One-dimensional symmetry and Liouville type results for the fourth order Allen-Cahn equation in ℝ ^N

Abstract

In this paper, the authors prove an analogue of Gibbons’ conjecture for the extended fourth order Allen-Cahn equation in ℝ ^N, as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. The authors also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.

Keywords

Fourth order elliptic equation / Allen-Cahn equation / Extended Fisher-Kolmogorov equation / One-dimensional symmetry / Liouville type results

References

[1] Ambrosio L., Cabré X.. Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi. J. Amer. Math. Soc., 2000, 13: 725-739

[2] Aronson, D. G. andWeinberger, H. F., Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30, 1978, 33–76.

[3] Barlow M. T., Bass R., Gui C.. The Liouville property and a conjecture of De Giorgi. Comm. Pure Appl. Math., 2000, 53: 1007-1038

[4] Berestycki H., Caffarelli L., Nirenberg L.. Further qualitative properties for elliptic equations in unbounded domains. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV, 1997, 25: 69-94

[5] Berestycki H., Hamel F., Monneau R.. One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J., 2000, 103: 375-396

[6] Bers L., John F., Schechter M.. Partial differential equations, Lectures in Applied Mathematics, 1964, New York: John Wiley Sons

[7] Bonheure D.. Multitransition kinks and pulses for fourth order equations with a bistable nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2004, 21: 319-340

[8] Bonheure D.. Heteroclinic solutions for a class of fourth order ordinary differential equations. Acad. Roy. Belg. Cl. Sci. Mém. Collect., 2006, 80(3): 188

[9] Bonheure, D., Földes, J. and Salda, A., On the higher-dimensional extended Allen-Cahn equation, preprint.

[10] Bonheure D., Habets P., Sanchez L.. Minimizers for fourth order symmetric bistable equation. Atti Sem. Mat. Fis. Univ. Modena e Reggio Emilia, 2004, 52: 213-227

[11] Bonheure, D. and Sanchez, L., Heteroclinic orbits for some classes of second and fourth order differential equations, Handbook of Differential Equations: Ordinary Differential Equations, Vol. III, 103–202, Elsevier/North-Holland, Amsterdam, 2006.

[12] Bonheure D., Sanchez L., Tarallo M., Terracini S.. Heteroclinic connections between nonconsecutive equilibria of a fourth order differential equation. Calc. Var. Partial Diff. Equations, 2003, 17: 341-356

[13] Carbou G.. Unicité et minimalité des solutions d’une équation de Ginzburg-Landau. Ann. Inst. H. Poincaré Anal. Non Linéaire, 1995, 12: 305-318

[14] Chermisi M., Dal Maso G., Fonseca I., Leoni G.. Singular perturbation models in phase transitions for second-order materials. Indiana Univ. Math. J., 2011, 60: 367-409

[15] Coleman B. D., Marcus M., Mizel V. J.. On the thermodynamics of periodic phases. Arch. Ration. Mech. Anal., 1992, 117: 321-347

[16] De Giorgi E.. Convergence problems for functionals and operators. Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, 1979 131-188

[17] Dee G. T., van Saarloos W.. Bistable systems with propagating fronts leading to pattern formation. Phys. Rev. Lett., 1988, 60: 2641-2644

[18] Del Pino M., Kowalczyk M., Wei J.. On the De Giorgi conjecture in dimension N = 9. Ann. Math., 2011, 174: 1485-1569

[19] Douglis A., Nirenberg L.. Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math., 1955, 8: 503-538

[20] Farina A.. Finite-energy solutions. quantization effects and Liouville-type results for a variant of the Ginzburg-Landau systems in RK,Diff. Int. Equations, 1998, 11: 875-893

[21] Farina A.. Symmetry for solutions of semilinear elliptic equations in Rn and related conjectures. Ric. Matematica, 1999, 48: 129-154

[22] Farina A.. Monotonicity and one-dimensional symmetry for the solutions of u + f(u) = 0 in RNwith possibly discontinuous nonlinearity. Adv. Math. Sci. Appl., 2001, 11: 811-834

[23] Farina A.. Rigidity and one-dimensional symmetry for semilinear elliptic equations in the whole of RNand in half spaces. Adv. Math. Sci. Appl., 2003, 13: 65-82

[24] Farina A.. Two results on entire solutions of Ginzburg-Landau system in higher dimensions. J. Funct. Anal., 2004, 214: 386-395

