Continuity of solutions of a nonlinear elliptic equation
ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 1, pp. 1-19.

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 a is radial: a ( ξ ) = l ( | ξ | ) | ξ | ξ for some increasing l:ℝ+ → ℝ+.

DOI: 10.1051/cocv/2011194
Classification: 35J20, 35J25, 35J60
Keywords: nonlinear elliptic equations, continuity of solutions, lower bounded slope condition, Lavrentiev phenomenon
@article{COCV_2013__19_1_1_0,
     author = {Bousquet, Pierre},
     title = {Continuity of solutions of a nonlinear elliptic equation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1--19},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {1},
     year = {2013},
     doi = {10.1051/cocv/2011194},
     mrnumber = {3023057},
     zbl = {1271.35028},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2011194/}
}
TY  - JOUR
AU  - Bousquet, Pierre
TI  - Continuity of solutions of a nonlinear elliptic equation
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 1
EP  - 19
VL  - 19
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2011194/
DO  - 10.1051/cocv/2011194
LA  - en
ID  - COCV_2013__19_1_1_0
ER  - 
%0 Journal Article
%A Bousquet, Pierre
%T Continuity of solutions of a nonlinear elliptic equation
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 1-19
%V 19
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2011194/
%R 10.1051/cocv/2011194
%G en
%F COCV_2013__19_1_1_0
Bousquet, Pierre. Continuity of solutions of a nonlinear elliptic equation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 1, pp. 1-19. doi : 10.1051/cocv/2011194. http://www.numdam.org/articles/10.1051/cocv/2011194/

[1] P. Bousquet, The lower bounded slope condition. J. Convex Anal. 1 (2007) 119 − 136. | MR | Zbl

[2] P. Bousquet, Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations. ESAIM Control Optim. Calc. Var. 13 (2007) 707 − 716. | Numdam | MR | Zbl

[3] P. Bousquet, Continuity of solutions of a problem in the calculus of variations. Calc. Var. Partial Differential Equations 41 (2011) 413 − 433. | MR | Zbl

[4] F. Clarke, Continuity of solutions to a basic problem in the calculus of variations. Ann. Scvola Norm. Super. Pisa Cl. Sci. (5) 4 (2005) 511 − 530. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[7] P. Hartman, On the bounded slope condition. Pac. J. Math. 18 (1966) 495 − 511. | MR | Zbl

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

[9] O.A. Ladyzhenskaya and N.N. Uraltseva, Linear and quasilinear elliptic equations. Academic Press, New York (1968). | MR | Zbl

[10] C.B. Morrey, Multiple integrals in the calculus of variations. Springer-Verlag, New York (1966). | MR | Zbl

Cited by Sources: