a nonzero subspace. The Euclidean norm is denoted by . For define
The dependence on is suppressed. These quantities are finite and strictly positive even for degenerate seminorms; a direct proof is given below. The definition does not require attainment in the degenerate case. Our torsion convention agrees with the integral of the solution of the torsion equation in the usual coercive setting. In particular, is the variational quotient before taking the root in (1.2).
The normalized product and its angular deficit are
The exponent is independent of the energy exponent . Since , decreasing the anisotropy lowers the eigenvalue and raises the torsion. Separate monotonicity statements therefore do not determine the product. Even convergence of optimizers to as tends to an endpoint would not establish that is the exact optimizer at a fixed finite .
Theorem 1.1 (Euclidean persistence and deficit equivalence). For every and , there are numbers
with the following properties. The Euclidean norm uniquely maximizes over whenever , and uniquely minimizes it whenever . For each fixed in these regimes, set
There are constants such that
The theorem applies to the full infinite-dimensional class of normalized symmetric seminorms, with no ellipticity constant prescribed in advance. In particular it includes polyhedral norms and seminorms of every positive rank. Explicit, though nonoptimal, choices of the thresholds are provided below.
The associated uniform distance is
Theorem 1.2 (optimal stability power). Under the assumptions and exponent restrictions of Theorem 1.1, there is such that
The power cannot be replaced by a smaller one, even if the competitors are restricted to genuine norms. More precisely, there are norms , for , with and
By scaling the spatial variable, the same normalized conclusions hold on any Euclidean ball. The thresholds do not depend on its radius: the eigenvalue scales by the inverse -th power of the radius and torsion by the power , and these factors cancel in .
Relation to previous work
Buttazzo and Fernandes Horta [BFH] study eigenvalue-torsion products while varying normalized seminorms on a fixed domain. Their Section 6 asks whether the norm optimizers in the small- maximum and large- minimum regimes are Euclidean. For balls, their Theorem 4.10 proves maximality for within the class of quadratic seminorms. Theorem 1.1 answers both questions on balls in the full symmetric-seminorm class, and treats the general -growth setting also proposed in their final remarks. It does not resolve their questions for arbitrary fixed domains or determine the optimal thresholds.
The optimization of the eigenvalue alone, including rigidity of Euclidean maximality, was studied by Fernandes Horta and Montenegro [HM]; see their Theorem 1.5. We do not claim that endpoint result as new. Our spectral lower estimate uses the convex symmetrization theorem of Alvino, Ferone, Trombetti and Lions [AFTL]. The body and normalization conventions for this input are stated explicitly in Section 4. The convex-geometric part uses support and radial deficits, quantities related to the intrinsic and dual volume deviations studied in [BHK]. We provide the geometric estimate needed here in full and make no separate priority claim for the elementary cap geometry.
The main analytic point is a common linear control, in , of the spectral loss and torsion gain near . Directional first variation alone gives no such uniform estimate over increasingly concentrated angular perturbations. The support-to-radial comparison in Section 3 supplies this missing control without smoothness assumptions. It also produces the sharp supremum norm stability power.
2. Convex representation and global variational bounds
Every is the support function of the compact centrally symmetric convex set
Indeed, the support representation follows from separation for finite continuous sublinear functions. Normalization gives and . To check the latter, the maximum of on equals the maximum of over . In particular, there is , and symmetry implies
Also . When has interior, write for its radial function and for the dual norm of . Then
Thus is the polar of the unit ball of , not the unit ball of itself.
Lemma 2.1 (global bounds). Put
For every ,
Moreover, , where this last integral depends only on .
Proof. For on an interval of length , the fundamental theorem of calculus and Hölder's inequality imply
Apply this on chords parallel to and integrate over the orthogonal variables. Using (2.2) and density in gives
This proves the eigenvalue lower bound. Combining (2.7) with proves the torsion upper bound in (2.5). In particular, none of the denominators in (1.2) vanish for nonzero admissible functions.
