Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones

Journal of Functional Analysis

Volumn 264, Issue 5, 2013, Pages 1131-1167

  Pacard, Frank ^a   Wei, Juncheng ^b  

^a CENTRE DE MATHÉMATIQUES LAURENT SCHWARTZ   (France)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Allen Cahn equation; De Giorgi conjecture; Minimal cones; Minimal hypersurfaces; Stable solutions

Indexed keywords

EID: 84872598703     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2012.03.010     Document Type: Article

References (31)

1. Alberti G., Ambrosio L., Cabré X. On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math. 2001, 65(1-3):9-33. (special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday).
2. m+n. Ann. Global Anal. Geom. 2005, 27:179-199.
3. Allen S.M., Cahn J.W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. Mater. 1979, 27:1085-1095.
4. 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.
5. Barlow M.T., Bass R.F., Gui C. The Liouville property and a conjecture of De Giorgi. Comm. Pure Appl. Math. 2000, 53(8):1007-1038.
6. Berestycki H., Hamel F., Monneau R. One-dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J. 2000, 103:375-396.
7. Bombieri E., De Giorgi E., Giusti E. Minimal cones and the Bernstein problem. Invent. Math. 1969, 7:243-268.
8. Cabré X. Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation preprint. arxiv:1102.3111.
9. 2m. J. Eur. Math. Soc. (JEMS) 2009, 11(4):819-843.
10. Cabré X., Terra J. Qualitative properties of saddle-shaped solutions to bistable diffusion equations. Comm. Partial Differential Equations 2010, 35(11):1923-1957.
11. Caffarelli L., Córdoba A. Uniform convergence of a singular perturbation problem. Comm. Pure Appl. Math. 1995, XLVII:1-12.
12. Caffarelli L., Hardt R., Simon L. Minimal surfaces with isolated singularities. Manuscripta Math. 1984, 48(1-3):1-18.
13. Chan C. Complete minimal hypersurfaces with prescribed asymptotics at infinity. J. Reine Angew. Math. 1997, 483:163-181.
14. N. Dancer, private communication.
15. Dang H., Fife P.C., Peletier L.A. Saddle solutions of the bistable diffusion equation. Z. Angew. Math. Phys. 1992, 43(6):984-998.
16. De Giorgi E. Convergence problems for functionals and operators. Proc. Int. Meeting on Recent Methods in Nonlinear Analysis 1979, 131-188. Pitagora, Bologna.
17. del Pino M., Kowalczyk M., Wei J. On De Giorgi conjecture in dimension n≥9. Ann. of Math. (3) 2011, 174:1485-1569.
18. Farina A. Symmetry for solutions of semilinear elliptic equations in RN and related conjectures. Ric. Mat. 1999, 48(suppl.):129-154.
19. Farina A., Sciunzi B., Valdinoci Enrico Bernstein and De Giorgi type problems: new results via a geometric approach. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2008, 7(4):741-791.
20. 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. World Sci. Publ., Hackensack, NJ.
21. Fischer-Colbrie D., Schoen R. The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 1980, 33:199-211.
22. Ghoussoub N., Gui C. On a conjecture of De Giorgi and some related problems. Math. Ann. 1998, 311:481-491.
23. Ghoussoub N., Gui C. On De Giorgi's conjecture in dimensions 4 and 5. Ann. of Math. (2) 2003, 157(1):313-334.
24. Hardt R., Simon L. Area minimizing hypersurfaces with isolated singularities. J. Reine Angew. Math. 1985, 362:102-129.
25. Kapouleas N., Yang S.D. Minimal surfaces in the three-sphere by doubling the Clifford torus. Amer. J. Math. 2010, 132(2):257-295.
26. Lawson B. The equivariant Plateau problem and interior regularity. Trans. Amer. Math. Soc. 1972, 173.
27. Nikiforov A.F., Uvarov V.B. Special Functions of Mathematical Physics 1988, Birkháuser Verlag, Basel.
28. Pacard F. Lectures on "Connected sum constructions in geometry and nonlinear analysis" preprint. http://www.math.polytechnique.fr/~pacard/Publications/Lecture-Part-I.pdf.
29. Pacard F., Ritoré M. From constant mean curvature hypersurfaces to the gradient theory of phase transitions. J. Differential Geom. 2003, 64:359-423.
30. Pitts J.T., Rubinstein J.H. Equivariant minimax and minimal surfaces in geometric three-manifolds. Bull. Amer. Math. Soc. (N.S.) 1988, 19(1):303-309.
31. Savin O. Regularity of flat level sets in phase transitions. Ann. of Math. (1) 2009, 169:41-78.