Applications of P-functions to Fully Nonlinear Elliptic equations: Gradient Estimates and Rigidity Results

We introduce the notion of $ P -$functions for fully nonlinear equations and establish a general criterion for obtaining such quantities for this class of equations. Some applications are gradient bounds, De Giorgi-type properties of entire solutions and rigidity results. In particular, we establish a pointwise gradient bound and a rigidity result for Pucci's equations. This pointwise gradient bound generalizes the Modica inequality in the case of fully nonlinear elliptic equations. Furthermore, we prove Harnack-type inequalities and local estimates for the gradient of solutions. In addition, we consider such quantities for higher order nonlinear equations and for equations of order greater than two we obtain Liouville-type theorems and pointwise estimates for the Laplacian.