Eigenvalue Inequalities for Fully Nonlinear Elliptic Equations via the Alexandroff-Bakelman-Pucci Method

In this work we establish eigenvalue inequalities for elliptic differential operators either for Dirichlet or for Robin eigenvalue problems, by using the technique introduced by Alexandroff, Bakelman and Pucci. These inequalities can be extended for fully nonlinear elliptic equations, such as for the Monge-Amp\`ere equation and for Pucci's equations. As an application we establish a lower bound for the $L^p -$norm of the Laplacian and this bound is sharp, in the sense that, when equality is achieved then a symmetry property is obtained. In addition, we obtain an $L^{\infty}$ bound for the gradient of solutions to fully nonlinear elliptic equations and as a result, a $C^3$ estimate.