ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)

ESAIM: COCV, Vol. 13, N°4, pp. 707-716
DOI: 10.1051/cocv:2007035

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

Pierre Bousquet

Institut Camille Jordan, Université Claude Bernard, Lyon 1, France; bousquet@math.univ-lyon1.fr

(Received March 14, 2006. Published online July 20, 2007.)

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 $u\in W^(\Omega)$ that assume given boundary values $\phi$ on $\partial\Omega.$ 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 $\phi$ 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 $\phi:$ the lower (or upper) bounded slope condition. This condition, which is less restrictive than the previous one, is satisfied if $\phi$ is the restriction to $\partial \Omega$ 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 $\Omega.$

Mathematics Subject Classification. 35J25, 35J60

Key words: Non-linear elliptic PDE's, Lipschitz continuous solutions, lower bounded slope condition

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