Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi

Authors:
Luigi Ambrosio and Xavier Cabré

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

Published electronically:
July 6, 2000

MathSciNet review:
1775735

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Abstract: In 1978 De Giorgi formulated the following conjecture. *Let ** be a solution of * * in all of * *such that * * and * * in * *. Is it true that all level sets * * of ** are hyperplanes, at least if *? Equivalently, does depend only on one variable? When , this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for . The question, however, remains open for . The results for and 3 apply also to the equation for a large class of nonlinearities .

**[AAC]**G. Alberti, L. Ambrosio and X. Cabré,*On a long-standing conjecture of E. De Giorgi: Old and recent results*, forthcoming.**[B]**M. T. Barlow,*On the Liouville property for divergence form operators*, Canad. J. Math.**50**(1998), 487-496. MR**99k:31010****[BBG]**M. T. Barlow, R. F. Bass and C. Gui,*The Liouville property and a conjecture of De Giorgi*, preprint.**[BCN]**H. Berestycki, L. Caffarelli and L. Nirenberg,*Further qualitative properties for elliptic equations in unbounded domains*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**25**(1997), 69-94. MR**2000e:35053****[BHM]**H. Berestycki, F. Hamel and R. Monneau,*One-dimensional symmetry of bounded entire solutions of some elliptic equations*, preprint.**[BDG]**E. Bombieri, E. De Giorgi and E. Giusti,*Minimal cones and the Bernstein problem*, Invent. Math.**7**(1969), 243-268. MR**40:3445****[CC]**L. Caffarelli and A. Córdoba,*Uniform convergence of a singular perturbation problem*, Comm. Pure Appl. Math.**48**(1995), 1-12. MR**95j:49015****[CGS]**L. Caffarelli, N. Garofalo and F. Segala,*A gradient bound for entire solutions of quasi-linear equations and its consequences*, Comm. Pure Appl. Math.**47**(1994), 1457-1473. MR**95k:35030****[DG]**E. De Giorgi,*Convergence problems for functionals and operators*, Proc. Int. Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), 131-188. MR**80k:49010****[F1]**A. Farina,*Some remarks on a conjecture of De Giorgi*, Calc. Var. Partial Differential Equations**8**(1999), 233-245. MR**2000d:35214****[F2]**A. Farina,*Symmetry for solutions of semilinear elliptic equations in**and related conjectures*, Ricerche di Matematica**XLVIII**(1999), 129-154.**[F3]**A. Farina, forthcoming.**[GG]**N. Ghoussoub and C. Gui,*On a conjecture of De Giorgi and some related problems*, Math. Ann.**311**(1998), 481-491. MR**99j:35049****[G]**E. Giusti,*Minimal Surfaces and Functions of Bounded Variation*, Birkhäuser Verlag, Basel-Boston (1984). MR**87a:58041****[LM]**S. Luckhaus and L. Modica,*The Gibbs-Thompson relation within the gradient theory of phase transitions*, Arch. Rational Mech. Anal.**107**(1989), 71-83. MR**90k:49041****[M1]**L. Modica,*A gradient bound and a Liouville theorem for nonlinear Poisson equations*, Comm. Pure Appl. Math.**38**(1985), 679-684. MR**87m:35088****[M2]**L. 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. 2, 843-850. MR**91a:35015****[MM1]**L. Modica and S. Mortola,*Un esempio di**-convergenza*, Boll. Un. Mat. Ital. B (5)**14**(1977), 285-299. MR**56:3704****[MM2]**L. Modica and S. Mortola,*Some entire solutions in the plane of nonlinear Poisson equations*, Boll. Un. Mat. Ital. B (5)**17**(1980), 614-622. MR**81k:35036****[Z]**W. P. Ziemer,*Weakly Differentiable Functions*, Springer Verlag, New York (1989). MR**91e:46046**

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Additional Information

**Luigi Ambrosio**

Affiliation:
Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa, Italy

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

DOI:
https://doi.org/10.1090/S0894-0347-00-00345-3

Keywords:
Nonlinear elliptic PDE,
symmetry and monotonicity properties,
energy estimates,
Liouville theorems

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.

Article copyright:
© Copyright 2000
American Mathematical Society