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

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 4, page 707-716
- ISSN: 1292-8119

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topBousquet, 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 -

## References

top- P. Bousquet, The lower bounded slope condition. J. Convex Anal.14 (2007) 119–136.
- P. Bousquet and F. Clarke, Local Lipschitz continuity of solutions to a problem in the calculus of variations. J. Differ. Eq. (to appear).
- 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.
- 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.
- P. Hartman, On the bounded slope condition. Pacific J. Math.18 (1966) 495–511.
- P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations. Acta Math.115 (1966) 271–310.
- G.M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary. Comm. Partial Differential Equations6 (1981) 437–497.
- G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with Hölder continuous boundary values. Arch. Rational Mech. Anal.79 (1982) 305–323.
- G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data. Comm. Partial Differential Equations11 (1986) 167–229.
- M. Miranda, Un teorema di esistenza e unicità per il problema dell'area minima in n variabili. Ann. Scuola Norm. Sup. Pisa19 (1965) 233–249.

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