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

G Barles (1, 2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 FDP - Fédération de recherche Denis Poisson

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

Mots clés: Second-order elliptic and parabolic equations; gradient bounds; weak Bernstein's method; viscosity solutions

Domaines: Equations aux dérivées partielles [math.AP]

G Barles. Local Gradient Estimates for Second-Order Nonlinear Elliptic and Parabolic Equations by the Weak Bernstein's Method. 2017. ⟨hal-01526740v1⟩