References & Citations
Mathematics > Analysis of PDEs
Title: Gradient estimates for the insulated conductivity problem: the non-umbilical case
(Submitted on 18 Mar 2022 (v1), last revised 6 Apr 2022 (this version, v2))
Abstract: We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb R^n$, for $n \ge 3$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-zero eigenvalue of a divergence form elliptic operator on $\mathbb S^{n-2}$.
Submission history
From: Hongjie Dong [view email][v1] Fri, 18 Mar 2022 17:50:25 GMT (30kb)
[v2] Wed, 6 Apr 2022 15:26:45 GMT (30kb)
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