Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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 div a(u)+F[u](x)=0, over the functions uW 1,1 (Ω) that assume given boundary values φ on Ω. The vector field a: n 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.

DOI: 10.1051/cocv:2007035
Classification: 35J25,  35J60
Keywords: non-linear elliptic PDE's, Lipschitz continuous solutions, lower bounded slope condition
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     title = {Local {Lipschitz} continuity of solutions of non-linear elliptic differential-functional equations},
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Bousquet, Pierre. Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 4, pp. 707-716. doi : 10.1051/cocv:2007035. http://www.numdam.org/articles/10.1051/cocv:2007035/

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