We propose a simple level set method for the Dirichlet k-partition problem which aims to partition an open domain into K different subdomains as to minimize the sum of the smallest eigenvalue of the Dirichlet Laplace operator in each subdomain. We first formulate the problem as a nested minimization problem of a functional of the level set function and the eigenfunction defined in each subdomain. As an approximation, we propose to simply replace the eigenfunction by the level set function so that the nested minimization can then be converted to a single minimization problem. We apply the standard gradient descent method so that the problem leads to a Hamilton–Jacobi type equation. Various numerical examples will be given to demonstrate the effectiveness of our proposed method.

我们为Dirichlet k分区问题提出了一种简单的水平集方法，该方法旨在将一个开放域划分为K个不同的子域，以使每个子域中Dirichlet Laplace算子的最小特征值之和最小。我们首先将该问题表述为在每个子域中定义的水平集函数和特征函数的嵌套最小化问题。作为近似，我们建议将简单的特征函数替换为水平集函数，以便将嵌套的最小化然后转换为单个最小化问题。我们应用标准梯度下降法，以便该问题导致汉密尔顿-雅各比型方程。将给出各种数值示例，以证明我们提出的方法的有效性。