On the nodal line of the second eigenfunction of the Laplacian over some concave domains in ${\mathbb{R}}^2$

In this paper we will prove the nodal line $N$ of the second eigenfunction of the Laplacian over some simply connected concave domain $\mathrm{\Omega }$ in ${\mathbb{R}}^2$ must intersect the boundary $\mathrm{\partial }\mathrm{\Omega }$ at exactly two points.