Optimal gradient estimates for conductivity problems with imperfect low-conductivity interfaces

TL;DR

The study addresses sharp gradient estimates for conductivity problems with imperfect (Robin) interfaces between closely spaced inclusions, illuminating how interfacial resistance via the parameter $\gamma$ interacts with the small gap $\varepsilon$. By reducing to constant-coefficient models, employing Campanato-type regularity, and decomposing the narrow-region problem, the authors obtain explicit, dimension-dependent gradient bounds that smoothly interpolate between the bounded- and blow-up regimes as $\gamma\to0$ or $\varepsilon\to0$. They prove optimal rates, show a dichotomy based on $\gamma$ relative to a local distance $\delta(x')$, and establish convergence to the perfect-conductor limit as $\gamma\to0$, with a global gradient bound outside the neck. The results yield precise local-field characterizations in composites with interfacial resistance, enhancing both theoretical understanding and material-design implications. The work also provides a unified framework applicable to related problems in elasticity and fluid-structure interfaces.

Abstract

This paper studies field concentration between two nearly touching conductors separated by imperfect low-conductivity interfaces, modeled by Robin boundary conditions. It is known that for any sufficiently small interfacial bonding parameter $γ> 0$, the gradient remains uniformly bounded with respect to the separation distance $\varepsilon$. In contrast, for the perfect bonding case ($γ= 0$, corresponding to the perfect conductivity problem), the gradient may blow up as $\varepsilon \to 0$ at a rate depending on the dimension. In this work, we establish optimal pointwise gradient estimates that explicitly depend on both $γ$ and $\varepsilon$ in the regime where these parameters are small. These estimates provide a unified framework that encompasses both the previously known bounded case ($γ> 0$) and the singular blow-up scenario ($γ= 0$), thus furnishing a complete and continuous characterization of the gradient behavior throughout the transition in $γ$. The key technical achievement is the derivation of new regularity results for elliptic equations as $γ\to0$, along with a case dichotomy based on the relative sizes of $γ$ and a distance function $δ(x')$. Our results hold for strictly relatively convex conductors in all dimensions $n \geq 2$.

Theorems & Definitions (49)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Corollary 2.2
• proof : Proof of Theorem \ref{['gradientsmallgamma3']}
• Lemma 2.3
• proof