]. S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, vol.27, issue.6, pp.1085-1095, 1979.
DOI : 10.1016/0001-6160(79)90196-2

H. Berestycki, Le nombre de solutions de certains probl??mes semi-lin??aires elliptiques, Journal of Functional Analysis, vol.40, issue.1, pp.1-29, 1981.
DOI : 10.1016/0022-1236(81)90069-0

URL : http://doi.org/10.1016/0022-1236(81)90069-0

H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Boletim da Sociedade Brasileira de Matem???tica, vol.43, issue.1, pp.1-37, 1991.
DOI : 10.1007/BF01244896

D. Bonheure, P. Habets, and L. Sanchez, Heteroclinics for fourth order symmetric bistable equations, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, vol.52, issue.2, pp.213-227, 2004.

D. Bonheure and F. Hamel, One-dimensional symmetry and liouville type results for the fourth order Allen-Cahn equation in R n, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01182688

D. Bonheure and L. Sanchez, Heteroclinic orbits for some classes of second and fourth order differential equations In Handbook of differential equations: ordinary differential equations, Handb. Differ. Equ, vol.III, pp.103-202, 2006.

D. Bonheure, L. Sanchez, M. Tarallo, and S. Terracini, Heteroclinic connections between nonconsecutive equilibria of a fourth order differential equation, Calculus of Variations and Partial Differential Equations, vol.17, issue.4, pp.341-356, 2003.
DOI : 10.1007/s00526-002-0172-y

X. Cabré, Uniqueness and stability of saddle-shaped solutions to the Allen???Cahn equation, Journal de Math??matiques Pures et Appliqu??es, vol.98, issue.3, pp.239-256, 2012.
DOI : 10.1016/j.matpur.2012.02.006

J. W. Cahn and J. E. Hilliard, Free Energy of a Nonuniform System. I. Interfacial Free Energy, The Journal of Chemical Physics, vol.28, issue.2, pp.258-267, 1958.
DOI : 10.1063/1.1744102

M. Chermisi, G. Dal-maso, I. Fonseca, and G. Leoni, Singular perturbation models in phase transitions for second-order materials, Indiana University Mathematics Journal, vol.60, issue.2, pp.367-409, 2011.
DOI : 10.1512/iumj.2011.60.4346

B. D. Coleman, M. Marcus, and V. J. , On the thermodynamics of periodic phases, Archive for Rational Mechanics and Analysis, vol.106, issue.4, pp.321-347, 1992.
DOI : 10.1007/BF00376187

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, itand linearized stability, Archive for Rational Mechanics and Analysis, vol.52, issue.2, pp.161-180, 1973.
DOI : 10.1007/BF00282325

H. Dang, P. C. Fife, and L. A. Peletier, Saddle solutions of the bistable diffusion equation, ZAMP Zeitschrift f???r angewandte Mathematik und Physik, vol.116, issue.6, pp.984-998, 1992.
DOI : 10.1007/BF00916424

G. T. Dee and W. Van-saarloos, Bistable Systems with Propagating Fronts Leading to Pattern Formation, Physical Review Letters, vol.60, issue.25, pp.2641-2644, 1988.
DOI : 10.1103/PhysRevLett.60.2641

URL : https://openaccess.leidenuniv.nl/bitstream/handle/1887/5456/850_033.pdf?sequence=1

E. J. Doedel and B. E. Oldeman, AUTO-07P : Continuation and bifurcation software for ordinary differential equations, 2012.

R. A. Fisher, THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES, Annals of Eugenics, vol.7, issue.4, pp.355-369, 1937.
DOI : 10.1111/j.1469-1809.1937.tb02153.x

I. Fonseca and C. Mantegazza, Second Order Singular Perturbation Models for Phase Transitions, SIAM Journal on Mathematical Analysis, vol.31, issue.5, pp.1121-1143, 2000.
DOI : 10.1137/S0036141099356830

F. Gazzola, H. Ch, G. Grunau, and . Sweers, Polyharmonic boundary value problems, Positivity preserving and nonlinear higher order elliptic equations in bounded domains, 1991.
DOI : 10.1007/978-3-642-12245-3

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, 2001.

