A remark on the regularity of the coefficient matrix appearing in the charge simulation method

The charge simulation method (CSM) is an effective technique for solving the boundary value problem of the Laplace operator numerically. In this note, we first present an alternative proof for Theorem 1 of Katsurada-Okamoto [3] which is the necessary and sufficient condition for the regularity of the matrix appearing in CSM when the domain is a two dimensional disk. In a similar way, we prove the regularity of the corresponding matrix when the domain is a three dimensional open ball.