Lipschitz regularity results for nonlinear strictly elliptic equations and applications

Olivier Ley (1) , Vinh Duc Nguyen (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 School of Mathematics [Cardiff]

Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.

Mots clés: nonlinear strictly elliptic equations; nonlinear parabolic equations; Hamilton-Jacobi equations; Lipschitz estimates; Asymptotic behavior; viscocity solutions

Domaines: Equations aux dérivées partielles [math.AP]

Olivier Ley, Vinh Duc Nguyen. Lipschitz regularity results for nonlinear strictly elliptic equations and applications. Journal of Differential Equations, 2017, 263 (7), pp.4324-4354. ⟨10.1016/j.jde.2017.05.020⟩. ⟨hal-01344438⟩