论文标题
在连续环境中的有效前线的不同性能在两个维度
Differentiability of effective fronts in the continuous setting in two dimensions
论文作者
论文摘要
我们研究与连续系数的周期性设置中的二维（$ n = 2 $）相关的有效前线。我们的主要结果表明，在每个非理性点上，有效前线的边界都是可区分的。同等地，与连续的$ \ mathbb {z}^2 $ - 周期性riemannian指标相关的稳定规范在非理性点上是可区分的。该结论是几十年前的平滑指标获得的（[3,5]）。据我们所知，我们的结果在连续环境中提供了有效阵线的第一个非平凡特性，这是PDE理论中的标准假设。与足够的结果相结合[12]，我们的结果意味着，对于连续系数，当且仅当它与有理顶点和非空内部对称时，多边形才能是一个有效的前部。
We study the effective front associated with first-order front propagations in two dimensions ($n=2$) in the periodic setting with continuous coefficients. Our main result says that that the boundary of the effective front is differentiable at every irrational point. Equivalently, the stable norm associated with a continuous $\mathbb{Z}^2$-periodic Riemannian metric is differentiable at irrational points. This conclusion was obtained decades ago for smooth metrics ([3,5]). To the best of our knowledge, our result provides the first nontrivial property of the effective fronts in the continuous setting, which is the standard assumption in the PDE theory. Combining with the sufficiency result in [12], our result implies that for continuous coefficients, a polygon could be an effective front if and only if it is centrally symmetric with rational vertices and nonempty interior.
