Accepted Paper

Inserted: 15 jan 2016
Last Updated: 15 jan 2016

Journal: Comm. in PDE
Year: 2015

Abstract:

We study bounded, monotone solutions of $\Delta u=W'(u)$ in the whole of $R^n$, 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 $4$.

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