Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi |
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Authors: | Luigi Ambrosio Xavier Cabré |
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Affiliation: | Scuola Normale Superiore di Pisa, Piazza dei Cavalieri, 7, 56126 Pisa, Italy ; Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Diagonal, 647, 08028 Barcelona, Spain |
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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 . |
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Keywords: | Nonlinear elliptic PDE symmetry and monotonicity properties energy estimates Liouville theorems |
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