论文标题
关于周期介质中二阶级别集PDE的均质化
On the homogenization of second order level set PDE in periodic media
论文作者
论文摘要
本文分析了抛物线缩放中的定期介质中的两类二阶级别设置PDE。首先，我们在尺寸$ d = 2 $的一般假设下研究完全非线性的几何算子，并证明在这种情况下，关联的方程式均质化。接下来，我们在任意维度中对待一类准线性几何操作员$ d \ geq 2 $。在这种情况下，通过调整参数构成振荡边界价值问题的研究，我们证明有效系数在所有维度上都是不连续的$ d \ geq 3 $。这需要对级别集的PDE进行研究，而PDE是由在球体上每个理性方向上不连续的操作员驱动的。我们证明，实际上，因此获得的有效运算符确实具有比较原理，因此发生了均质化。最后，我们研究了在准线性情况下获得的有效迁移率与线性响应之间的联系，从而在我们的结果与双曲线缩放中获得的连接之间建立了联系。
This paper analyzes two classes of second order level set PDE in periodic media in the parabolic scaling. First, we study fully nonlinear geometric operators under general assumptions in dimension $d = 2$ and prove that the associated equations homogenize in this case. Next, we treat a class of quasi-linear geometric operators in arbitrary dimensions $d \geq 2$. In this setting, by adapting arguments form the study of oscillating boundary value problems, we prove that the effective coefficients are generically discontinuous in all dimensions $d \geq 3$. This necessitates a study of level set PDE driven by operators that are discontinuous at every rational direction on the sphere. We prove that, in fact, the effective operators so obtained do have a comparison principle and, thus, homogenization occurs. Finally, we investigate the connection between the effective mobility obtained in the quasi-linear case and linear response, drawing a connection between our results and those obtained in the hyperbolic scaling.
