Title: Local Gradient Estimates for Second-Order Nonlinear Elliptic and Parabolic Equations by the Weak Bernstein's Method,
Abstract: In the theory of second-order, nonlinear elliptic and parabolic equations,
obtaining local or global gradient bounds is often a key step for proving the
existence of solutions but it may be even more useful in many applications, for
example to singular perturbations problems. The classical Bernstein's method is
the well-known tool to obtain these bounds but, in most cases, it has the
defect of providing only a priori estimates. The "weak Bernstein's method" ,
based on viscosity solutions' theory, is an alternative way to prove the global
Lipschitz regularity of solutions together with some estimates but it is not so
easy to perform in the case of local bounds. The aim of this paper is to
provide an extension of the "weak Bernstein's method" which allows to prove
local gradient bounds with reasonnable technicalities. The classical
Bernstein's method is a well-known tool for obtaining gradient estimates for
solutions of second-order, elliptic and parabolic equations