A counterexample to a conjecture by De Giorgi in large dimensions
DelPino_Manuel.pdf (273.4Kb)Abstract
We consider the Allen-Cahn equation
Delta u + u(1 - u(2)) = 0 in R-N.
A celebrated conjecture by E. De Giorgi (1978) states that if u is it bounded Solution to this problem Such that partial derivative(xN) u > 0, then the level sets {u =lambda}, lambda is an element of 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 ¯rst 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.
Identifier
URI: https://repositorio.uchile.cl/handle/2250/125143
DOI: 10.1016/j.crma.2008.10.010
ISSN: 1631-073X
Quote Item
COMPTES RENDUS MATHEMATIQUE Volume: 346 Issue: 23-24 Pages: 1261-1266 Published: DEC 2008
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