A counterexample to a conjecture by De Giorgi in large dimensions
Pino_Mdel.pdf (313.8Kb)Abstract
We consider the Allen–Cahn equation
u +u
1− u2
=0 inRN.
A celebrated conjecture by E. De Giorgi (1978) states that if u is a bounded solution to this problem such that ∂xNu>0, then the
level sets {u=λ}, λ ∈R, must be hyperplanes at least if N 8. We construct a family of solutions which shows that this statement
does not hold true for N 9.
Patrocinador
The first author has been partly supported by research grants Fondecyt 1070389 and FONDAP, Chile. The second
author has been supported by Fondecyt grant 1050311, Nucleus Millennium grant P04-069-F, and FONDAP, Chile.
The research of the third author is partially supported by an Earmarked Grant from RGC of Hong Kong and a Direct
Grant from CUHK.
Quote Item
COMPTES RENDUS MATHEMATIQUE Volume: 346 Issue: 23-24 Pages: 1261-1266 Published: DEC 2008
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