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.

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
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] P. Bousquet, The lower bounded slope condition. J. Convex Anal. 14 (2007) 119-136. | Zbl

[2] P. Bousquet and F. Clarke, Local Lipschitz continuity of solutions to a problem in the calculus of variations. J. Differ. Eq. (to appear). | MR | Zbl

[3] F. Clarke, 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] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR | Zbl

[5] P. Hartman, On the bounded slope condition. Pacific J. Math. 18 (1966) 495-511. | Zbl

[6] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations. Acta Math. 115 (1966) 271-310. | Zbl

[7] G.M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary. Comm. Partial Differential Equations 6 (1981) 437-497. | Zbl

[8] G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with Hölder continuous boundary values. Arch. Rational Mech. Anal. 79 (1982) 305-323. | Zbl

[9] G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data. Comm. Partial Differential Equations 11 (1986) 167-229. | Zbl

[10] M. Miranda, Un teorema di esistenza e unicità per il problema dell’area minima in n variabili. Ann. Scuola Norm. Sup. Pisa 19 (1965) 233-249. | Numdam | Zbl

Cité par Sources :