1D symmetry for semilinear PDEs from the limit interface of the solution. Issue 4 (2nd April 2016)

• Record Type: Journal Article
• Authors: Farina, Alberto; Valdinoci, Enrico
• Is Part Of: Communications in partial differential equations. Volume 41:Issue 4(2016)
• Page Start: 665
• Page End: 682
• Publication Date: 2016-04-02
• Subjects: Limit interface -- phase transitions -- symmetry results; 35J61 -- 35J15; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2015.1135165
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: We study bounded, monotone solutions of Δ 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 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 Γ-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.