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$ 1$D symmetry for solutions of semilinear and quasilinear elliptic equations


Authors: Alberto Farina and Enrico Valdinoci
Journal: Trans. Amer. Math. Soc. 363 (2011), 579-609
MSC (2010): Primary 35J92, 35J91, 35J20
DOI: https://doi.org/10.1090/S0002-9947-2010-05021-4
Published electronically: September 21, 2010
MathSciNet review: 2728579
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Abstract: Several new $ 1$D results for solutions of possibly singular or degenerate elliptic equations, inspired by a conjecture of De Giorgi, are provided. In particular, $ 1$D symmetry is proven under the assumption that either the profiles at infinity are $ 2$D, or that one level set is a complete graph, or that the solution is minimal or, more generally, $ Q$-minimal.


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Additional Information

Alberto Farina
Affiliation: Faculté des Sciences, LAMFA – CNRS UMR 6140, Université de Picardie Jules Verne, 33, rue Saint-Leu, 80039 Amiens CEDEX 1, France
Email: alberto.farina@u-picardie.fr

Enrico Valdinoci
Affiliation: Dipartimento di Matematica, Università di Roma Tor Vergata, via della ricerca scientifica, 1, I-00133 Rome, Italy
Email: enrico@mat.uniroma3.it

DOI: https://doi.org/10.1090/S0002-9947-2010-05021-4
Received by editor(s): April 7, 2008
Published electronically: September 21, 2010
Additional Notes: The second author was supported by MIUR Metodi variazionali ed equazioni differenziali nonlineari and FIRB Analysis and Beyond. We thank an anonymous referee whose advice improved the exposition of this paper.
Article copyright: © Copyright 2010 American Mathematical Society

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