Estimates of the first eigenvalue of minimal hypersurfaces of   $\mathbb{S}^{n+1}$
  We consider a solution f of a certain Dirichlet Problem on a domain in $S^{(n+1)}$ whose boundary is a minimal hypersurface and we prove a Poincare type inequality for f. One have equality iff Yau's conjecture about the first non-zero eigenvalue of closed minimal hypersurfaces of $S^{(n+1)}$ is true. 
Abdenago Barros, G. Pacelli Bessa