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 $4$.