Asymptotic analysis of transmission problems with parameter-dependent Robin conditions

TL;DR

This work analyzes a two-domain diffusion model coupled by Robin-type transmission across a fixed interface $S$, with permeability $α$ controlling inter-domain exchange. Using a subdifferential framework and the energy functional $φ_α$, it establishes well-posedness for $α>0$ and derives rigorous asymptotic limits as $α\to 0$ (decoupled domains) and $α\to +∞$ (single-structure), including convergence rates under natural regularity assumptions. The results connect the limit processes to Mosco convergence by showing $φ_α$ converges to $φ_0$ and $φ_∞$ in the Mosco sense, corresponding to the two extreme transmission regimes. The analysis is extended to non-autonomous permeability $α(t)$, including scenarios where $α$ blows up in finite time and the system continues in the single-domain regime beyond the blow-up. Collectively, the findings illuminate how a single parameter can unify distinct biological and mathematical mechanisms, with broader implications for dynamic boundary conditions and related diffusion models such as the GMS, LW, and KLLM frameworks.

Abstract

We study a transmission problem of {N}eumann--{R}obin type involving a parameter $α$ and perform an asymptotic analysis with respect to $α$. The limits $α\to 0$ and $α\to +\infty$ correspond respectively to complete decoupling and full unification of the problem, and we obtain rates of convergence for both regimes. Biologically, the model describes two cells connected by a gap junction with permeability $α$: the case $α\to 0$ corresponds to a situation where the gap junction is closed, leaving only tight junctions between the cells so that no substance exchange occurs, while $α\to +\infty$ corresponds to a situation that can be interpreted as the cells forming a single structure. We also clarify the relationship between the asymptotic analysis with respect to the parameter $α$ and the asymptotics of the system in connection with the convergence of convex functionals known as {M}osco convergence. Finally, we consider time-dependent permeability and analyze the case where $α$ blows up in finite time. Under suitable regularity assumptions, we show that the solution can be extended beyond the blow-up time, remaining in the single structure regime.

Theorems & Definitions (12)

• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Lemma 2.8
• Definition 3.1
• Theorem 3.2