The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\mathrm{div}\phantom{\rule{0.166667em}{0ex}}a\left(\nabla u\right)+F\left[u\right]\left(x\right)=0,$ over the functions $u\in {W}^{1,1}\left(\Omega \right)$ that assume given boundary values $\phi $ on $\partial \Omega .$ The vector field $a:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$ 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 $\Omega .$

Keywords: non-linear elliptic PDE's, Lipschitz continuous solutions, lower bounded slope condition

@article{COCV_2007__13_4_707_0, author = {Bousquet, Pierre}, title = {Local {Lipschitz} continuity of solutions of non-linear elliptic differential-functional equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {707--716}, publisher = {EDP-Sciences}, volume = {13}, number = {4}, year = {2007}, doi = {10.1051/cocv:2007035}, mrnumber = {2351399}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007035/} }

TY - JOUR AU - Bousquet, Pierre TI - Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 707 EP - 716 VL - 13 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007035/ DO - 10.1051/cocv:2007035 LA - en ID - COCV_2007__13_4_707_0 ER -

%0 Journal Article %A Bousquet, Pierre %T Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 707-716 %V 13 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007035/ %R 10.1051/cocv:2007035 %G en %F COCV_2007__13_4_707_0

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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