[25] Farina A., Valdinoci E.. The state of the art for a conjecture of De Giorgi and related problems. Recent progress on reaction-diffusion systems and viscosity solutions, 2009 74-96

[26] Farina A., Valdinoci E.. 1D symmetry for solutions of semilinear and quasilinear elliptic equations. Trans. Amer. Math. Soc., 2011, 363: 579-609

[27] Farina A., Valdinoci E.. Rigidity results for elliptic PDEs with uniform limits: An abstract framework with applications. Indiana Univ. Math. J., 2011, 60: 121-141

[28] Fife P. C., McLeod J. B.. The approach of solutions of non-linear diffusion equations to traveling front solutions. Arch. Ration. Mech. Anal., 1977, 65: 335-361

[29] Fisher R. A.. The advance of advantageous genes. Ann. Eugenics, 1937, 7: 335-369

[30] Fonseca I., Mantegazza C.. Second order singular perturbation models for phase transitions. SIAM J. Math. Anal., 2000, 31: 1121-1143

[31] Ghoussoub N., Gui C.. On a conjecture of De Giorgi and some related problems. Math. Ann., 1998, 311: 481-491

[32] Gibbons G. W., Townsend P. K.. Bogomolnyi equation for intersecting domain walls. Phys. Rev. Lett., 1999, 83: 1727-1730

[33] Gomper G., Schick M.. Phase transitions and critical phenomena, 1994, London: Academic Press

[34] Hilhorst D., Peletier L. A., Schätzle R.. G-limit for the extended Fisher-Kolmogorov equation. Proc. Royal Soc. Edinburgh A, 2002, 132: 141-162

[35] Kalies W. D., Kwapisz J., VanderVorst R. C. A. M.. Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria. Comm. Math. Phys., 1998, 193: 337-371

[36] Kalies W. D., VanderVorst R. C. A. M.. Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation. J. Diff. Equations, 1996, 131: 209-228

[37] Kawakatsu T., Andelman D., Kawasaki K., Taniguchi T.. Phase-transitions and shapes of twocomponent membranes and vesicles I: Strong segregation limit. J. Phys. II (France), 1993, 3: 971-997

[38] Kolmogorov A., Petrovski I., Piscounov N.. Etude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull. Univ. Moscou Ser. Internat. Sec. A, 1937, 1: 1-25

[39] Leibler S., Andelman D.. Ordered and curved meso-structures in membranes and amphiphilic films. J. Phys. (France), 1987, 48: 2013-2018

[40] Leizarowitz A., Mizel V. J.. One-dimensional infinite-horizon variational problems arising in continuum mechanics. Arch. Ration. Mech. Anal., 1989, 106: 161-194

[41] Mizel, V. J., Peletier, L. A. and Troy, W. C., Periodic phases in second-order materials, Arch. Ration. Mech. Anal., 145, 1998, 343–382.

[42] Peletier, L. A. and Troy, W. C., Spatial patterns described by the extended Fisher-Kolmogorov (EFK) equation: Kinks, Diff. Int. Equations, 8, 1995, 1279–1304.

[43] Peletier L. A., Troy W. C.. A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation. Topol. Meth. Nonlin. Anal., 1995, 6: 331-355

[44] Peletier L. A., Troy W. C.. Chaotic spatial patterns described by the extended Fisher-Kolmogorov equation. J. Diff. Equations, 1996, 129: 458-508

[45] Peletier L. A., Troy W. C.. Spatial patterns, higher order models in physics and mechanics, 2001

[46] Peletier L. A., Van der Vorst R. K. A. M., Troy W. C.. Stationary solutions of a fourth order nonlinear diffusion equation. Diff. Uravneniya, 1995, 31: 327-337

[47] Savin O.. Regularity of flat level sets in phase transitions. Ann. Math., 2009, 169: 41-78

[48] Swift, J. B. and Hohenberg, P. C., Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15, 1977, 319–328.

[49] Van den Berg J. B.. Branches of heteroclinic, homoclinic and periodic solutions in a fourth order bi-stable system, a numerical study, 1996

[50] Van den Berg, J. B., The phase plane picture for a class of fourth order conservative differential equations, J. Diff. Equations, 161, 2000, 110–153.

[51] Wheeler A. A.. Phase-field theory of edges in an anisotropic crystal. Proc. Royal Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2006, 462: 3363-3384