  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
a 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.

Analysis of PDEs