Entire solutions in 2 for a class of Allen-Cahn equations
ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 4, pp. 633-672.

We consider a class of semilinear elliptic equations of the form

-ε 2 Δu(x,y)+a(x)W ' (u(x,y))=0,(x,y) 2
where ε>0, a: is a periodic, positive function and W: is modeled on the classical two well Ginzburg-Landau potential W(s)=(s 2 -1) 2 . We look for solutions to (1) which verify the asymptotic conditions u(x,y)±1 as x± uniformly with respect to y. We show via variational methods that if ε is sufficiently small and a is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

DOI : 10.1051/cocv:2005023
Classification : 34C37, 35B05, 35B40, 35J20, 35J60
Mots clés : heteroclinic solutions, elliptic equations, variational methods
@article{COCV_2005__11_4_633_0,
     author = {Alessio, Francesca and Montecchiari, Piero},
     title = {Entire solutions in $\mathbb {R}^{2}$ for a class of {Allen-Cahn} equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {633--672},
     publisher = {EDP-Sciences},
     volume = {11},
     number = {4},
     year = {2005},
     doi = {10.1051/cocv:2005023},
     mrnumber = {2167878},
     zbl = {1084.35020},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2005023/}
}
TY  - JOUR
AU  - Alessio, Francesca
AU  - Montecchiari, Piero
TI  - Entire solutions in $\mathbb {R}^{2}$ for a class of Allen-Cahn equations
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2005
SP  - 633
EP  - 672
VL  - 11
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2005023/
DO  - 10.1051/cocv:2005023
LA  - en
ID  - COCV_2005__11_4_633_0
ER  - 
%0 Journal Article
%A Alessio, Francesca
%A Montecchiari, Piero
%T Entire solutions in $\mathbb {R}^{2}$ for a class of Allen-Cahn equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2005
%P 633-672
%V 11
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2005023/
%R 10.1051/cocv:2005023
%G en
%F COCV_2005__11_4_633_0
Alessio, Francesca; Montecchiari, Piero. Entire solutions in $\mathbb {R}^{2}$ for a class of Allen-Cahn equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 4, pp. 633-672. doi : 10.1051/cocv:2005023. http://www.numdam.org/articles/10.1051/cocv:2005023/

[1] S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in 2 for an Allen-Cahn system with multiple well potential. Calc. Var. 5 (1997) 359-390. | Zbl

[2] G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math. 65 (2001) 9-33. | Zbl

[3] F. Alessio, L. Jeanjean and P. Montecchiari, Stationary layered solutions in 2 for a class of non autonomous Allen-Cahn equations. Calc. Var. Partial Differ. Equ. 11 (2000) 177-202. | Zbl

[4] F. Alessio, L. Jeanjean and P. Montecchiari, Existence of infinitely many stationary layered solutions in 2 for a class of periodic Allen Cahn Equations. Commun. Partial Differ. Equ. 27 (2002) 1537-1574. | Zbl

[5] L. Ambrosio and X. Cabre, Entire solutions of semilinear elliptic equations in 3 and a conjecture of De Giorgi. J. Am. Math. Soc. 13 (2000) 725-739. | Zbl

[6] V. Bargert, On minimal laminations on the torus. Ann. Inst. H. Poincaré Anal. Nonlinéaire 6 (1989) 95-138. | Numdam | Zbl

[7] M.T. Barlow, R.F. Bass and C. Gui, The Liouville property and a conjecture of De Giorgi. Comm. Pure Appl. Math. 53 (2000) 1007-1038. | Zbl

[8] H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry for some bounded entire solutions of some elliptic equations. Duke Math. J. 103 (2000) 375-396. | Zbl

[9] E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis. Rome, E. De Giorgi et al. Eds. (1978). | Zbl

[10] A. Farina, Symmetry for solutions of semilinear elliptic equations in N and related conjectures. Ricerche Mat. (in memory of Ennio De Giorgi) 48 (1999) 129-154. | Zbl

[11] N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems. Math. Ann. 311 (1998) 481-491. | Zbl

[12] J. Moser, Minimal solutions of variational problem on a torus. Ann. Inst. H. Poincaré Anal. NonLinéaire 3 (1986) 229-272. | Numdam | Zbl

[13] P.H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation. Commun. Pure Appl. Math. 56 (2003) 1078-1134.

[14] P.H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, II. Calc. Var. Partial Differ. Equ. 21 (2004) 157-207.

[15] P.H. Rabinowitz, Heteroclinic for reversible Hamiltonian system. Ergod. Th. Dyn. Sys. 14 (1994) 817-829. | Zbl

[16] P.H. Rabinowitz, Solutions of heteroclinic type for some classes of semilinear elliptic partial differential equations. J. Math. Sci. Univ. Tokio 1 (1994) 525-550. | Zbl

Cité par Sources :