A Euclidean first eigenfunction can be chosen radial and nonnegative by Schwarz symmetrization of a minimizer. Existence follows by the direct method and the compact embedding into . For every radial Sobolev function , homogeneity and polar coordinates give
Testing the eigenvalue with a radial Euclidean minimizer proves its upper bound. The Euclidean torsion function and its integral are
In fact , so in distributions. Testing this identity with gives
Hölder applied to the same weak identity shows that attains the Euclidean torsion quotient. Equations (2.8)-(2.10) then give its lower bound in (2.5). The final assertion follows by integrating (2.2) on . ∎
We shall use the fixed constants
They give the global bounds and . Their sharp values are unnecessary.
3. An integral comparison of support and radial deficits
For and , let be a chordal cap. The notation denotes the cap with the same center and radius multiplied by .
Lemma 3.1 (support-to-radial comparison). There is such that every compact convex set with satisfies
Symmetry is not required in this lemma.
Proof. We first record a cap estimate. Suppose that , where . Since , its support function at a direction making angle with satisfies
For completeness, because . Thus the unconstrained support point is excluded by the cutting plane, and maximizing the linear functional on the truncated ball puts the maximizing point on that plane. Decomposition into the and components gives the first line of (3.2). The last estimates use and .
Set . On the angle is less than , so this cap lies in .
Now fix . Let be its metric projection onto , and put
The projection inequality gives , hence . Therefore
The cap from (3.2) consequently has .
Fix and write . For each choose the cap just constructed; its associated exceeds . Every compact subset is covered by finitely many of the open caps . Order the corresponding original caps by decreasing radius, and retain a cap precisely when it is disjoint from every previously retained cap. The retained caps are disjoint. Any discarded cap of radius intersects a retained cap of radius , so its twelvefold enlargement lies in , and hence in . Thus .
Spherical cap measure is doubling: there is such that
This follows directly by writing cap area in polar coordinates. The ratio tends to as and is bounded on every remaining compact interval; a cap whose chordal radius exceeds is the entire sphere. Each retained cap is contained in by (3.2). Disjointness and (3.5) give
By inner regularity this holds with . Integrating in yields
The gauge is -Lipschitz since contains . Indeed, subadditivity gives , and one can interchange . For the projection above, , so
Since , we have . Equations (3.7)-(3.8) prove (3.1) with . ∎
The estimate transfers angular support loss to the radial moments needed by both variational problems. A supremum norm comparison alone would not give the linear dependence on used below.
4. Uniform estimates near the Euclidean norm
Set
which is positive and independent of . If , the -Lipschitz property implies on a cap of radius . Thus . In particular, if , then on and contains .
For the remainder of this section write . Since on , Lemma 3.1 gives
Polar integration and for yield
Similarly, on the derivative of has magnitude at most , and therefore
Fix
Lemma 4.1 (spectral and torsion control). If , then
and
Proof. Only the lower spectral and upper torsion bounds remain. Convex symmetrization [AFTL, Theorem 3.1 and equation (3.5)] implies, for and its Euclidean Schwarz rearrangement ,
Here one first replaces by ; evenness of preserves the energy. To specify the normalization in (4.8), let . The energy of the convex rearrangement has the level-set expression
in the usual rearrangement interpretation. The corresponding expression for replaces by . The convex rearrangement theorem decreases the original energy; comparison of these expressions gives (4.8), with the general Sobolev statement supplied by that theorem. The body denoted in Section 2 of [AFTL] is the unit ball of ; its polar is the body called here. Their auxiliary volume normalization can be removed by scaling : both sides of (4.8) scale by its -th power. Their hypotheses allow nonsmooth convex norms with positive coercivity, which holds in the present neighborhood. No regularity approximation of or equality case is needed.
Equimeasurability preserves the denominator in the Rayleigh quotient. Taking the infimum in (4.8) gives , and (4.3) proves (4.6).
