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

Bousquet, Pierre. "Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations." ESAIM: Control, Optimisation and Calculus of Variations 13.4 (2007): 707-716. <http://eudml.org/doc/249985>.

@article{Bousquet2007,
abstract = {The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\textrm\{div\}\,a(\nabla u)+F[u](x)=0,$ over the functions $u\in W^\{1,1\}(\Omega)$ that assume given boundary values ϕ on ∂Ω. The vector field $a:\{\mathbb R\}^n\to \{\mathbb R\}^n$ 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. },
author = {Bousquet, Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Non-linear elliptic PDE's; Lipschitz continuous solutions; lower bounded slope condition; nonlinear elliptic PDE's},
language = {eng},
month = {7},
number = {4},
pages = {707-716},
publisher = {EDP Sciences},
title = {Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations},
url = {http://eudml.org/doc/249985},
volume = {13},
year = {2007},
}

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
DA - 2007/7//
PB - EDP Sciences
VL - 13
IS - 4
SP - 707
EP - 716
AB - The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\textrm{div}\,a(\nabla u)+F[u](x)=0,$ over the functions $u\in W^{1,1}(\Omega)$ that assume given boundary values ϕ on ∂Ω. The vector field $a:{\mathbb R}^n\to {\mathbb R}^n$ 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.
LA - eng
KW - Non-linear elliptic PDE's; Lipschitz continuous solutions; lower bounded slope condition; nonlinear elliptic PDE's
UR - http://eudml.org/doc/249985
ER -