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

Bousquet, Pierre

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 707-716.

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\mathrm{div}a\left(\nabla u\right)+F\left[u\right]\left(x\right)=0,$ over the functions $u\in W^{1,1}\left(\Omega \right)$ that assume given boundary values $\phi$ on $\partial \Omega .$ The vector field $a:{ℝ}^n\to {ℝ}^n$ satisfies an ellipticity condition and for a fixed $x,F\left[u\right]\left(x\right)$ 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 $\Omega .$

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