For torsion use the Euclidean flux
For every , integration by parts, duality of norms and Hölder's inequality give
Take the supremum and the power in (1.2). Radiality of and (2.10) imply
Equation (4.4) proves the upper torsion bound. Notice that the duality test uses the flux , whose Euclidean -energy equals , rather than the gradient with an incorrect exponent. ∎
Corollary 4.2. For and ,
Proof. For the first estimate use and . For the second use , valid because , and . Apply (4.6)-(4.7) to the logarithm of the product. ∎
5. Persistence and quantitative equivalence
With the constants from (2.11) and (4.5), choose
and
These constants satisfy (1.4).
Proof of Theorem 1.1. Suppose first that . Corollary 4.2 gives
and
For , Lemma 2.1 and (2.11) imply
whereas
These estimates include all degenerate seminorms. Finally, forces everywhere on by continuity, so . Equations (5.3)-(5.6) prove uniqueness and existence of the stated optimizers directly.
For completeness, we establish both sides of (1.6). In the small- regime and for , (5.3) gives
Since , the other side follows from (4.6):
The last inequality follows from Bernoulli's inequality if and from otherwise. In the large- regime, (5.4) implies . For its upper bound, and (4.7) give
where one may take .
It remains to treat . We always have . The uniform gaps in (5.5)-(5.6) therefore give a positive constant times as a lower bound for . In the maximum regime . In the minimum regime , so . Combining these constants with (5.7)-(5.9) proves (1.6). ∎
6. Optimal supremum norm stability
Lemma 6.1 (angular concentration bound). There is such that every satisfies
Proof. If , both assertions are immediate. Otherwise choose with . The cap calculation (3.2), which holds for every without an inball assumption, applies with . On we have , and hence . Its measure is at least a dimensional constant times . This proves the lower bound. The upper bound follows from for . ∎
Proof of Theorem 1.2. The lower bound follows at once by combining (1.6) and Lemma 6.1. For sharpness, take
This centrally symmetric body contains and, since , touches on the equator perpendicular to . Consequently is a genuine normalized norm. At its support equals , so .
Whenever , the point itself belongs to , and thus . The angular deficit is supported on two spherical caps of angular radius . Their total measure is at most . Also , whence
The upper bound in (1.6) proves (1.9). If a uniform estimate held for some , evaluating it at and letting would force a fixed positive to be zero. This proves optimality. ∎
7. Scope and further questions
The argument identifies exact optimizers at finite exponents and gives a common angular measure of their stability. The two regimes and the nonlinear extension follow from the same comparison mechanism. None of the constants in the construction is intended to be optimal.
The natural next question is to determine the full interval of for which is extremal. For , the quadratic-seminorm conclusion of [BFH, Theorem 4.10] reaches in the maximizing problem; extending that endpoint to arbitrary norms requires information not supplied by (4.13). Another question is whether analogous persistence holds on fixed domains other than balls. Our radial trial identities and dual flux calculation depend on the ball geometry, and the present theorem makes no assertion for general domains. Finally, constants uniform as or are not obtained here.
References
[AFTL] A. Alvino, V. Ferone, G. Trombetti and P.-L. Lions, Convex symmetrization and applications, Annales de l'Institut Henri Poincaré C, Analyse non linéaire 14 (1997), 275-293. Primary article and full text.
[BFH] G. Buttazzo and R. Fernandes Horta, On the relations between fundamental frequency and torsional rigidity in the case of anisotropic energies, arXiv:2603.09851v1 (2026). Versioned full text.
[BHK] F. Besau, S. Hoehner and G. Kur, Intrinsic and dual volume deviations of convex bodies and polytopes, International Mathematics Research Notices 2021 (22), 17456-17513. DOI; author preprint.
[HM] R. Fernandes Horta and M. Montenegro, Sharp isoanisotropic estimates for fundamental frequencies of membranes and connections with shapes, Calculus of Variations and Partial Differential Equations 64 (2025), article 273. Published article; arXiv:2406.18683v4.