Interior H\"older and Calder\'on-Zygmund estimates for fully nonlinear equations with natural gradient growth

Abstract

We establish local H\"older estimates for viscosity solutions of fully nonlinear second order equations with quadratic growth in the gradient and unbounded right-hand side in $L^q$ spaces, for an integrability threshold $q$ guaranteeing the validity of the maximum principle. This is done through a nonlinear Harnack inequality for nonhomogeneous equations driven by a uniformly elliptic Isaacs operator and perturbed by a Hamiltonian term with natural growth in the gradient. As a byproduct, we derive a new Liouville property for entire $L^p$ viscosity solutions of fully nonlinear equations as well as a nonlinear Calder\'on-Zygmund estimate for strong solutions of such equations.