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Recovering fractal boundary dimensions from Robin eigenvalues

Habib Ammari, Yat Tin Chow, Fuqun Han

arXiv:2610.34284v1math.NAcs.NA

TL;DR

The paper asks how Robin spectra behave when a small local coefficient competes with the growing measure of a rough prefractal boundary. It combines fixed-domain expansions with depth-uniform estimates based on total Robin boundary mass. Under boundary-resolution and scale assumptions, ground-state measurements recover lower and upper Minkowski dimensions.

  • Problem

    The paper asks how to analyze Robin spectra when shrinking local coefficients interact with growing prefractal boundary measure, since fixed-domain remainders need not be uniform in depth.

    Source: §1 · pp. 1–2

  • Method

    It combines fixed-domain layer-potential expansions with normalized-trace estimates in total Robin mass, then applies boundary-resolution and scale-spacing assumptions to ground-state measurements.

    Source: §1 · pp. 2–3

  • Results

    Under boundary resolution, logarithmic scale spacing, and bounded total Robin boundary mass, ground-state measurements recover the lower and upper Minkowski dimensions.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The product τ_m = ε_m|Γ_m| links the local Robin coefficient to the spectral effect of an increasingly rough boundary.

    Source: §5 · p. 21

    Dimension reconstruction requires known Robin coefficients; with unknown coefficients, leading measurements determine only total Robin boundary mass.

    Source: §1 · p. 3

Abstract

from arXiv · show

We study Robin problems in which a small local boundary coefficient competes with the increasing surface measure of a finite-depth fractal approximation. For a fixed smooth domain, layer potentials give Green function and eigenpair expansions; contour residues and a finite-rank reduction yield a quantitative spectral-gap-dependent eigenvalue remainder. For varying Lipschitz domains, a uniform Poincar\'e inequality and a trace estimate with respect to normalized surface measure give a uniform ground-state expansion and a rank-one resolvent singularity as the total Robin boundary mass tends to zero. Under the boundary-resolution hypothesis and the condition on the approximation scales, ground-state measurements with known Robin coefficients recover the lower and upper Minkowski dimensions for sequences with bounded total Robin boundary mass. We describe Neumann, generalized Robin, and Dirichlet-type operator limits under the corresponding compact Mosco convergence hypotheses. Koch snowflake computations illustrate the reconstruction formulas, and an exact finite-stage area correction improves the adjacent-stage dimension estimator.

1 Motivation and main results

The paper studies how increasingly rough prefractal boundaries interact with a small Robin coefficient, whose accumulated effect is measured by total Robin boundary mass. It develops depth-uniform spectral estimates and uses ground-state measurements to recover boundary dimensions under stated scale and coefficient assumptions.

1 Motivation and main results

  • The corrosion motivation is idealized by finite-depth prefractal boundaries and a scalar local-exchange Robin coefficient, without asserting exact self-similarity for every corroded surface.The paper also cautions that electrochemical response is not determined by fractal dimension alone.Source: §1 · p. 1
  • For a fixed smooth domain, layer-potential formulas and contour residues produce Green-function and eigenpair expansions with a spectral-gap-dependent remainder.The analysis also treats multiple eigenvalues and the zero mode, and proves convergence of the coefficient series.Source: §1 · p. 2
  • A normalized Poincaré-trace estimate yields depth-uniform eigenvalue bounds and a first-order ground-state expansion as τ_m tends to zero.The corresponding resolvent estimate identifies the same constant-mode singularity uniformly in prefractal depth.Source: §1 · p. 2
  • Under boundary resolution, logarithmic scale spacing, and bounded total Robin boundary mass, ground-state measurements recover the lower and upper Minkowski dimensions.The reconstruction uses the logarithmic growth of Λ_m(ε_m)/ε_m; standard self-similar constructions provide examples of boundary resolution.Source: §1 · pp. 2–3
  • If Robin coefficients are unknown, the leading measurements determine total Robin boundary mass rather than separating the coefficient from boundary measure.The reconstruction does not establish uniqueness for joint recovery of an arbitrary shape and unknown coefficient from a single eigenvalue.Source: §1 · p. 3
  • Under compact Mosco convergence hypotheses, τ_m tending to zero gives a Neumann limit, a finite positive limit gives generalized Robin, and τ_m tending to infinity gives a Dirichlet-type limit.The Dirichlet-type regime also requires the stated zero-trace and recovery conditions.Source: §1 · p. 3

2 Fixed smooth domain

For a fixed smooth domain, the paper develops small-Robin-coefficient Green-function and eigenpair expansions, then gives spectral-gap-dependent control of their remainders. The Green-function resolvent’s singular part is rank one and comes from perturbing the Neumann zero mode.

2.1 Green function

  • The Green-function construction subtracts the constant singular term before solving for the harmonic correction on a fixed C1,α domain.Source: §2.1 · pp. 3–4

2.2 Eigenvalue problem

  • Robin eigenvalues and eigenfunctions admit small-ε expansions for fixed modes, with an eigenspace matrix treating multiple eigenvalues.Source: §2.2 · p. 7

2.3 Uniform high frequency asymptotics

  • At high frequency, εl^(2/d)/γ_l→0 is the smallness condition up to domain-dependent constants, and it does not ensure uniformity in prefractal depth.Fixed-index remainder constants can depend on the mode, so quantitative Taylor control requires spectral separation.Source: §2.3 · pp. 7, 9

2.4 Spectral decomposition of the Green function

  • The Green-function resolvent’s singular part is rank one, generated by perturbation of the Neumann zero mode.Source: §2.4 · p. 9

3 Finite-depth prefractal boundary

This section examines finite-depth prefractal boundary problems, relating Robin ground-state measurements to lower and upper Minkowski dimensions under suitable geometric, scale, and mass assumptions and classifying power-law couplings. It also describes Neumann, generalized Robin, or zero-trace operator limits under compact Mosco convergence, including convergence of zero-extended resolvents and fixed-index eigenvalues.

3.1 Uniform estimates and bounded-mass recovery

  • Uniform-domain geometry and normalized boundary-measure growth provide trace and Poincaré estimates, yielding a uniform ground-state expansion with eigenvalue τ/|Ωm| + O(τ^2).Here τ = ε|Γm|, and the estimates hold uniformly over approximation stages for sufficiently small τ.Source: §3.1 · p. 12
  • For subcritical sequences, the signed Green operator has leading term −(|Ωm|/τm)P0,m, a rank-one singularity associated with projection onto constants.The result holds uniformly for sufficiently small total Robin boundary mass under (R1)–(R2).Source: §3.1 · p. 13
  • Under (R1)–(R3), the scale condition (3.4), and bounded total Robin mass, ground-state measurements recover the lower and upper Minkowski dimensions; subcritical couplings yield a stronger recovery form.When the Minkowski dimension exists, the lower and upper limits coincide; condition (3.4) ensures the approximation scales recover dimensions defined over all sufficiently small scales.Source: §3.1 · pp. 10, 13
  • At a fixed eigenvalue level below the uniform bound, there is a unique Robin coefficient attaining that level, and the corresponding total boundary masses remain bounded above and away from zero.The result does not assert convergence of those masses to a critical value.Source: §3.1 · p. 14
  • For power-law couplings, distinct lower and upper Minkowski dimensions can produce an intermediate regime with subsequences of total mass tending to zero and infinity; nondegeneracy reduces the transition to ac = 1/θ.At a = ac, masses stay bounded above and away from zero, but strict criticality requires δm^θ|Γm| to converge to a positive finite value.Source: §3.1 · p. 14
  • For the standard Koch snowflake approximations, (R1)–(R3) and the scale condition hold with s = 1, while the limiting boundary has dimension log 4/ log 3.Thus the dimension-recovery theorem applies to this family.Source: §3.1 · p. 14

3.2 Operator interpretation of the three regimes

  • The operator-limit discussion assumes nested inner approximations and additional measure convergence, since total Robin boundary mass alone does not determine a limiting operator.These additional hypotheses are not needed for the dimension-recovery results.Source: §3.2 · pp. 15–16
  • The assumed Mosco convergence requires a lower-bound condition and recovery sequences; uniform extension bounds and nested exhaustion give compactness of the zero extensions.Bounded norm and energy imply relative compactness.Source: §3.2 · p. 16
  • For a specific approximation family, verifying the boundary step uses (R2) for finite limiting mass; zero-trace recovery follows from ker T = H1_0(Ω), otherwise it remains to be checked.Under (R2), a fixed H1 extension supplies recovery for finite limiting mass; the zero-trace recovery construction uses compactly supported smooth approximants inside Ωm.Source: §3.2 · pp. 16–17
  • Under the variational hypotheses, vanishing total Robin mass gives the Neumann Laplacian, while finite limiting mass gives the generalized Robin Laplacian for τ* bσ.These operator interpretations require the stated Mosco-convergence hypotheses.Source: §3.2 · p. 17
  • When total Robin mass diverges, the limit is the operator of the zero-trace form, and it is the usual Dirichlet Laplacian precisely when ker T = H1_0(Ω).Mass divergence alone does not determine the limit.Source: §3.2 · p. 17
  • Compact Mosco convergence gives operator-norm convergence of zero-extended resolvents for every α > 0 and convergence of eigenvalues at each fixed index, counting multiplicities.The passage also points to [25, Theorem 6.1].Source: §3.2 · p. 17

4 Applications and numerical illustrations

The applications analyze self-similar boundary scaling and use Koch snowflake stages to illustrate dimension recovery, finite-element estimates, and the effects of correction and mesh refinement.

4.1 Scaling for self-similar boundaries

  • For equal-ratio self-similar boundaries with n pieces and ratio r, the boundary measure scales by ρ = nr^(d−1), with D = −log n/log r.At a stage where ε|Γ| is comparable to one, the generation index is related to the local coefficient and the scaling parameters up to O(1).Source: §4.1 · p. 17
  • For |Γδ| ∼ CΓδ^-θ and δ ∼ ε^a, the critical exponent is a_c = θ^-1; at equality the total Robin mass has a finite nonzero limit.With only two-sided comparability of surface growth, endpoint masses are bounded above and away from zero.Source: §4.1 · p. 18

4.2 Numerical illustrations

  • The Koch illustration uses unit-side stages m = 0,…,6, whose exact area varies by generation and enters adjacent-stage estimates.The finite-element mesh resolves each fixed polygon separately from its geometric prefractal scale.Source: §4.2 · p. 18, §4.2.1 · p. 18

4.2.1 Finite-element formulation

  • Conforming piecewise-linear discretization on the exact stage polygon gives 0 ≤ Λ_m,h(ε) − Λ_m(ε) ≤ τ|Ω_m| − Λ_m(ε) ≤ Cτ^2 for small τ.The eigenvalue bounds presuppose the exact polygon and exact bilinear forms.Source: §4.2.1 · pp. 18–19

4.2.2 Dimension recovery and the finite-stage area correction

  • For the Koch family, an adjacent-stage estimator separates area variation from the small-mass remainder, and the area-corrected estimator removes the area contribution without another eigenvalue solve.The exact area change matters for finite-stage estimates even though it does not affect the logarithmic limit.Source: §4.2 · p. 18, §4.2.2 · p. 19
  • The area-corrected generation-six estimate is 1.261908051, reducing discrepancy from D to about 4.85 × 10^-5 from 3.26 × 10^-3 before correction.The exact finite-stage area correction uses the original rounded data and adds no eigenvalue solve.Source: §4.2.2 · p. 19

4.2.3 Mesh refinement

  • The finest mesh refinement changes the eigenvalue by at most 7.03 × 10^-7, while the generation-five and generation-six constant-test upper bounds are 0.006802429150 and 0.004674698200.The finest computed eigenvalues lie below those bounds by relative amounts 1.75 × 10^-4 and 1.20 × 10^-4, respectively.Source: §4.2.3 · p. 20, Table 1 · p. 21

4.2.4 Interpretation and error analysis of the numerical results

  • At exact criticality τ_m = 1 at every stage; subcritical eigenvalues tend to zero, while critical and supercritical operator limits require the corresponding Section 3.2 hypotheses.The latter limits are generalized Robin and Dirichlet-type, respectively.Source: §4.2.4 · p. 21

5 Concluding remarks

The conclusion connects total Robin boundary mass to spectral effects and dimension recovery, while distinguishing these results from the stronger assumptions needed for operator limits and joint recovery.

