Alberto Farina and Enrico Valdinoci
1D symmetry for semilinear PDEs from the limit interface of the solution
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ABSTRACT.  We study bounded, monotone solutions of $\Delta u=W'(u)$ 
in the whole of ${f{R}}^n$, 
where $W$ is a double-well potential. We prove that 
under suitable assumptions on the limit interface 
and on the energy growth, $u$ is $1$D. 
In particular, differently from the previous literature, the solution is not 
assumed to have minimal properties and the cases studied lie outside 
the range of $\Gamma$-convergence methods. 
We think that this approach could be fruitful in concrete situations, 
where one can observe the phase separation at a large scale and whishes 
to deduce the values of the state parameter in the vicinity of the interface. 
As a simple example of the results obtained with this point of view, 
we mention that monotone solutions with energy bounds, 
whose limit interface does not contain a vertical line through the origin, 
are $1$D, at least up to dimension $4$.