具有不完美键合界面的电导率问题的梯度估计
Gradient estimates for the conductivity problem with imperfect bonding interfaces
AI总结：
本文研究具有不完美键合界面的电导率问题中的场集中现象，发现梯度有界性依赖于键合参数，并确定了阈值与最优爆破速率，提出了适用于多种椭圆方程的通用估计框架。
摘要：
我们研究了两个紧密间隔的完美导体之间，在低电导率类型的不完美键合界面情况下的场集中现象。这些界面上的边界条件由Robin型边界条件给出。我们发现了一个依赖于键合参数$\gamma$的场集中的\textit{新的}二分法。具体而言，我们证明了当$\gamma$足够小时，解的梯度一致有界，且与$\varepsilon$（两个夹杂物之间的距离）无关。然而，当$\gamma$较大时，梯度可能会爆破。此外，我们在某些对称性假设下确定了$\gamma$的阈值和最优爆破速率。证明依赖于两个夹杂物之间薄颈部的一个关键的各向异性梯度估计。我们建立了一个用于此类估计的一般框架，该框架适用于广泛的椭圆方程和边界条件。
英文摘要：
We study the field concentration phenomenon between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type. The boundary condition on these interfaces is given by a Robin-type boundary condition. We discover a \textit{new} dichotomy for the field concentration depending on the bonding parameter $γ$. Specifically, we show that the gradient of solution is uniformly bounded independent of $\varepsilon$ (the distance between two inclusions) when $γ$ is sufficiently small. However, the gradient may blow up when $γ$ is large. Moreover, we identify the threshold of $γ$ and the optimal blow-up rates under certain symmetry assumptions. The proof relies on a crucial anisotropic gradient estimate in the thin neck between two inclusions. We develop a general framework for establishing such estimate, which is applicable to a wide range of elliptic equations and boundary conditions.
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