A Level Set Method for the Dirichlet k-Partition Problem

Abstract

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.