2 5 Ju n 20 08 On the Allen - Cahn equation in the Grushin plane : a monotone entire solution that is not one - dimensional
نویسنده
چکیده
We consider solutions of the Allen-Cahn equation in the whole Grushin plane and we show that if they are monotone in the vertical direction, then they are stable and they satisfy a good energy estimate. However, they are not necessarily one-dimensional, as a counter-example shows.
منابع مشابه
Two-end solutions to the Allen-Cahn equation in R
has been studied for several decades and is an important nonlinear PDE due to the fact that it lies at the interface of several different mathematical fields. The famous De Giorgi conjecture states that any entire solution to (1) in R which is monotone in one direction should be one dimensional, at least for n ≤ 8. The conjecture was proved in dimension n = 2 by Ghoussoub-Gui ([13]) and dimensi...
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