1D symmetry for semilinear PDEs from the limit interface of the solution
  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. 
Alberto Farina and Enrico Valdinoci
math.AP