The α-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem
Abstract
We study the singular limit of a family of linear degenerate interface transmission problems arising from a regularization procedure in the newly proposed Two-Parameter Diffuse Domain Method (DDM2p). For α>0, the regularized problem admits a strictly convex variational formulation on H1(Ω). In the limit α→0, the problem degenerates to a weakly coupled interface system with a nonstandard energy structure. To characterize the limit, we introduce a closed Hilbert subspace ℋ⊂ H1(Ω), defined through an auxiliary Helmholtz problem on an annular subdomain Ω2⊂ Ω, and identify the limiting energy functional ℰ0 on ℋ. We prove that the regularized energies ℰα Γ-converge to ℰ0 in the strong L2(Ω) topology, using the standard framework. Consequently, minimizers of ℰα converge to the unique minimizer of ℰ0, which is shown to be equivalent to the solution of the limiting interface problem. We further prove strong convergence uα→ u0 in H1(Ω) and establish an O(α) convergence rate. Numerical experiments in one spatial dimension confirm the predicted first-order convergence rate and suggest that this rate is sharp.
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BibTeX
@misc{luong2026αlimit,
title = {{The $α$-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem}},
author = {Toai Luong and Tadele Mengesha and Kerrek Stinson and Steven M. Wise and Ming Hei Wong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01237},
eprint = {2609.01237},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01237},
}
RIS
TY - UNPB
TI - The α-Limit Problem: Convergence of a Linear Degenerate Interface Transmission Problem
AU - Toai Luong
AU - Tadele Mengesha
AU - Kerrek Stinson
AU - Steven M. Wise
AU - Ming Hei Wong
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01237
UR - https://arxiv.org/abs/2609.01237
AB - We study the singular limit of a family of linear degenerate interface transmission problems arising from a regularization procedure in the newly proposed Two-Parameter Diffuse Domain Method (DDM2p). For α>0, the regularized problem admits a strictly convex variational formulation on H^1(Ω). In the limit α→0, the problem degenerates to a weakly coupled interface system with a nonstandard energy structure. To characterize the limit, we introduce a closed Hilbert subspace ℋ⊂ H^1(Ω), defined through an auxiliary Helmholtz problem on an annular subdomain Ω_2⊂ Ω, and identify the limiting energy functional ℰ_0 on ℋ. We prove that the regularized energies ℰ_α Γ-converge to ℰ_0 in the strong L^2(Ω) topology, using the standard framework. Consequently, minimizers of ℰ_α converge to the unique minimizer of ℰ_0, which is shown to be equivalent to the solution of the limiting interface problem. We further prove strong convergence u_α→ u_0 in H^1(Ω) and establish an O(α) convergence rate. Numerical experiments in one spatial dimension confirm the predicted first-order convergence rate and suggest that this rate is sharp.
ER -