The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form over the functions that assume given boundary values on The vector field satisfies an ellipticity condition and for a fixed denotes a non-linear functional of 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 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 the lower (or upper) bounded slope condition. This condition, which is less restrictive than the previous one, is satisfied if is the restriction to of a convex function. We show that if and 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
Mots clés : 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, Tome 13 (2007) no. 4, pp. 707-716. doi : 10.1051/cocv:2007035. http://www.numdam.org/articles/10.1051/cocv:2007035/
[1] The lower bounded slope condition. J. Convex Anal. 14 (2007) 119-136. | Zbl
,[2] Local Lipschitz continuity of solutions to a problem in the calculus of variations. J. Differ. Eq. (to appear). | MR | Zbl
and ,[3] Continuity of solutions to a basic problem in the calculus of variations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 511-530. | Numdam | Zbl
,[4] Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR | Zbl
and ,[5] On the bounded slope condition. Pacific J. Math. 18 (1966) 495-511. | Zbl
,[6] On some non-linear elliptic differential-functional equations. Acta Math. 115 (1966) 271-310. | Zbl
and ,[7] The quasilinear Dirichlet problem with decreased regularity at the boundary. Comm. Partial Differential Equations 6 (1981) 437-497. | Zbl
,[8] The Dirichlet problem for quasilinear elliptic equations with Hölder continuous boundary values. Arch. Rational Mech. Anal. 79 (1982) 305-323. | Zbl
,[9] The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data. Comm. Partial Differential Equations 11 (1986) 167-229. | Zbl
,[10] Un teorema di esistenza e unicità per il problema dell’area minima in variabili. Ann. Scuola Norm. Sup. Pisa 19 (1965) 233-249. | Numdam | Zbl
,Cité par Sources :