On the nodal set conjecture for the $p$-Laplacian in circularly symmetric domains

In 1990, Pütter shown that the nodal line of any second eigenfunction of the Dirichlet Laplacian on a planar bounded simply connected domain $\mathrm{\Omega }$ intersects the boundary $\mathrm{\partial }\mathrm{\Omega }$ provided $\mathrm{\Omega }$ has the circular symmetry. By adopting the method of moving polarization, we establish similar information on the nodal set of second eigenfunctions of the Dirichlet $p$-Laplacian on circularly symmetric domains in arbitrary higher dimension.