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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Sharp counterexamples related to the De Giorgi conjecture in dimensions $ 4\leq n \leq 8$


Author: Amir Moradifam
Journal: Proc. Amer. Math. Soc. 142 (2014), 199-203
MSC (2010): Primary 35B53, 35J60
DOI: https://doi.org/10.1090/S0002-9939-2013-12040-X
Published electronically: September 20, 2013
MathSciNet review: 3119195
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Abstract: In this note, we show that in dimensions $ n\geq 4$ there exists a smooth bounded potential $ V$ such that $ (\Delta +V)w=0$ has a positive solution $ u$ as well as a bounded sign-changing solution $ v$ satisfying

$\displaystyle \int _{B_{R}}v^2\leq CR^3 \ \ \ \ \forall R>0,$    

for some $ C>0$ independent of $ R$. This in particular implies that the
Ambrosio-Cabré proof of the De Giorgi conjecture in dimension $ n=3$ cannot be extended to dimensions $ 4\leq n \leq 8$. We also answer an open question of
L. Moschini [L. Moschini, New Liouville theorems for linear second order
degenerate elliptic equations in divergence form, Ann. Inst. H. Poincarè Anal. Non Linéaire 22 (2005), 11-23].

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Additional Information

Amir Moradifam
Affiliation: Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027
Address at time of publication: Department of Mathematics, University of Toronto, Toronto, ON M5S 1A1, Canada
Email: am3937@columbia.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-12040-X
Received by editor(s): November 3, 2011
Received by editor(s) in revised form: February 29, 2012
Published electronically: September 20, 2013
Additional Notes: The author was supported by MITACS and NSERC Postdoctoral Fellowships
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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