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

Pierre Bousquet

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

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

Abstract

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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 u W 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.

How to cite

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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 -

References

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  1. P. Bousquet, The lower bounded slope condition. J. Convex Anal.14 (2007) 119–136.  Zbl1132.49031
  2. P. Bousquet and F. Clarke, Local Lipschitz continuity of solutions to a problem in the calculus of variations. J. Differ. Eq. (to appear).  Zbl1141.49033
  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.  Zbl1127.49001
  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.  Zbl1042.35002
  5. P. Hartman, On the bounded slope condition. Pacific J. Math.18 (1966) 495–511.  Zbl0149.32001
  6. P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations. Acta Math.115 (1966) 271–310.  Zbl0142.38102
  7. G.M. Lieberman, The quasilinear Dirichlet problem with decreased regularity at the boundary. Comm. Partial Differential Equations6 (1981) 437–497.  Zbl0458.35039
  8. G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with Hölder continuous boundary values. Arch. Rational Mech. Anal.79 (1982) 305–323.  Zbl0497.35010
  9. G.M. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data. Comm. Partial Differential Equations11 (1986) 167–229.  Zbl0589.35036
  10. 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.  Zbl0137.08201

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