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I am interested in the Allen-Cahn equation in $ R^N$ and one can consider the related energy functional $$ E(u):= \frac{1}{2}\int_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int_{R^N} (u^2-1)^2dx.$$ There has been a lot of work on this equation and in particular on the DeGiorgi conjecture. My question is related to whether any of the solutions have finite energy. So here is my exact question. Lets take $N=9$ and suppose $ x=(x',x_9)$.

Question. Does there exist a function $u$ with $ -1 <u<1$ with a $ u_{x_9}>0$ and $\lim_{x_9 \rightarrow \infty} u(x',x_9)=1$ and $ \lim_{x_9 \rightarrow -\infty} u(x',x_9)=-1$. Furthermore $ E(u)<\infty$.

The reason I ask this is I see some results about 'finite energy solutions' yet they just impose growth on the energy in terms of $B_R$ (ball radius $R$ centered at the origin) so I thought if the full energy is finite maybe something is trivial (but I just can't see it..)

Thanks for the comments.

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    $\begingroup$ Maybe lower bound |grad u| by |du/d_{x_9}| and then the integral over x_9 for any fixed x' is bounded away from zero. $\endgroup$ – user36212 Jan 21 at 22:27
  • $\begingroup$ @Willie Wong. I saw a comment that seems to have disappeared (or maybe I accidentily erased something). So at this point I am just looking for a function as described (it does not need to be a critical point) $\endgroup$ – Math604 Jan 21 at 22:31
  • $\begingroup$ @user36212 I will need to think a bit about what you are suggesting. $\endgroup$ – Math604 Jan 21 at 22:32
  • $\begingroup$ It's Willie's answer below, no need to think further... $\endgroup$ – user36212 Jan 22 at 21:28
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Sketch of an argument:

For a fixed $x'$, let $f(x')$ denote the measure of the set $\{ u(x',x_9) \in (-1/2,1/2) \}$.

Note that for fixed $x'$ you have $$ \int |\nabla u(x',x_9)|^2 d x_9 \geq \int |\partial_{x_9} u(x',x_9)|^2 d x_9 \geq 1 / f(x') $$ (on the ends of the interval (it is an interval by monotonicity) defining $f(x')$ the function takes values $-1/2$ and $1/2$ respectively, and so the minimizer is the linear function with slope $1/f(x')$.)

For fixed $x'$ you also have $$ \int (u^2 - 1)^2 dx_9 \geq \frac9{16} f(x') $$

Finiteness of energy requires both be integrable in $x'$, which is not possible.

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  • $\begingroup$ This argument doesn't seem to depend on the dimension, provided $N \geq 2$. Also, isn't it the case that the De Giorgi conjecture only asks for $C^2$ solutions to the Euler-Lagrange equation, and not actually solutions with finite energy? $\endgroup$ – Willie Wong Jan 22 at 4:55
  • $\begingroup$ That you very much for your answer. Regarding your question; you are correct, they don't ask for finite energy. I was just adding this for some other reasons. $\endgroup$ – Math604 Jan 22 at 17:47

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