论文标题
绝缘电导率问题的梯度估计：非野性情况
Gradient estimates for the insulated conductivity problem: the non-umbilical case
论文作者
论文摘要
我们研究了绝缘的电导率问题，其中包括$ \ Mathbb r^n $中的有界域中的夹杂物，以$ n \ ge 3 $。解决方案的梯度可能会爆炸为$ \ varepsilon $，夹杂物之间的距离，即$ 0 $。我们在最近对包括球在内的一类夹杂物的最佳梯度估计中建立了最佳梯度估计。在本文中，我们证明了一般严格凸成分的一般梯度估计。与完美的导电性问题不同，估计值取决于包含物的主要曲率，我们表明这些估计值的特征是在$ \ mathbb s^{n-2} $上的第一个差异形式的椭圆形算子的第一个非零特征值。
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 $\mathbb S^{n-2}$.
