Comparison inequalities for Dirichlet-to-Neumann maps

Abstract: We prove comparison inequalities for Dirichlet-to-Neumann maps corresponding to different non-positive Helmholtz parameters. For convex domains our bounds are sharp, and the resulting eigenvalue inequalities partially confirm an earlier conjecture, which we show does not hold in full generality. We further obtain geometry-dependent versions for arbitrary sufficiently regular domains, together with extensions to compact Riemannian manifolds with boundary. We also discuss analogous questions for metric graphs.

• The paper proves the sharp bound 0≤σ_k^(Λ₁)−σ_k^(Λ₂)≤√(Λ₂−Λ₁) for every eigenvalue on convex domains, using an exponential distance-function substitution.
• For smooth non-convex domains, explicit curvature- and torsion-dependent estimates provide Hölder control, while weak mean-convexity preserves the sharp leading constant in dimensions at least three.
• The paper disproves broad conjectures with annular and Swiss-cheese counterexamples and extends the framework to Robin eigenvalues, Riemannian manifolds, and metric graphs.

Setting and motivation

For a bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$ and a Helmholtz parameter $\Lambda\le 0$, the Dirichlet-to-Neumann map $\mathcal{D}_\Lambda$ sends boundary data $u\in H^{1/2}(\partial\Omega)$ to the normal derivative of its $\Lambda$-harmonic extension, with quadratic form

$$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$

Its eigenvalues $\sigma_k^{(\Lambda)}$ are monotone decreasing in $\Lambda$ along analytic branches. The paper (2608.18678) addresses a conjecture from an earlier arXiv version of Grebenkov–Levitin–Polterovich (Grebenkov et al., 13 Apr 2026), asserting that for all $\Lambda\le 0$ and every real-analytic branch (Conjecture A), or at least for every ordered eigenvalue (Conjecture B),

$$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}.$$

The conjectures were known to hold for balls and disks, for the principal eigenvalue on arbitrary domains, and asymptotically in the leading term as $\Lambda\le 0$0; Girouard–Karpukhin–Levitin–Polterovich had also established a uniform $\Lambda\le 0$1 comparison between $\Lambda\le 0$2 and $\Lambda\le 0$3, where $\Lambda\le 0$4 are Laplace–Beltrami eigenvalues of the boundary.

The main comparison theorem for convex domains

The central result is a quadratic-form inequality: if $\Lambda\le 0$5 is bounded and convex and $\Lambda\le 0$6, then

$\Lambda\le 0$7

with the sharp H\"older constant one. By min–max this yields $\Lambda\le 0$8 for every $\Lambda\le 0$9, confirming Conjecture B — and hence Conjecture A along branches that respect ordering — for convex domains. Sharpness is immediate: for a ball, $\mathcal{D}_\Lambda$0 as $\mathcal{D}_\Lambda$1, so no smaller constant can hold uniformly.

The proof rests on a single substitution idea. Writing $\mathcal{D}_\Lambda$2 and testing against $\mathcal{D}_\Lambda$3 in the Dirichlet principle for $\mathcal{D}_\Lambda$4, integration by parts reduces everything to a distributional lower bound $\mathcal{D}_\Lambda$5. For convex domains $\mathcal{D}_\Lambda$6 is concave, so $\mathcal{D}_\Lambda$7, and choosing $\mathcal{D}_\Lambda$8 annihilates the residual term exactly. This isolates the true analytic input: superharmonicity of the distance function, not convexity per se.

Geometry-dependent bounds for general smooth domains

For $\mathcal{D}_\Lambda$9 boundaries two complementary estimates hold. First, with $u\in H^{1/2}(\partial\Omega)$0 (the negative part of mean curvature) and $u\in H^{1/2}(\partial\Omega)$1,

$u\in H^{1/2}(\partial\Omega)$2

The constant $u\in H^{1/2}(\partial\Omega)$3 is controlled explicitly via the torsion function $u\in H^{1/2}(\partial\Omega)$4: one has $u\in H^{1/2}(\partial\Omega)$5, with both inequalities equalities for a ball. Notably, $u\in H^{1/2}(\partial\Omega)$6 coincides with the minimal tension parameter in the method of particular solutions, connecting the estimate to numerical spectral inclusion theory.

Second, choosing instead $u\in H^{1/2}(\partial\Omega)$7 as the non-negative root of $u\in H^{1/2}(\partial\Omega)$8 gives

$u\in H^{1/2}(\partial\Omega)$9

Neither bound implies the other: the first is stronger near the diagonal and provides genuine $\Lambda$0-H\"older continuity in $\Lambda$1; the second is stronger away from it and retains the sharp leading constant one, since $\Lambda$2. The additive defect $\Lambda$3 is optimal, as shown by annuli where the upper radial branch satisfies $\Lambda$4 in the iterated limit. In dimension $\Lambda$5, weak mean-convexity ($\Lambda$6, strictly weaker than convexity) suffices to recover the sharp constant-one inequality. Both results extend to $\Lambda$7 boundaries with $\Lambda$8 replacing $\Lambda$9.

