# Estimate of the Hausdorff measure of the singular set of a solution for a semi-linear elliptic equation associated with superconductivity

## Junichi Aramaki

Address: Department of Mathematical Sciences, Faculty of Science and Engineering, Tokyo Denki University, Hatoyama-machi, Saitama 350-0394, Japan

E-mail: aramaki@mail.dendai.ac.jp

Abstract: We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean space ${\mathbb{R}}^n$. In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation. In particular, we showed that the singular set is $(n-2)$-rectifiable. In this paper, we shall show that under some additive smoothness assumptions, the $(n-2) $-dimensional Hausdorff measure of singular set of any solution is locally finite.

AMSclassification: primary 82D55; secondary 47F05, 35J60.

Keywords: singular set, semi-linear elliptic equation, Ginzburg-Landau system.