Entire solutions of semilinear elliptic equations in $\mathbb {R}^{3}$ and a conjecture of De Giorgi
HTML articles powered by AMS MathViewer
- by Luigi Ambrosio and Xavier Cabré
- J. Amer. Math. Soc. 13 (2000), 725-739
- DOI: https://doi.org/10.1090/S0894-0347-00-00345-3
- Published electronically: July 6, 2000
- HTML | PDF | Request permission
Abstract:
In 1978 De Giorgi formulated the following conjecture. Let $u$ be a solution of $\Delta u=u^{3}-u$ in all of $\mathbb {R}^{n}$ such that $\vert u\vert \le 1$ and $\partial _{n} u >0$ in $\mathbb {R}^{n}$. Is it true that all level sets $\{ u=\lambda \}$ of $u$ are hyperplanes, at least if $n\le 8$? Equivalently, does $u$ depend only on one variable? When $n=2$, this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for $n=3$. The question, however, remains open for $n\ge 4$. The results for $n=2$ and 3 apply also to the equation $\Delta u=F’(u)$ for a large class of nonlinearities $F$.References
- [AAC]AAC G. Alberti, L. Ambrosio and X. Cabré, On a long-standing conjecture of E. De Giorgi: Old and recent results, forthcoming.
- Martin T. Barlow, On the Liouville property for divergence form operators, Canad. J. Math. 50 (1998), no. 3, 487–496. MR 1629807, DOI 10.4153/CJM-1998-026-9 [BBG]BBG M. T. Barlow, R. F. Bass and C. Gui, The Liouville property and a conjecture of De Giorgi, preprint.
- Henri Berestycki, Luis Caffarelli, and Louis Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69–94 (1998). Dedicated to Ennio De Giorgi. MR 1655510 [BHM]BHM H. Berestycki, F. Hamel and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, preprint.
- E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. MR 250205, DOI 10.1007/BF01404309
- Luis A. Caffarelli and Antonio Córdoba, Uniform convergence of a singular perturbation problem, Comm. Pure Appl. Math. 48 (1995), no. 1, 1–12. MR 1310848, DOI 10.1002/cpa.3160480101
- Luis Caffarelli, Nicola Garofalo, and Fausto Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences, Comm. Pure Appl. Math. 47 (1994), no. 11, 1457–1473. MR 1296785, DOI 10.1002/cpa.3160471103
- Ennio De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) Pitagora, Bologna, 1979, pp. 131–188. MR 533166
- Alberto Farina, Some remarks on a conjecture of De Giorgi, Calc. Var. Partial Differential Equations 8 (1999), no. 3, 233–245. MR 1688549, DOI 10.1007/s005260050124 [F2]F2 A. Farina, Symmetry for solutions of semilinear elliptic equations in $\mathbb {R}^{N}$ and related conjectures, Ricerche di Matematica XLVIII (1999), 129–154. [F3]F3 A. Farina, forthcoming.
- N. Ghoussoub and C. Gui, On a conjecture of De Giorgi and some related problems, Math. Ann. 311 (1998), no. 3, 481–491. MR 1637919, DOI 10.1007/s002080050196
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
- Stephan Luckhaus and Luciano Modica, The Gibbs-Thompson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal. 107 (1989), no. 1, 71–83. MR 1000224, DOI 10.1007/BF00251427
- Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 679–684. MR 803255, DOI 10.1002/cpa.3160380515
- Luciano Modica, Monotonicity of the energy for entire solutions of semilinear elliptic equations, Partial differential equations and the calculus of variations, Vol. II, Progr. Nonlinear Differential Equations Appl., vol. 2, Birkhäuser Boston, Boston, MA, 1989, pp. 843–850. MR 1034031
- Luciano Modica and Stefano Mortola, Un esempio di $\Gamma ^{-}$-convergenza, Boll. Un. Mat. Ital. B (5) 14 (1977), no. 1, 285–299 (Italian, with English summary). MR 0445362
- Luciano Modica and Stefano Mortola, Some entire solutions in the plane of nonlinear Poisson equations, Boll. Un. Mat. Ital. B (5) 17 (1980), no. 2, 614–622 (English, with Italian summary). MR 580544
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
Bibliographic Information
- Luigi Ambrosio
- Affiliation: Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
- MR Author ID: 25430
- Email: luigi@ambrosio.sns.it
- Xavier Cabré
- Affiliation: Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Diagonal, 647, 08028 Barcelona, Spain
- Email: cabre@ma1.upc.es
- Received by editor(s): October 8, 1999
- Published electronically: July 6, 2000
- Additional Notes: The authors would like to thank Mariano Giaquinta for several useful discussions. Most of this work was done while the second author was visiting the University of Pisa. He thanks the Department of Mathematics for its hospitality.
- © Copyright 2000 American Mathematical Society
- Journal: J. Amer. Math. Soc. 13 (2000), 725-739
- MSC (2000): Primary 35J60, 35B05, 35B40, 35B45
- DOI: https://doi.org/10.1090/S0894-0347-00-00345-3
- MathSciNet review: 1775735