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Entire solutions of semilinear elliptic equations in $\mathbb {R}^{3}$ 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
Published electronically: July 6, 2000
MathSciNet review: 1775735
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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 $F$.

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Additional Information

Luigi Ambrosio
Affiliation: Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
MR Author ID: 25430

Xavier Cabré
Affiliation: Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Diagonal, 647, 08028 Barcelona, Spain

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