1D symmetry for semilinear PDEs from the limit interface of the solution

TL;DR

This paper proves that bounded, monotone solutions of certain semilinear PDEs are one-dimensional under specific conditions on the interface and energy growth, extending previous results beyond minimal solutions.

Contribution

It introduces a new approach to establish 1D symmetry for solutions without assuming minimality, applicable to large-scale phase separation scenarios.

Findings

Monotone solutions with energy bounds are 1D if their limit interface lacks a vertical line in dimensions up to 4.

The method applies to solutions outside the scope of Γ-convergence techniques.

The approach can be useful in practical phase separation problems.

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.