5 Concluding remarks

  • Under boundary-resolution and approximation-scale conditions, ground-state measurements with known Robin coefficients recover lower and upper Minkowski dimensions for sequences with bounded total Robin mass.A uniform Poincaré-trace estimate also gives a uniform first-order ground-state expansion and a rank-one resolvent singularity as the mass tends to zero.Source: §5 · p. 21
  • Operator limits require measure-convergence and compact Mosco hypotheses; the Dirichlet-type limit additionally requires zero-trace and recovery conditions.Vanishing mass gives a Neumann limit, finite positive limiting mass a generalized Robin limit, and diverging mass a Dirichlet-type limit under the stated conditions.Source: §5 · p. 21
  • The exact Koch area formula corrects adjacent-stage estimates, while extending high-frequency estimates uniformly across prefractal families and separating geometry from unknown Robin coefficients remain open questions.The fixed-domain eigenpair remainder is controlled by the spectral gap.Source: §5 · p. 21

Statements and declarations

The authors report funding, data and code availability, and their use of generative AI during manuscript preparation.

Statements and declarations

A Proofs of the asymptotic formulas

The appendix proves fixed-domain Robin eigenpair formulas using layer potentials and spectral methods, with scope and boundary regularity conditions specified for the arguments.

A Proofs of the asymptotic formulas

  • The fixed-mode expansions use layer potentials and the Gohberg–Sigal method; eigenvalue coefficients are contour residues and eigenfunction corrections come from a finite-rank factorization.The appendix also treats the zero mode, multiple eigenvalues, and spectral-gap dependence of the remainder.Source: A Proofs of the asymptotic formulas · p. 22
  • The free-space layer-potential characterization assumes a bounded smooth connected domain with connected exterior, and asserts no constants uniform over prefractal approximations.The transfer-operator argument also applies to smooth connected domains with holes.Source: A Proofs of the asymptotic formulas · p. 22
  • The spectral transfer-operator argument proves fixed-mode formulas under the C1,α boundary assumption, whereas the sharp boundary restriction estimate requires C∞ regularity.Source: A Proofs of the asymptotic formulas · p. 22

A.1 Robin eigenvalue characterization

For a connected exterior, Robin eigenvalues correspond to positive real characteristic values of the boundary operator A, with the boundary trace as the unknown. On smooth boundaries A is Fredholm of index zero and locally holomorphic near positive frequencies.

A.1 Robin eigenvalue characterization

  • A Robin eigenvalue ω² corresponds exactly to a positive real characteristic value of A(·, ε), with boundary unknown ϕ = u|∂Ω.The correspondence is stated for connected ℝᵈ \ Ω and real ε.Source: §A.1 · p. 23
  • A nonzero Robin eigenfunction must have nonzero boundary trace, and a nonzero null vector of A reconstructs a nonzero interior Robin field.The converse uses exterior Dirichlet uniqueness and the radiation condition, together with the layer-potential trace jumps.Source: §A.1 · p. 23
  • For smooth ∂Ω, compactness of the boundary layer operators makes A Fredholm of index zero and holomorphic near each positive frequency.In dimension two, this local holomorphy assertion excludes the logarithmic branch at ω = 0.Source: §A.1 · p. 23
  • Bounded components of the complement can add characteristic values of the free-space operator that are not Robin eigenvalues in Ω.The boundary Neumann transfer operator is intrinsic to Ω, and its regularized block pencil avoids these exterior artifacts.Source: §A.1 · p. 23

A.2 Gohberg–Sigal contour formula and normalization of the pole

A contour argument isolates a simple characteristic value and yields an expansion whose residues are finite rank. The pole normalization and the change from frequency to eigenvalue fix the coefficients of that expansion.

A.2 Gohberg–Sigal contour formula and normalization of the pole

  • A small disk around the simple Neumann characteristic value, containing no other characteristic value, permits generalized Rouché and argument-principle counting of nearby values with multiplicity.In two dimensions the disk also avoids the logarithmic cut.Source: §A.2 · p. 23
  • The contour expression gives holomorphic dependence on ε, and the nearby eigenvalue is real for real ε.Source: §A.2 · p. 24
  • The pole denominator is nonzero by simplicity, and the adjoint in the pole formula is the Hilbert-space adjoint of A₀(ω_l⁰).The pole expansion converges in operator norm on ∂V.Source: §A.2 · p. 24
  • The normalized pole expansion yields the AKL expansion through cyclicity of contour-residue traces and integration by parts.Its first-order frequency term is distinguished from the eigenvalue coefficient by F(ω) = ω² and F′(ω) = 2ω.Source: §A.2 · p. 24
  • Taking the trace after contour integration is valid because the integral is a finite-rank residue, although the operator-valued integrand need not be trace class.The spectral representation is used to evaluate the residues.Source: §A.2 · p. 24

A.3 The Neumann-to-Dirichlet operator and its spectral residues

The boundary Neumann-to-Dirichlet operator and its interior potential have spectral representations with operator-norm convergence. Their pole separation and residue weights support the expansions used in the analysis.

