Asymptotic analysis of Dirichlet problems in supercritically perforated domains via quantification by torsional rigidity
Abstract
We study the Dirichlet problem −div(Aε∇ uε) = f on varying open sets Dε, with uniformly elliptic coefficients Aε, in the regime where the solutions vanish in the limit for every source. For periodically perforated domains, this is the supercritical regime. We show that the torsional rigidity T(Dε) is the natural normalization: up to a subsequence, uε/T(Dε) converges weakly as measures to kf for every continuous source f, where k is a nonzero finite measure independent of f, which we call the effective measure. The key ingredient is an estimate comparing uε with the product of f and the torsion function, whose rate depends on the domains only through T(Dε). We also show that this rate is optimal. No structural assumption on the perforation is needed, and the results hold for relaxed Dirichlet problems. Every nonzero finite measure arises as an effective measure, so the continuity of the source cannot be dropped in general. For rougher sources, we characterize the convergence by a condition on the domains alone, expressed through their torsion functions or, equivalently, their Poincaré constants. We also prove that the effective measure is local, satisfies optimal two-sided bounds and is monotone with respect to the coefficients, and we study the role of the antisymmetric parts of the coefficients.
Related
- A new localization approach for Willmore flows: Asymptotic analysis for the Dirichlet problem
- Asymptotic-Preserving Tensor Neural Networks for 3D High-Field Kinetic Equations with Applications to Uncertainty Quantification
- A sharp lower bound for torsional rigidity of convex domains
- Proof of the local version of the Pólya--Szegö conjecture for the torsional rigidity of polygons
- Effective Hamiltonians in Perforated Domains: Inf-Max formula and Stability
BibTeX
@misc{sato2026asymptotic,
title = {{Asymptotic analysis of Dirichlet problems in supercritically perforated domains via quantification by torsional rigidity}},
author = {Naoto Sato},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33621},
eprint = {2610.33621},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33621},
}
RIS
TY - UNPB
TI - Asymptotic analysis of Dirichlet problems in supercritically perforated domains via quantification by torsional rigidity
AU - Naoto Sato
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33621
UR - https://arxiv.org/abs/2610.33621
AB - We study the Dirichlet problem −div(A^ε∇ u^ε) = f on varying open sets D^ε, with uniformly elliptic coefficients A^ε, in the regime where the solutions vanish in the limit for every source. For periodically perforated domains, this is the supercritical regime. We show that the torsional rigidity T(D^ε) is the natural normalization: up to a subsequence, u^ε/T(D^ε) converges weakly as measures to kf for every continuous source f, where k is a nonzero finite measure independent of f, which we call the effective measure. The key ingredient is an estimate comparing u^ε with the product of f and the torsion function, whose rate depends on the domains only through T(D^ε). We also show that this rate is optimal. No structural assumption on the perforation is needed, and the results hold for relaxed Dirichlet problems. Every nonzero finite measure arises as an effective measure, so the continuity of the source cannot be dropped in general. For rougher sources, we characterize the convergence by a condition on the domains alone, expressed through their torsion functions or, equivalently, their Poincaré constants. We also prove that the effective measure is local, satisfies optimal two-sided bounds and is monotone with respect to the coefficients, and we study the role of the antisymmetric parts of the coefficients.
ER -