Effective Hamiltonians in Perforated Domains: Inf-Max formula and Stability
Abstract
We study the dependence of the effective Hamiltonian H̅Ω(p) on the geometry of periodic perforated domains for convex Hamilton--Jacobi equations with state-constraint boundary conditions. We establish an inf--max representation over smooth test functions for domains with C1 boundary. Using this representation, we derive a quantitative stability estimate in terms of the Hausdorff distance between domains and the reach of one boundary. This estimate yields convergence for non-nested domain approximations whose Hausdorff error is negligible relative to their boundary reach. Examples demonstrate the sharpness of the continuity modulus and show that Hausdorff convergence alone can fail to preserve the effective Hamiltonian, even for smooth domains and the quadratic Hamiltonian.
Related
- Bulk-surface spectral homogenization with indefinite mass in periodic perforated domains
- Surface-to-Volume Homogenization of NonLinear Steklov Eigencouples with NonHomogeneous Orlicz Growth in Perforated Domains
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- Critical homogenization of the Navier--Stokes--Cahn--Hilliard system in perforated domains
BibTeX
@misc{han2026effective,
title = {{Effective Hamiltonians in Perforated Domains: Inf-Max formula and Stability}},
author = {Yuxi Han and Son N. T. Tu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34910},
eprint = {2610.34910},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34910},
}
RIS
TY - UNPB
TI - Effective Hamiltonians in Perforated Domains: Inf-Max formula and Stability
AU - Yuxi Han
AU - Son N. T. Tu
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34910
UR - https://arxiv.org/abs/2610.34910
AB - We study the dependence of the effective Hamiltonian H̅_Ω(p) on the geometry of periodic perforated domains for convex Hamilton--Jacobi equations with state-constraint boundary conditions. We establish an inf--max representation over smooth test functions for domains with C^1 boundary. Using this representation, we derive a quantitative stability estimate in terms of the Hausdorff distance between domains and the reach of one boundary. This estimate yields convergence for non-nested domain approximations whose Hausdorff error is negligible relative to their boundary reach. Examples demonstrate the sharpness of the continuity modulus and show that Hausdorff convergence alone can fail to preserve the effective Hamiltonian, even for smooth domains and the quadratic Hamiltonian.
ER -