G. Gompper, C. Domb, M. S. Green, M. Schick, and J. L. Lebowitz, Phase Transitions and Critical Phenomena: Self-assembling amphiphilic systems. Phase transitions and critical phenomena, 1994.
DOI : 10.1063/1.2807946

F. Hecht, New development in freefem++, Journal of Numerical Mathematics, vol.20, issue.3-4, pp.251-265, 2012.
DOI : 10.1515/jnum-2012-0013

D. Hilhorst, L. A. Peletier, and R. Schätzle, ??-limit for the extended Fisher???Kolmogorov equation, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.132, issue.01, pp.141-162, 2002.
DOI : 10.1017/S0308210500001566

T. Kawakatsu, D. Andelman, K. Kawasaki, and T. Taniguchi, Phase transitions and shapes of two component membranes and vesicles I: strong segregation limit, Journal de Physique II, vol.3, issue.7, pp.971-997, 1993.
DOI : 10.1051/jp2:1993177

URL : https://hal.archives-ouvertes.fr/jpa-00247892

S. Kesavan, Symmetrization & applications, volume 3 of Series in Analysis, 2006.

H. Kielhöfer, Bifurcation theory, Applied Mathematical Sciences, vol.156

A. Kolmogorov, I. Petrovskii, and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos. Univ, vol.1, issue.6, pp.1-25, 1937.

N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol.12, 1996.

N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, vol.96, 2008.
DOI : 10.1090/gsm/096

S. Leibler and D. Andelman, Ordered and curved meso-structures in membranes and amphiphilic films, Journal de Physique, vol.48, issue.11, pp.2013-2018, 1987.
DOI : 10.1051/jphys:0198700480110201300

URL : https://hal.archives-ouvertes.fr/jpa-00210644

A. Leizarowitz and V. J. , One-dimensional infinite-horizon variational problems arising in continuum mechanics, Arch. Rational Mech. Anal, vol.106, issue.2, pp.161-194, 1989.

E. Mitidieri and G. Sweers, Weakly Coupled Elliptic Systems and Positivity, Mathematische Nachrichten, vol.116, issue.1, pp.259-286, 1995.
DOI : 10.1002/mana.19951730115

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.5790

V. J. Mizel, L. A. Peletier, and W. C. Troy, Periodic Phases in Second-Order Materials, Archive for Rational Mechanics and Analysis, vol.145, issue.4, pp.343-382, 1998.
DOI : 10.1007/s002050050133

L. A. Peletier and W. C. Troy, Spatial patterns Progress in Nonlinear Differential Equations and their Applications, 45, Higher order models in physics and mechanics, 2001.

J. Serrin, A symmetry problem in potential theory, Archive for Rational Mechanics and Analysis, vol.43, issue.4, pp.304-318, 1971.
DOI : 10.1007/BF00250468

G. Sweers, Strong positivity in $$C(\bar \Omega )$$ for elliptic systemsfor elliptic systems, Mathematische Zeitschrift, vol.116, issue.1, pp.251-271, 1992.
DOI : 10.1007/BF02570833

W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, Journal of Differential Equations, vol.42, issue.3, pp.400-413, 1981.
DOI : 10.1016/0022-0396(81)90113-3

J. D. Van-der-waals, Thermodynamische theorie der capillariteit in de onderstelling van continue dichtheidsverandering, Verhand. Kon. Akad. Wetensch. Amsterdam Sect J. Stat. Phys, vol.1, issue.20, 1893.

A. A. Wheeler, Phase-field theory of edges in an anisotropic crystal, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.462, issue.2075, pp.3363-3384, 2006.
DOI : 10.1098/rspa.2006.1721