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Recovering fractal boundary dimensions from Robin eigenvalues

by H. Ammari and Y.T. Chow and F. Han

(Report number 2026-43)

Abstract
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 Poincaré 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.

Keywords: Robin Laplacian; prefractal boundary; normalized boundary measure; Green function; spectral perturbation; Minkowski dimension; inverse problem.

@Techreport{ACH26_1198,
  author = {H. Ammari and Y.T. Chow and F. Han},
  title = {Recovering fractal boundary dimensions from Robin eigenvalues},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2026-43},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2026/2026-43.pdf },
  year = {2026}
}

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