A.3 The Neumann-to-Dirichlet operator and its spectral residues

  • The rank-one operator Π_l = f_l ⊗ f_l is generally not a projection: Π_l² = a_lΠ_l and tr Π_l = a_l.Source: §A.3 · p. 24
  • The boundary operator N is the Neumann-to-Dirichlet map, while P is its interior Neumann potential.Source: §A.3 · p. 24
  • The assembled transfer operators N and P are single-valued meromorphic functions of z, including near z = 0, unlike each individual outgoing layer there.Source: §A.3 · p. 25
  • Away from the Neumann spectrum, finite spectral truncations converge in operator norm for the boundary series and in the Neumann spectral energy norm for the H¹(Ω) target series.The argument uses compactness of T or T* and boundedness of the diagonal multiplier on compact spectral-complement subsets.Source: §A.3 · p. 25
  • Near a fixed simple Neumann eigenvalue, the pole is separated from holomorphic regular parts, and the residue weights ensure absolute convergence of B_l and H¹ convergence of the eigenfunction correction.For fixed l, each weight is bounded by a constant depending on l times (1 + λ_j⁰)^−1.Source: §A.3 · p. 25

A.4 First two terms in the asymptotic expansions of the eigenvalues

The first two contour residues give the stated second-order eigenvalue expansion, while converting back to frequency produces a distinct second coefficient. The double-pole term contributes no contour integral.

A.4 First two terms in the asymptotic expansions of the eigenvalues

  • The frequency calculation changes variables by z = ω² before evaluating the contour contributions at z = λ_l⁰.Source: §A.4 · p. 25
  • The double pole has zero contour integral, so the first two residues determine the second-order expansion.The passage states that substitution into the preceding residue expression yields the result.Source: §A.4 · p. 26
  • The first two AKL contour residues prove (2.22), and expanding the square root gives the corresponding frequency expansion.Source: §A.4 · p. 26
  • The second coefficients for the eigenvalue and frequency expansions are different and must not be identified.Source: §A.4 · p. 26

A.5 Finite-rank factorization and the eigenfunction expansion

A finite-rank reduction turns the layer-potential problem into a scalar characteristic equation for a simple mode and supports reconstruction of its eigenfunction. Separate treatments handle the zero mode and multiplicities, while the Taylor remainder is controlled in terms of the spectral gap.

A.5 Finite-rank factorization and the eigenfunction expansion

  • A finite-rank reduction gives a scalar characteristic equation whose unique holomorphic zero is the perturbed eigenvalue for a simple mode.The reduction uses holomorphic invertible factors to obtain diag(Q_l, s_l), and Rouché’s theorem gives the unique zero for small ε.Source: §A.5 · p. 26
  • Normalizing the reconstructed eigenfunction changes its expansion only at order ε², while the scalar remainder check gives an O_l(ε³) remainder.The eigenfunction correction is orthogonal to the unperturbed mode.Source: §A.5 · p. 27

A.6 The zero Neumann eigenvalue

  • The zero Neumann mode has frequency of order √ε, so pole subtraction yields a holomorphic block pencil at (z, ε) and avoids a zero-frequency branch ambiguity.The method does not require A(√z, ε) itself to be holomorphic at z = 0.Source: §A.6 · p. 27

A.7 Splitting of a multiple eigenvalue

  • A multiple eigenvalue splits as λ_l,i^ε = λ_l^0 + ε μ_i + O(ε²), where the μ_i are eigenvalues of the effective Hermitian matrix C.The simple-mode eigenfunction formula cannot be applied to an arbitrary vector in the multiple eigenspace; unresolved multiplicities require the next effective matrix.Source: §A.7 · p. 28

A.8 A quantitative remainder controlled by the spectral gap

  • For each simple Neumann mode, the Taylor remainder is controlled on a mode-dependent disk ρ_l = c/H_l, with constants depending on the fixed domain but not l.The gap-dependent estimate controls the full analytic remainder, not only its second coefficient.Source: §A.8 · pp. 28–29
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