A lower bound for the principal eigenvalue of fully nonlinear elliptic operators

In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We ilustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that $\lim_{p\to \infty}\lambda_{1,p}=\lambda_{1,\infty}=\left(\frac{\pi}{2R}\right)^2$where$\lambda_{1,p}$and$\lambda_{1,\infty}$ are the principal eigenvalue for the homogeneous $p$-laplacian and the homogeneous infinity laplacian respectively.