On the Hersch-Weinberger inequality in higher dimensions

Abstract: We investigate a reverse Faber-Krahn type inequality for the Robin Laplacian in a bounded smooth domain $Ω\subset \mathbb{R}<sup>N$ whose boundary has two connected components. We prove that a concentric spherical shell maximizes the first eigenvalue over a class of such domains under perimeter and volume constraints, and under an additional convexity assumption when $N \geq 3$. This result generalizes to a wider class, and extends to higher dimensions, the inequality of Hersch [20], whose approach was substantially based on a construction of the so-called effectless cut by Weinberger [35], so that we call it the Hersch-Weinberger inequality. Our method is based on the analysis of the gradient flow of the first eigenfunction and several approximation procedures, without relying on the effectless cut itself. The effectless cut being a complicated object related to the attractor of the gradient flow, we describe its most fundamental topological properties. In particular, we show that it does not necessarily have to be a hypersurface.

• The paper proves that concentric spherical shells maximize the first Robin Laplacian eigenvalue among admissible doubly connected domains in [?] dimensions, using quermassintegral, perimeter, volume, and convexity constraints.
• The authors replace Weinbergers planar effectless-cut method with Morse-function approximations and gradient flows, enabling the result for arbitrary positive Robin parameters, including Dirichlet conditions.
• The paper establishes sharpness through elongated-domain counterexamples and shows that higher-dimensional effectless cuts can be topologically singular, clarifying why the planar proof does not extend directly.

The Hersch–Weinberger inequality and its higher-dimensional extension

This paper establishes a reverse Faber–Krahn type inequality for the first eigenvalue of the Robin Laplacian on bounded doubly connected domains in $\mathbb{R}^N$, $N \geq 2$. For a domain $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$ with two boundary components carrying Robin parameters $\beta_1, \beta_2 \in (0, +\infty]$ (with $+\infty$ meaning Dirichlet), the authors prove that among domains satisfying prescribed quermassintegral, perimeter, and volume constraints, a concentric spherical shell maximizes $\lambda_1$. This generalizes to arbitrary dimension and to general Robin parameters the classical planar inequalities of Payne–Weinberger and Hersch, whose proofs relied on Weinberger's construction of an "effectless cut" — a curve along which the annular domain can be split without changing its first eigenvalue. The new proof avoids the effectless cut entirely, replacing it with an analysis of gradient flows of Morse approximations of the first eigenfunction.

Main result and the admissible class

The central theorem states: if $\Omega \in \mathcal{K}_{\alpha,\beta}^N$ for some $\beta > \alpha > 0$, then

$$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$

for all Robin parameters in $(0, +\infty]$. The class $N \geq 2$0 is defined by three constraints relating $N \geq 2$1 to the reference shell $N \geq 2$2:

Dimension	Constraints defining $N \geq 2$3
$N \geq 2$4	$N \geq 2$5, $N \geq 2$6, $N \geq 2$7
$N \geq 2$8	$N \geq 2$9, $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$0, $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$1, plus convexity of both $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$2 and $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$3

Here $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$4 denotes the $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$5-th quermassintegral from the Steiner formula. In dimension two the quermassintegral constraint reduces to the perimeter constraint; in higher dimensions it is strictly stronger. Any member of $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$6 is homeomorphic to the spherical shell, and the radii $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$7 are uniquely determined by the domain. Notable members include $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$8 for any convex inner hole (via the Alexandrov–Fenchel inequality), eccentric shells, and — in the plane — parallel neighborhoods $$\Omega = \Omega_{\mathrm{out}} \setminus \overline{\Omega_{\mathrm{in}}}$$9 of convex planar domains. In dimensions $\beta_1, \beta_2 \in (0, +\infty]$0, nontrivial examples beyond shells require Minkowski combinations as constructed following Amato–Gavitone–de Giovanni.

An important structural point is that the volume constraint is an inequality rather than an equality, and the paper shows this cannot be dropped.

Sharpness via counterexample

The authors demonstrate that the inequality reverses if the volume constraint is violated while the quermassintegral and perimeter constraints hold. Specifically, for $\beta_1, \beta_2 \in (0, +\infty]$1 with $\beta_1, \beta_2 \in (0, +\infty]$2 and $\beta_1, \beta_2 \in (0, +\infty]$3 chosen so that $\beta_1, \beta_2 \in (0, +\infty]$4, one has $\beta_1, \beta_2 \in (0, +\infty]$5. Since $\beta_1, \beta_2 \in (0, +\infty]$6, domain monotonicity gives

$\beta_1, \beta_2 \in (0, +\infty]$7

whereas the explicit Dirichlet spectrum of the parallelepiped yields $\beta_1, \beta_2 \in (0, +\infty]$8. Hence for large $\beta_1, \beta_2 \in (0, +\infty]$9,

$+\infty$0

the strict reverse of the Hersch–Weinberger inequality. Although $+\infty$1 is only Lipschitz, stability of Dirichlet eigenvalues allows regularization, and convergence of Robin eigenvalues to Dirichlet ones extends the counterexample to large finite Robin parameters. This confirms that Theorem's hypotheses are, in this precise sense, optimal.

Proof strategy: Morse approximation and gradient flow

The proof circumvents the regularity problems inherent in the effectless cut through two approximation procedures.

