# Local Gradient Estimates for Second-Order Nonlinear Elliptic and Parabolic Equations by the Weak Bernstein's Method

## TL;DR

This paper extends the weak Bernstein's method, using viscosity solutions, to establish local gradient bounds for second-order nonlinear elliptic and parabolic equations, facilitating solution regularity analysis.

## Contribution

It introduces an extension of the weak Bernstein's method to obtain local gradient estimates, overcoming previous limitations of the approach.

## Key findings

- Established local gradient bounds for nonlinear elliptic equations.
- Provided a new technique for local regularity in parabolic equations.
- Enhanced the applicability of the weak Bernstein's method in PDE analysis.

## 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 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.