Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations

Abstract

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\textrm{div}\,a(\nabla u)+F[u](x)=0,$ over the functions $u\in W^{1,1}(\Omega)$ that assume given boundary values ϕ on ∂Ω. The vector field $a:{\mathbb R}^n\to {\mathbb R}^n$ 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.