Approximation by Morse eigenfunctions. The key technical device is Proposition showing that the first eigenfunction $+\infty$2 can be approximated in $+\infty$3 by first eigenfunctions $+\infty$4 of perturbed problems $+\infty$5 with compactly supported potentials $+\infty$6 such that: $+\infty$7 exactly; each $+\infty$8 is a positive Morse function with no interior minimum points; and $+\infty$9. The construction perturbs $\lambda_1$0 by a linear term $\lambda_1$1 with generic $\lambda_1$2 (Sard's theorem ensures non-degeneracy of critical points), localizes via a cutoff, and defines $\lambda_1$3. This is necessary because Uhlenbeck-type genericity results are available for pure Dirichlet problems but not, prior to this work, for mixed Robin problems.

Gradient flow argument. For fixed $\lambda_1$4, the authors flow the boundary components backward under the gradient-descent system $\lambda_1$5, obtaining $\lambda_1$6 subdomains $\lambda_1$7 and $\lambda_1$8 analogously. A Green's identity comparison shows the strict inequality

$\lambda_1$9

using the positivity of the normal derivative of $\Omega \in \mathcal{K}_{\alpha,\beta}^N$0 on the flowed boundaries, which follows from transversality of orbits to the flowed hypersurfaces. Known reverse Faber–Krahn inequalities for the mixed Robin–Neumann and Neumann–Robin problems (due to Della Pietra–Piscitelli, Paoli–Piscitelli–Trani, Anoop–Ghosh, and others) then bound these eigenvalues by those of spherical shells $\Omega \in \mathcal{K}_{\alpha,\beta}^N$1 and $\Omega \in \mathcal{K}_{\alpha,\beta}^N$2. A measure-theoretic lemma, proved using the Stable Manifold Theorem, shows $\Omega \in \mathcal{K}_{\alpha,\beta}^N$3 as $\Omega \in \mathcal{K}_{\alpha,\beta}^N$4 and $\Omega \in \mathcal{K}_{\alpha,\beta}^N$5, with equality iff the volume constraint is tight. Combined with monotonicity of shell eigenvalues, this yields the desired bound up to errors $\Omega \in \mathcal{K}_{\alpha,\beta}^N$6, which vanish by choosing $\Omega \in \mathcal{K}_{\alpha,\beta}^N$7 large and then $\Omega \in \mathcal{K}_{\alpha,\beta}^N$8 very negative.

Notably, the Morse property enters only in proving the measure-theoretic lemma: for degenerate saddles, stable manifolds may have full dimension $\Omega \in \mathcal{K}_{\alpha,\beta}^N$9, so the argument breaks down. The authors state they believe the lemma holds even without the Morse assumption but lack a proof.

Topology of the effectless cut in higher dimensions

The second major contribution is a dynamical-systems study of the effectless cut itself. Defining, for a Morse–Smale function $\beta > \alpha > 0$0 on a domain diffeomorphic to $\beta > \alpha > 0$1, the sets $\beta > \alpha > 0$2 swept out by forward flow from the two boundary components, the authors define the effectless cut as $\beta > \alpha > 0$3 and prove:

• $\beta > \alpha > 0$4, where $\beta > \alpha > 0$5 collects critical points whose stable manifold closures contain both sources;
• $\beta > \alpha > 0$6 has topological dimension exactly $\beta > \alpha > 0$7;
• $\beta > \alpha > 0$8 is an attractor of the flow;
• $\beta > \alpha > 0$9 is closed, connected, and separates $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$0 into two connected subdomains abutting the respective boundary components.

In the plane, $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$1 is necessarily a simple closed curve homeomorphic to $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$2, recovering the classical picture. In dimension three, however, the authors construct an explicit Morse–Smale function (glued from a polynomial model $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$3 across two balls with four transverse heteroclinic intersections) for which $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$4 is not a two-dimensional manifold — it is not homeomorphic to $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$5. This substantiates the claim that naive generalizations of Hersch's splitting argument fail in higher dimensions, since the eigenvalue problem on subdomains bounded by such a wild set need not be well posed. The authors also strengthen their approximation result: the first eigenfunction can be approximated by first eigenfunctions of the perturbed problem that are not merely Morse but Morse–Smale.

Limitations and open questions

Several restrictions are acknowledged explicitly. First, for $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$6 the theorem requires convexity of both $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$7 and $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$8 together with the quermassintegral constraint; whether the quermassintegral constraint can be weakened to a perimeter constraint on $$\lambda_1^{\mathcal{RR}}(\Omega) \leq \lambda_1^{\mathcal{RR}}(B_\beta \setminus \overline{B_\alpha}),$$9 remains open, though numerical evidence for the corresponding stronger Dirichlet–Neumann inequality exists in the literature. Second, the extension to negative Robin parameters is open in higher dimensions; the inner parallel method used here fails when $(0, +\infty]$0 under only the perimeter constraint, whereas the planar case has been settled recently. Third, the measure-theoretic lemma relies on the Morse property, and its validity for degenerate critical points is conjectured but unproved. Finally, it is natural to ask whether eigenfunctions of the unperturbed problem are generically Morse–Smale under potential or domain perturbations; this question is left open.

Conclusion

The paper delivers a definitive higher-dimensional version of the Hersch–Weinberger inequality for doubly connected domains under quermassintegral, perimeter, and volume constraints, proves its sharpness by explicit counterexample, and develops an approximation-by-Morse-eigenfunctions technique that bypasses the irregularity of the effectless cut. The accompanying topological analysis shows that in dimension three the effectless cut can fail to be a manifold, explaining why the original planar method does not extend directly and validating the alternative approach presented here.