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

作者：Farina Alberto; Valdinoci Enrico^*

来源：Communications in Partial Differential Equations, 2016, 41(4): 665-682.

DOI：10.1080/03605302.2015.1135165

• Limit interface
• phase transitions
• symmetry results

We study bounded, monotone solutions of u=W(u) in the whole of (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 1D.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 -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 wishes 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 1D, at least up to dimension 4.

• 出版日期2016