Continuity of solutions of a nonlinear elliptic equation

Pierre Bousquet

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 1, page 1-19
  • ISSN: 1292-8119

Abstract

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We consider a nonlinear elliptic equation of the form div [a(∇u)] + F[u] = 0 on a domain Ω, subject to a Dirichlet boundary condition tru = φ. We do not assume that the higher order term a satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and φ satisfies a one-sided bounded slope condition, or when ais radial: a ( ξ ) = l ( | ξ | ) | ξ | ξ a ( ξ ) = l ( | ξ | ) | ξ | ξ for some increasingl:ℝ+ → ℝ+.

How to cite

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Bousquet, Pierre. "Continuity of solutions of a nonlinear elliptic equation." ESAIM: Control, Optimisation and Calculus of Variations 19.1 (2013): 1-19. <http://eudml.org/doc/272871>.

@article{Bousquet2013,
abstract = {We consider a nonlinear elliptic equation of the form div [a(∇u)] + F[u] = 0 on a domain Ω, subject to a Dirichlet boundary condition tru = φ. We do not assume that the higher order term a satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and φ satisfies a one-sided bounded slope condition, or when ais radial: $a(\xi )=\{l(|\xi |)\}\{|\xi |\} \xi $ a ( ξ ) = l ( | ξ | ) | ξ | ξ for some increasingl:ℝ+ → ℝ+.},
author = {Bousquet, Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {nonlinear elliptic equations; continuity of solutions; lower bounded slope condition; Lavrentiev phenomenon},
language = {eng},
number = {1},
pages = {1-19},
publisher = {EDP-Sciences},
title = {Continuity of solutions of a nonlinear elliptic equation},
url = {http://eudml.org/doc/272871},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Bousquet, Pierre
TI - Continuity of solutions of a nonlinear elliptic equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 1
SP - 1
EP - 19
AB - We consider a nonlinear elliptic equation of the form div [a(∇u)] + F[u] = 0 on a domain Ω, subject to a Dirichlet boundary condition tru = φ. We do not assume that the higher order term a satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and φ satisfies a one-sided bounded slope condition, or when ais radial: $a(\xi )={l(|\xi |)}{|\xi |} \xi $ a ( ξ ) = l ( | ξ | ) | ξ | ξ for some increasingl:ℝ+ → ℝ+.
LA - eng
KW - nonlinear elliptic equations; continuity of solutions; lower bounded slope condition; Lavrentiev phenomenon
UR - http://eudml.org/doc/272871
ER -

References

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  1. [1] P. Bousquet, The lower bounded slope condition. J. Convex Anal. 1 (2007) 119 − 136. Zbl1132.49031MR2310433
  2. [2] P. Bousquet, Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations. ESAIM Control Optim. Calc. Var. 13 (2007) 707 − 716. Zbl1142.35094MR2351399
  3. [3] P. Bousquet, Continuity of solutions of a problem in the calculus of variations. Calc. Var. Partial Differential Equations 41 (2011) 413 − 433. Zbl1227.49043MR2796238
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  5. [5] M. Degiovanni and M. Marzocchi, On the Euler-Lagrange equation for functionals of the calculus of variations without upper growth conditions. SIAM J. Control Optim. 48 (2009) 2857 − 2870. Zbl1201.49022MR2558324
  6. [6] 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. Zbl1042.35002MR1814364
  7. [7] P. Hartman, On the bounded slope condition. Pac. J. Math. 18 (1966) 495 − 511. Zbl0149.32001MR197640
  8. [8] P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations. Acta Math. 115 (1966) 271 − 310. Zbl0142.38102MR206537
  9. [9] O.A. Ladyzhenskaya and N.N. Uraltseva, Linear and quasilinear elliptic equations. Academic Press, New York (1968). Zbl0164.13002MR244627
  10. [10] C.B. Morrey, Multiple integrals in the calculus of variations. Springer-Verlag, New York (1966). Zbl1213.49002MR202511

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