Local Lipschitz continuity of solutions of non-linear elliptic differentialfunctional equations

Abstract. The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form div a(∇u) + F [u](x) = 0, over the functions u ∈ W 1,1(Ω) that assume given boundary values φ on ∂Ω. The vector field a: Rn → Rn satisfies an ellipticity condition and for a fixed x, F [u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math. 115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions when φ satisfies the classical bounded slope condition. In a variational context, Clarke [Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 511–530] has introduced a new type of hypothesis on the boundary condition φ: the lower (or upper) bounded slope condition. This condition, which is less restrictive than the previous one, is satisfied if φ is the restriction to ∂Ω of a convex function. We show that if a and F satisfy hypotheses similar to those of Hartman and Stampacchia, the lower bounded slope condition implies the existence of solutions in the space of locally Lipschitz continuous functions on Ω