Recovering fractal boundary dimensions from Robin eigenvalues
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.
Related
- A spectral-element method for computing many eigenvalues and their asymptotics of the Schrödinger operator with Robin boundary condition
- Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes
- Optimal location of small favourable regions for Robin eigenvalues with indefinite weights
- Fractal dimensions and quasisymmetric packing-minimalities of some homogeneous Moran sets
- Generalized Discrete Fractal Dimensions of Graph Families
BibTeX
@misc{ammari2026recovering,
title = {{Recovering fractal boundary dimensions from Robin eigenvalues}},
author = {Habib Ammari and Yat Tin Chow and Fuqun Han},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34284},
eprint = {2610.34284},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34284},
}
RIS
TY - UNPB
TI - Recovering fractal boundary dimensions from Robin eigenvalues
AU - Habib Ammari
AU - Yat Tin Chow
AU - Fuqun Han
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34284
UR - https://arxiv.org/abs/2610.34284
AB - 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.
ER -