Counterexamples to the conjectures

The sharp result does not extend to arbitrary domains, and both conjectures fail without geometric hypotheses:

Annuli. Let $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$0 be the root of $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$1. For any planar annulus $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$2 with $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$3, the upper radial eigenvalue branch satisfies $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$4 as $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$5, contradicting Conjecture A for all sufficiently negative $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$6. Numerically, $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$7 already violates the conjecture for $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$8.

Swiss cheese domains. Removing $$\mathfrak{d}_\Lambda[u]=\|\nabla \mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}-\Lambda\|\mathcal{E}_\Lambda u\|^2_{L^2(\Omega)}.$$9 tiny disks of radius $\sigma_k^{(\Lambda)}$0 from the unit disk produces smooth counterexamples to Conjecture B itself: for small $\sigma_k^{(\Lambda)}$1 there exists $\sigma_k^{(\Lambda)}$2 with $\sigma_k^{(\Lambda)}$3. The proof combines Glazman's lemma with Neumann bracketing: logarithmic cutoff functions localized near each hole give $\sigma_k^{(\Lambda)}$4 Robin eigenvalues below zero at parameter $\sigma_k^{(\Lambda)}$5, while bracketing shows fewer than $\sigma_k^{(\Lambda)}$6 eigenvalues below $\sigma_k^{(\Lambda)}$7 at parameter $\sigma_k^{(\Lambda)}$8, with $\sigma_k^{(\Lambda)}$9. Analogous three-dimensional examples exist, even with connected boundary; whether the conjectures hold for simply connected non-convex planar domains remains open.

Extensions

Robin eigenvalues. Via the Robin–Dirichlet-to-Neumann duality ($\Lambda$0 iff $\Lambda$1), the comparison inequalities translate into lower bounds on Robin eigenvalue gaps. For convex or weakly mean-convex $\Lambda$2 domains, whenever $\Lambda$3,

$\Lambda$4

with coefficient one again sharp by the known asymptotics $\Lambda$5.

Riemannian manifolds. For compact manifolds with smooth boundary, the same argument yields $\Lambda$6, where $\Lambda$7 quantifies the failure of $\Lambda$8. Non-negative Ricci curvature plus weak mean convexity of the boundary forces $\Lambda$9 (via Kasue's Laplacian comparison theorem), recovering the sharp constant-one inequality. The framework makes clear that these curvature assumptions are merely convenient sufficient conditions; any hypothesis implying distributional superharmonicity of the distance function would serve equally.

Metric graphs. On compact metric graphs with Dirichlet leaves, the analogue holds whenever the vertex sums $\Lambda\le 0$0 are non-positive at every Kirchhoff vertex — a discrete mean-convexity condition covering subdivided intervals and equilateral rooted trees with leaves at equal depth. The condition can fail even for trees: a three-leaf star with edge lengths $\Lambda\le 0$1 violates both graph conjectures. Without degree assumptions one obtains the general bound with additive defect $\Lambda\le 0$2.

Limitations and open questions

Several caveats are stated plainly in the paper. The regularity threshold is genuine: for "wiggly" domains bounded by curves $\Lambda\le 0$3, which converge to the disk in Hausdorff distance but have curvature oscillating with amplitude $\Lambda\le 0$4, the constants $\Lambda\le 0$5 and $\Lambda\le 0$6 blow up like $\Lambda\le 0$7, so the estimates become vacuous below $\Lambda\le 0$8 regularity — although these domains are not themselves counterexamples to the conjectures, since their fixed-$\Lambda\le 0$9 Steklov spectra converge to those of the disk. The counterexamples leave open whether Conjectures A and B hold for simply connected non-convex planar domains, and the metric-graph analysis identifies superharmonicity of the boundary distance only as a sufficient, not necessary, condition. Finally, the Riemannian sharp result depends on the joint assumption $$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}.$$0 and $$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}.$$1; the paper does not determine the minimal geometric hypothesis yielding $$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}.$$2.

Conclusion

This paper resolves the status of the $$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}.$$3 comparison conjecture for Dirichlet-to-Neumann maps: the inequality holds with sharp constant for convex domains, survives under the weaker condition of weak mean convexity in dimensions $$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}.$$4, degrades gracefully through explicit geometry-dependent constants for $$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}.$$5 (indeed $$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}.$$6) boundaries, and fails outright for general domains, with explicit annular and Swiss cheese counterexamples. The unifying mechanism — exponential substitution in the distance function governed by the distributional sign of $$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}.$$7 — transfers verbatim to Riemannian manifolds and, in modified form, to metric graphs, and yields new sharp lower bounds on Robin eigenvalue gaps as a by-product of the duality between the two problems.