First eigenvalues of free boundary hypersurfaces in the unit ball along the inverse mean curvature flow

Abstract

In this note, we consider the first nonzero eigenvalue ${\lambda }_{p,1}$ of the p-Laplacian on free boundary proper hypersurfaces in the unit ball evolving along the inverse mean curvature flow. We show that ${\lambda }_{p,1}$ is monotone decreasing along the flow. Using the convergence of free boundary disks in the unit ball, we give a lower bound of ${\lambda }_{p,1}$ of a free boundary disk type hypersurface in the unit ball.